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Adaptive independent sticky MCMC algorithms
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 5 (2018)
Abstract
Monte Carlo methods have become essential tools to solve complex Bayesian inference problems in different fields, such as computational statistics, machine learning, and statistical signal processing. In this work, we introduce a novel class of adaptive Monte Carlo methods, called adaptive independent sticky Markov Chain Monte Carlo (MCMC) algorithms, to sample efficiently from any bounded target probability density function (pdf). The new class of algorithms employs adaptive nonparametric proposal densities, which become closer and closer to the target as the number of iterations increases. The proposal pdf is built using interpolation procedures based on a set of support points which is constructed iteratively from previously drawn samples. The algorithm’s efficiency is ensured by a test that supervises the evolution of the set of support points. This extra stage controls the computational cost and the convergence of the proposal density to the target. Each part of the novel family of algorithms is discussed and several examples of specific methods are provided. Although the novel algorithms are presented for univariate target densities, we show how they can be easily extended to the multivariate context by embedding them within a Gibbstype sampler or the hit and run algorithm. The ergodicity is ensured and discussed. An overview of the related works in the literature is also provided, emphasizing that several wellknown existing methods (like the adaptive rejection Metropolis sampling (ARMS) scheme) are encompassed by the new class of algorithms proposed here. Eight numerical examples (including the inference of the hyperparameters of Gaussian processes, widely used in machine learning for signal processing applications) illustrate the efficiency of sticky schemes, both as standalone methods to sample from complicated onedimensional pdfs and within Gibbs samplers in order to draw from multidimensional target distributions.
1 Introduction
Markov chain Monte Carlo (MCMC) methods [1, 2] are very important tools for Bayesian inference and numerical approximation, which are widely employed in signal processing [3–7] and other related fields [1, 8]. A crucial issue in MCMC is the choice of a proposal probability density function (pdf), as this can strongly affect the mixing of the MCMC chain when the target pdf has a complex structure, e.g., multimodality and/or heavy tails. Thus, in the last decade, a remarkable stream of literature focuses on adaptive proposal pdfs, which allow for selftuning procedures of the MCMC algorithms, flexible movements within the state space, and improved acceptance rates [9, 10].
Adaptive MCMC algorithms are used in many statistical applications and several schemes have been proposed in the literature [8–11]. There are two main families of methods: parametric and nonparametric. The first strategy consists in adapting the parameters of a parametric proposal pdf according to the past values of the chain [10]. However, even if the parameters are perfectly adapted, a discrepancy between the target and the proposal pdfs remains. A second strategy attempts to adapt the entire shape of the proposal density using nonparametric procedures [12, 13]. Most authors have payed more attention to the first family, designing local adaptive randomwalk algorithms [9, 10], due to the difficulty of approximating the full target distribution by nonparametric schemes with any degree of generality.
In this work, we describe a general framework to design suitable adaptive MCMC algorithms with nonparametric proposal densities. After describing the different building blocks and the general features of the novel class, we introduce two specific algorithms. Firstly, we describe the adaptive independent sticky Metropolis (AISM) algorithm to draw efficiently from any bounded univariate target distribution.^{Footnote 1} Then, we also propose a more efficient scheme that is based on the multiple try Metropolis (MTM) algorithm: the adaptive independent sticky Multiple Try Metropolis (AISMTM) method. The ergodicity of the adaptive sticky MCMC methods is ensured and discussed. The underlying theoretical support is based on the approach introduced in [14]. The new schemes are particularly suitable for sampling from complicated fullconditional pdfs within a Gibbs sampler [5–7].
Moreover, the new class of methods encompasses different wellknown algorithms available in literature: the griddy Gibbs sampler [15], the adaptive rejection Metropolis sampler (ARMS) [12, 16], and the independent doubly adaptive Metropolis sampler (IA^{2}RMS) [13, 17]. Other related or similar approaches are also discussed in Section 6. The main contributions of this paper are the following:

1.
A very general framework, that allows practitioners to design proper adaptive MCMC methods by employing a nonparametric proposal.

2.
Two algorithms (AISM and AISMTM), that can be used offtheshelf in signal processing applications.

3.
An exhaustive overview of the related algorithms proposed in the literature, showing that several wellknown methods (such as ARMS) are encompassed by the proposed framework.

4.
A theoretical analysis of the AISM algorithm, proving its ergodicity and the convergence of the adaptive proposal to the target.
The structure of the paper is the following. Section 2 introduces the generalities of the class of sticky MCMC methods and the AISM scheme. Sections 3 and 4 present the general properties, altogether with specific examples, of the proposal constructions and the update control tests. Section 5 introduces some theoretical results. Section 6 discusses several related works and highlights some specific techniques belonging to the class of sticky methods. Section 7 introduces the AISMTM method. Section 8 describes the range of applicability of the proposed framework, including its use within other Monte Carlo methods (like the Gibbs sampler or the hit and run algorithm) to sample from multivariate distributions. Eight numerical examples (including the inference of hyperparameters of Gaussian processes) are then provided in Section 9.^{Footnote 2} Finally, Section 10 contains some conclusions and possible future lines.^{Footnote 3}
2 Adaptive independent sticky MCMC algorithms
Let \(\widetilde {\pi }(x) \propto \pi (x)>0\), with \(x\in \mathcal {X}\subseteq \mathbb {R}\), be a bounded^{Footnote 4} target density known up to a normalizing constant, \(c_{\pi }=\int _{\mathcal {X}} \pi (x) dx\), from which direct sampling is unfeasible. In order to draw from it, we employ an MCMC algorithm with an independent adaptive proposal,
where t is the iteration index of the corresponding MCMC algorithm, and \(\mathcal {S}_{t}=\{s_{1},\ldots,s_{m_{t}}\}\) with m_{ t }>0 is the set of support points used for building \(\widetilde {q}_{t}\). At the tth iteration, an adaptive independent sticky MCMC method is conceptually formed by three stages (see Fig. 1):

1.
Construction of the nonparametric proposal: given the nodes in \(\mathcal {S}_{t}\), the function q_{ t } is built using a suitable non parametric procedure that provides a function which is closer and closer to the target as the number of points m_{ t } increases. Section 3 describes the general properties that must be fulfilled by a suitable proposal construction, as well as specific procedures to build this proposal.

2.
MCMC stage: some MCMC method is applied in order to produce the next state of the chain, x_{ t }, employing \(\widetilde {q}_{t}(x\mathcal {S}_{t})\) as (part of the) proposal pdf. This stage produces the next state of the chain, x_{t+1}, and an auxiliary variable z (see Tables 1 and 4), used in the following update stage.

3.
Update stage: A statistical test on the auxiliary variable z is performed in order to decide whether to increase the number of points in \(\mathcal {S}_{t}\) or not, defining a new support set, \(\mathcal {S}_{t+1}\), which is used to construct the proposal at the next iteration. The update stage controls the computational cost and ensures the ergodicity of the generated chain (see Appendix A). Section 4 is devoted to the design of different suitable update rules.
In the following section, we describe the simplest possible sticky method, obtained by using the MH algorithm, whereas in Section 7 we consider a more sophisticated technique that employs the MTM scheme.^{Footnote 5}
2.1 Adaptive independent sticky Metropolis
The simplest adaptive independent sticky method is the adaptive independent sticky Metropolis (AISM) technique, outlined in Table 1. In this case, the proposal pdf \(\widetilde {q}_{t}(x\mathcal {S}_{t})\) changes along the iterations (see step 1 of Table 1) following an adaptation scheme that relies upon a suitable interpolation given the set of support points \(\mathcal {S}_{t}\) (see Section 3). Step 3 of Table 1 applies a statistical control to update the set \(\mathcal {S}_{t}\). The point z, rejected at the current iteration of the algorithm in the MH test, is added to \(\mathcal {S}_{t}\) with probability
where \( \eta _{t}(z,d): \mathcal {X}\times \mathbb {R}^{+}\rightarrow [0,1] \)is an increasing test function w.r.t. the variable d, such that η_{ t }(z,0)=0, and \( d=d_{t}(z)=\left \pi (z)q_{t}(z\mathcal {S}_{t})\right .\) is the point distance between π and q_{ t } at z. The rationale behind this test is to use information from the target density in order to include in the support set only those points where the proposal pdf differs substantially from the target value at z. Note that, since z is always different from the current state (i.e., z≠x_{ t } for all t), then the proposal pdf is independent from the current state according to Holden’s definition [14], and thus the theoretical analysis is greatly simplified.
3 Construction of the sticky proposals
There are many alternatives available for the construction of a suitable sticky proposal (SP). However, in order to be able to provide some theoretical results in Section 5, let us define precisely what we understand here by a sticky proposal.
Definition 1
(Valid Adaptive Proposal) Let us consider a target density, \(\widetilde {\pi }(x)\propto \pi (x)>0\) for any \(x \in \mathcal {X} \subseteq \mathbb {R}\) (the target’s support), and a set of \(m_{t}=\mathcal {S}_{t}\) support points, \(\mathcal {S}_{t}=\{s_{1},\ldots,s_{m_{t}}\}\) with \(s_{i}\in \mathcal {X}\) for all i=1,…,m_{ t }. An adaptive proposal built using \(\mathcal {S}_{t}\) via some nonparametric interpolation approach is considered valid if the following four properties are satisfied:

1.
The proposal function is positive, i.e., \(q_{t}(x\mathcal {S}_{t})>0\) for all \(x\in \mathcal {X}\) and for all possible sets \(\mathcal {S}_{t}\) with \(t\in \mathbb {N}\).

2.
Samples can be drawn directly and easily from the resulting proposal, \(\widetilde {q}_{t}(x\mathcal {S}_{t})\propto q_{t}(x\mathcal {S}_{t})\), using some exact sampling procedure.

3.
For any bounded target, π(x), the resulting proposal function, \(q_{t}(x\mathcal {S}_{t})\), is also bounded. Furthermore, defining \(\mathcal {I}_{t} = (s_{1},s_{m_{t}}]\), we have
$$ \max_{x \in \mathcal{I}_{t}} q_{t}(x\mathcal{S}_{t}) \le \max_{x \in \mathcal{I}_{t}} \pi(x). $$ 
4.
The proposal function, \(q_{t}(x\mathcal {S}_{t})\), has heavier tails than the target, i.e., defining \(\mathcal {I}_{t}^{c} = (\infty,s_{1}] \cup (s_{m_{t}},\infty)\), we have
$$ q_{t}(x\mathcal{S}_{t}) \ge \pi(x) \qquad \forall x \in \mathcal{I}_{t}^{c}. $$
Condition 1 guarantees that the function \(q_{t}(x\mathcal {S}_{t})\) leads to a valid pdf, \(\widetilde {q}_{t}(x\mathcal {S}_{t})\), that covers the entire support of the target.
Condition 2 is required from a practical point of view to obtain efficient algorithms. Finally, conditions 3 and 4 are required by the proofs of Theorems 3 and 1, respectively, and also make sense from a practical point of view: if the target is bounded, we would expect the proposal learnt from it to be also bounded and this proposal should be heavier tailed than the target in order to avoid undersampling the tails. Now we can define precisely what we understand by a “sticky” proposal.
Definition 2
(Sticky Proposal (SP)) Let us consider a valid proposal pdf according to Definition 1. Let us assume also that the ith support point is distributed according to p_{ i }(x) (i.e., s_{ i }∼p_{ i }(x)) such that p_{ i }(x)>0 for any \(x \in \mathcal {X}\)and i=1,…,m_{ t }. Then, a sticky proposal is any valid proposal pdf s.t. the L_{1} distance between q_{ t }(x)and π(x) vanishes to zero when the number of support points increases, i.e., if m_{ t }→∞,
where \(d_{t}(z) = \pi (z)q_{t}(z\mathcal {S}_{t})\) is the L_{1} distance between π(x) and q_{ t }(x) evaluated at \(z \in \mathcal {X}\), and (2) implies almost everywhere (a.k.a., almost surely) convergence of q_{ t }(x) to π(x).
In the following, we provide some examples of constructions that fulfill all the conditions in Definitions 1 and 2. All of them approximate the target pdf by interpolating points that belong to the graph of the target function π.
3.1 Examples of constructions
Given \(\mathcal {S}_{t}=\{s_{1},\ldots,s_{m_{t}}\}\) at the tth iteration, let us define a sequence of m_{ t }+1 intervals: \(\mathcal {I}_{0}=(\infty,s_{1}]\), \(\mathcal {I}_{j}=(s_{j},s_{j+1}]\) for j=1,…,m_{ t }−1, and \(\mathcal {I}_{m_{t}}=(s_{m_{t}},+\infty)\). The simplest possible procedure uses piecewise constant (uniform) pieces in \(\mathcal {I}_{i}\), 1≤i≤m_{ t }−1, with two exponential tails in the first and last intervals [13, 15, 18]. Mathematically,
where 1≤i≤m_{ t }−1 and E_{0}(x), \(E_{m_{t}}(x)\) represent two exponential pieces. These two exponential tails can be obtained simply constructing two straight lines in the logdomain as shown in [12, 13, 19]. For instance, defining V(x)= log[π(x)], we can build the straight line w_{0}(x) passing through the points (s_{1},V(s_{1})) and (s_{2},V(s_{2})), and the straight line \(w_{m_{t}}(x)\) passing through the points \((s_{m_{t}1},V(s_{m_{t}1}))\) and \((s_{m_{t}},V(s_{m_{t}}))\). Hence, the proposal function is defined as E_{0}(x)= exp(w_{0}(x)) for \(x\in \mathcal {I}_{0}\) and \(E_{m_{t}}(x)=\exp \left (w_{m_{t}}(x)\right)\) for \(x\in \mathcal {I}_{m_{t}}\). Other kinds of tails can be built, e.g., using Pareto pieces as shown in Appendix E.2 Heavy tails. Alternatively, we can use piecewise linear pieces [20]. The basic idea is to build straight lines, L_{i,i+1}(x), passing through the points (s_{ i },π(s_{ i })) and (s_{i+1},π(s_{i+1})) for i=1,…,m_{ t }−1, and two exponential pieces, E_{0}(x) and \(E_{m_{t}}(x)\), for the tails:
with i=1,…,m_{ t }−1. Note that drawing samples from these trapezoidal pdfs inside \(\mathcal {I}_{i}=(s_{i},s_{i+1}]\) is straightforward [20, 21]. Figure 2 shows examples of the construction of \(q_{t}(x\mathcal {S}_{t})\) using Eq. (3) or (4) with different number of points, m_{ t }=6,8,9,11. See Appendix A for further considerations.
A more sophisticated and costly construction has been proposed for the ARMS method in [12]. However, note that this construction does not fulfill Condition 3 in Definition 1. A similar construction based on Bspline interpolation methods has been proposed in [22, 23] to build a nonadaptive random walk proposal pdf for an MH algorithm. Other alternative procedures can also be found in the literature [13, 16, 18–20].
4 Update of the set of support points
In AISM, a suitable choice of the function η_{ t }(z,d) is required. Although more general functions could be employed, we concentrate on test functions that fulfill the conditions provided in the following definition.
Definition 3
(Test Function) Let us denote the L_{1} distance between the target and the proposal at the tth iteration, for any \(z \in \mathcal {X}\), as d=d_{ t }(z)=π(z)−q_{ t }(z). A valid test function, η_{ t }(z,d), is any function that fulfills all of the following properties:

1.
\(\eta _{t}(z,d): \mathcal {X}\times \mathbb {R}^{+}\rightarrow [0,1]\).

2.
η_{ t }(z,0)=0 for all \(z\in \mathcal {X}\) and \(t\in \mathbb {N}\).

3.
\(\lim \limits _{d\rightarrow \infty }\eta _{t}(z,d)=1\) for all \(z\in \mathcal {X}\) and \(t\in \mathbb {N}\).

4.
η_{ t }(z,d) is a strictly increasing function w.r.t. d, i.e., η_{ t }(z,d_{2})>η_{ t }(z,d_{1}) for any d_{2}>d_{1}.
The first condition ensures that we obtain a valid probability for the addition of new support points, P_{ a }(z)=η_{ t }(z,d), whereas the remaining three conditions imply that support points are more likely to be added in those areas where the proposal is further away from the target, with a nonnull probability of adding new points in places where d>0. In particular, Condition 4 is required by several theoretical results provided in the Appendix. However, update rules that do not fulfill this condition can also be useful, as discussed in the following. Figure 3 depicts an example of function η_{ t } when η_{ t }(z,d)=η_{ t }(d). Note that, for a given value of z, η_{ t } satisfies all the properties of a continuous distribution function (cdf) associated to a positive random variable. Therefore, any pdf for positive random variables can be used to define a valid test function η_{ t } through its corresponding cdf.
4.1 Examples of update rules
Below, we provide three different possible update rules. First of all, we consider the simplest case: η_{ t }(z,d)=η(d). As a first example, we propose
where β>0 is a constant parameter. Note that this is the cdf associated to an exponential random variable.
A second possibility is
where 0<ε_{ t }<M_{ π }, with \(M_{\pi }=\max \limits _{z\in \mathcal {X}}\{\pi (z)\}\),^{Footnote 6} is some appropriate timevarying threshold that can either follow some user predefined rule or be updated automatically.^{Footnote 7} Alternatively, we can also set this threshold to a fixed value, ε_{ t }=ε, as done in the simulations. In this case, setting ε≥M_{ π } implies that the update of \(\mathcal {S}_{t}\) never happens (i.e., new support points are never added to the support set), whereas candidate nodes would be incorporated to \(\mathcal {S}_{t}\) almost surely by setting ε→0. For any other value of ε(i.e., 0<ε<M_{ π }), the adaptation would eventually stop and no support points would be added after some random number of iterations. Note that this update rule does not fulfill Condition 4 in Definition 3, implying that some of the theoretical results of Section 5(e.g., Conjecture 1) are not applicable. However, we have included it here because it is a very simple rule that has shown a good performance in practice and can be useful to limit the number of support points by using a fixed value of ε. Finally, note also that Eq. (6) corresponds to the cdf associated to a Dirac’s delta located at ε_{ t }.
A third alternative is
for \(z \in \mathcal {X}\) and \(0 \le d \le \max \{\pi (z),q_{t}(z\mathcal {S}_{t})\}\), since
This rule appears in other related algorithms, as discussed in Section 6.1. Furthermore, it corresponds to the cdf of a uniform random variable defined in the interval \([\!0,\max \{\pi (z),q_{t}(z\mathcal {S}_{t})\}]\). Hence, for a given value of z, the update test can be implemented as follows: (a) draw a samples v^{′} uniformly distributed in the interval \(\left [0,\max \{\pi (z),q_{t}(z\mathcal {S}_{t})\}\right ]\); (b) if v^{′}≤d_{ t }(z), add z to the set of support points. A graphical representation of this rule is given in Fig. 4, whereas Table 2 summarizes all the previously described update rules.
5 Theoretical results
In this section, we provide some theoretical results regarding the ergodicity of the proposed approach, the convergence of a sticky proposal to the target, and the expected growth of the number of support points of the proposal. First of all, regarding the ergodicity of the AISM, we have the following theorem.
Theorem 1
(Ergodicity of AISM) Let x_{1},x_{2},…,x_{T−1} be the set of states generated by the AISM algorithm in Table 1, using a valid adaptive proposal function, \(\widetilde {q}_{t}(x\mathcal {S}_{t}) = \frac {1}{c_{t}} q_{t}(x\mathcal {S}_{t})\), constructed according to Definition 1, and a test rule fulfilling the conditions in Definition 3. The pdf of x_{ t }, p_{ t }(x), converges geometrically in total variation (TV) norm to the target, \(\widetilde {\pi }(x) = \frac {1}{c_{\pi }} \pi (x)\), i.e.,
where
with c_{ π } and c_{ ℓ } denoting the normalizing constants of π(x) and \(q_{\ell }(x\mathcal {S}_{\ell })\), respectively.
Proof
See Appendix A. □
Theorem 1 ensures that the pdf of the states of the Markov chain becomes closer and closer to the target pdf as t increases, since 0≤1−a_{ t }≤1 and thus the product in the right hand side of (9) is a decreasing function of t. This theorem is a direct consequence of Theorem 2 in [14], and ensures the ergodicity of the proposed adaptive MCMC approach. Regarding the convergence of a sticky proposal to the target, we consider the following conjecture.
Conjecture 1
(Convergence of SP to the target) Let \(\widetilde {\pi }(x) = \frac {1}{c_{\pi }} \pi (x)\) be a continuous and bounded target pdf that has bounded first and second derivatives for all \(x \in \mathcal {X}\). Let \(\widetilde {q}_{t}(x\mathcal {S}_{t}) = \frac {1}{c_{t}} q_{t}(x\mathcal {S}_{t})\) be a sticky proposal pdf, constructed according to Definition 1 by using either a piecewise constant (PWC) or piecewise linear (PWL) approximation (given by Eqs. (3) and (4), respectively). Let us also assume that the support points have been obtained by applying a test rule according to Definition 3 within the AISM algorithm described in Table 1. Then, it is reasonable to assume that \(q_{t}(x\mathcal {S}_{t})\) converges in L_{1} distance to π(x) as t increases (i.e., as the number of support points grows), i.e., as t→∞
An intuitive argumentation is provided in Appendix A.
Note that Conjecture 1 essentially shows that the “sticky” condition is fulfilled for PWC and PWL proposals and continuous, bounded targets with bounded first and second derivatives. Note also that this conjecture implies that \(q_{t}(x\mathcal {S}_{t}) \to \pi (x)\) almost everywhere. Combining Theorem 1 and Conjecture 1 we get the following corollary.
Corollary 2
Let x_{1},x_{2},…,x_{T−1} be the set of states generated by the AISM algorithm in Table 1, using either a PWC or a PWL sticky proposal function, \(\widetilde {q}_{t}(x\mathcal {S}_{t}) = \frac {1}{c_{t}} q_{t}(x\mathcal {S}_{t})\), constructed according to Definition 2 and a test rule fulfilling the conditions in Definition 3. Let \(\widetilde {\pi }(x) = \frac {1}{c_{\pi }} \pi (x)\) be a continuous and bounded target pdf that has bounded first and second derivatives for all \(x \in \mathcal {X}\). Then,
Proof
By Theorem 1 we have
with the a_{ ℓ } given by Eq. (10). Now, since \(q_{\ell }(x\mathcal {S}_{\ell }) \to \pi (x)\) almost everywhere by Conjecture 1, we have c_{ ℓ }→c_{ π } and thus a_{ ℓ }→1 as ℓ→∞. Consequently, ∥π(x)−q_{ t }(x)∥_{ TV }→0 as t→∞. □
Finally, we also have a bound on the expected growth of the number of support points, as provided by the following theorem.
Theorem 3
(Expected rate of growth of the number of support points) Let d_{ t }(z)=π(z)−q_{ t }(z) be the L_{1} distance between the bounded target, π(x), and an arbitrary sticky proposal function, q_{ t }(x), constructed according to Definition 2. Let also η_{ t }(z,d)=η_{ t }(d)be an acceptance function that only depends on z through d=d_{ t }(z) and fulfills the conditions in Definition 3. The expected probability of adding a new support point in the AISM algorithm of Table 1 at the tth iteration is
where \(D_{1}(\pi, q_{t}) = \int _{\mathcal {X}}{d_{t}(z) dz}\), and \(C = \max _{z\in \mathcal {X}} \widetilde {q}_{t}(z\mathcal {S}_{t})\) is a constant that depends on the sticky proposal used. Furthermore, under the conditions of Conjecture 1, \(E[P_{a}x_{t1}, \mathcal {S}_{t}] \to 0\) as t→∞.
Proof
See Appendix C.1. □
Theorem 3 sets a bound on the expected probability of adding new support points, and thus on the expected rate of growth of the number of support points. Furthermore, under certain smoothness assumptions for the target (i.e., that π(x) is twice continuously differentiable), it also guarantees that this expectation tends to zero as the number of iterations increases, hence implying that less points are added as the algorithm evolves.Finally, note that Theorem 3 has been derived only for η_{ t }(z,d)=η_{ t }(d). However, under certain mild assumptions, it can be easily extended to more general test functions, as stated in the following corollary.
Corollary 4
Let \(\eta _{t}(z,d_{t}(z)) = \eta _{t}(\widetilde {d}_{t}(z))\), where \(\widetilde {d}_{t}(z)=\widetilde {d}_{t}(\pi (z),q_{t}(z))\) is some valid semimetric and \(\eta _{t}(\widetilde {d}_{t}(z))\)is a concave function of \(\widetilde {d}_{t}(z)\). Then, if the rest of the conditions in Theorem 3 are satisfied, the expected probability of adding a new support point in the AISM algorithm of Table 1 at the tth iteration is
where \(\widetilde {D}_{1}(\pi, q_{t}) = \int _{\mathcal {X}}{\widetilde {d}_{t}(z) dz}\) and \(C = \max _{z\in \mathcal {X}} \widetilde {q}_{t}(z\mathcal {S}_{t})\). Furthermore, under the conditions of Conjecture 1, \(E\left [P_{a}x_{t1}, \mathcal {S}_{t}\right ] \to 0\) as t→∞.
Proof
See Appendix C.2. □
Note that Corollary 4 allows us to extend the results of Theorem 3 to update rule 3, which corresponds to \(\eta _{t}(z,d_{t}(z)) = \widetilde {d}_{t}(z)\) with \(\widetilde {d}_{t}(z) = \frac {d}{\max \{\pi (z),q_{t}(z\mathcal {S}_{t})\}}\) and d denoting the L_{1} norm.
6 Related works
6.1 Other examples of sticky MCMC methods
The novel class of adaptive independent MCMC methods encompasses several existing algorithms already available in the literature, as shown in Table 3. We denote the proposal pdf employed in these methods as p_{ t }(x) and, for simplicity, we have removed the dependence on \(\mathcal {S}_{t}\) in the function q_{ t }(x). The Griddy Gibbs Sampler [15] builds a proposal pdf as in Eq. (3), which is never adapted later. ARMS [12] and IA^{2}RMS [13] use as proposal density
where q_{ t }(x) is built using different alternative methods [12, 13, 16, 18]. Note that it is possible to draw easily from p_{ t }(x)∝ min{q_{ t }(x),π(x)} using the rejection sampling principle [24, 25], i.e., using the following procedure (in order to draw one sample x_{ a }):

1.
Draw \(x'\sim {\widetilde q}_{t}(x) \propto q_{t}(x)\) and \(u' \sim \mathcal {U}([0,1]\!)\).

2.
If \(u'\leq \frac {\pi (x')}{q_{t}(x')}\), then set x_{ a }=x^{′}.

3.
Otherwise, if \(u' > \frac {\pi (x')}{q_{t}(x')}\), repeat from 1.
The accepted sample x_{ a } has pdf p_{ t }(x)∝ min{q_{ t }(x),π(x)}. Moreover, ARMS adds new points to \(\mathcal {S}_{t}\) using the update Rule 3, only when q_{ t }(z)≥π(z), so that
Otherwise, if q_{ t }(z)<π(z), ARMS does not add new nodes (see the discussion in [13] about the issues in ARMS mixing). Then, the update rule for ARMS can be written as
Furthermore, the double update check used in IA^{2}RMS coincides exactly with Rule 3 when
is employed as proposal pdf. Finally, note that ARMS and IA^{2}RMS contain ARS in [19] as special case when q_{ t }(x)≥π(x), \(\forall x\in \mathcal {X}\) and \(\forall t\in \mathbb {N}\). Hence, ARS can be considered also a special case of the new class of algorithms.
6.2 Related algorithms
Other related methods, using nonparametric proposals, can be found in the literature. Samplers for drawing from univariate pdfs, using similar proposal constructions, has been proposed in [20] but the sequence of adaptive proposals does not converge to the target. Interpolation procedures for building the proposal pdf are also employed in [22, 23]. The authors in [22, 23] suggest to build the proposal by bspline procedures. However, in this case, the resulting proposal is a random walktype (not independent) and the resulting algorithm is not adaptive. Furthermore, there is not a convergence of the shape of proposal to the shape to target, but only local approximations via bspline interpolation. The methods [12, 13, 15] are included in the sticky class of algorithms, as pointed out in Section 6.1. In [16], the authors suggest an alternative proposal construction considering pieces of second order polynomial, in order to be used with the ARMS structure [12].
The adaptive rejection sampling (ARS) method [19, 26] is not an MCMC technique, but it is strongly related to the sticky approach, since it also employs an adaptive nonparametric proposal pdf. ARS needs to be able to build a proposal such that q_{ t }(x)≥π(x), \(\forall x\in \mathcal {X}\) and \(\forall t\in \mathbb {N}\). This is possible only when more requirements about the target are assumed (for instance, logconcavity). For this reason, several extensions of the standard ARS have been also proposed [25, 27, 28], for tackling wider classes of target distributions. In [29], the nonparametric proposal is still adapted by in this case the number of support points remains constant, fixed in advance by the user. Different construction nonparametric procedures in order to address multivariate distributions have been also presented [21, 30, 31].
Other techniques have been developed to be applied specifically for Monte CarlowithininGibbs scenario when it is possible to draw directly from the fullconditional pdfs. In [32], an importance sampling approximation of the univariate target pdf is employed and a resampling step is performed in order to provide an “approximate” sample from the fullconditional. In [18], the authors suggest a nonadaptive strategy for building a suitable nonparametric proposal via interpolation. In this work, the interpolation procedure is first performed using a huge amount of nodes and then many of them are discarded, according to a suitable criteria. Several other alternatives involving MHtype algorithms have been used for sampling efficiently from the fullconditional pdfs within a Gibbs sampler [5–7, 15, 33–35].
7 Adaptive independent sticky MTM
In this section, we consider an alternative MCMC structure for the second stage described in Section 2: using a multipletry Metropolis (MTM) approach [36, 37]. The resulting technique, Adaptive Independent Sticky MTM (AISMTM), is an extension of AISM that considers multiple candidates as possible new state, at each iteration. This improves the ability of the chain to explore the state space [37]. At iteration t, AISMTM builds the proposal density \(q_{t}(x\mathcal {S}_{t}) \) (step 1 of Table 4) using the current set of support points \(\mathcal {S}_{t}\). Let x_{ t }=x be the current state of the chain and xj′ (j=1,…,M) a set of i.i.d. candidates simulated from \(q_{t}(x\mathcal {S}_{t})\) (see step 2 of Table 4). Note that AISMTM uses an independent proposal [2], just like AISM. As a consequence, the auxiliary points in step 2.3 of Table 4 can be deterministically set ([1], pp. 119120), [37].
In step 2, a sample x^{′} is selected among the set of candidates {x1′,…,xM′}, with probability proportional to the importance sampling weights,
The selected candidate is then accepted or rejected according to the acceptance probability α given in step 2. Finally, step 3 updates the set \(\mathcal {S}_{t}\),including a new point
with probability P_{ a }(z^{′})=η_{ t }(z^{′},d_{ t }(z^{′})). Note that \(x_{t}\notin \mathcal {Z}\), and thus AISMTM is an independent MCMC algorithm according to Holden’s definition [14]. For the sake of simplicity, we only consider the case where a single point can be added to \(\mathcal {S}_{t}\) at each iteration. However, this update step can be easily extended to allow for more than one sample to be included into the set of support points. Note also that AISMTM becomes AISM for M=1.
AISMTM provides a better choice of the new support points than AISM (see Section 9). The price to pay for this increased efficiency is the higher computational cost per iteration. However, since the proposal quickly approaches the target, it is possible to design strategies with a decreasing number of tries (M_{1}≥M_{2}≥⋯≥M_{ t }≥⋯≥M_{ T }) in order to reduce the computational cost.
7.1 Update rules for AISMTM
The update rules presented above require changes that take into account the multiple samples available, when used in AISMTM. As an example, let us consider the update scheme in Eq. (7). Considering for simplicity that only a single point can be incorporated to \(\mathcal {S}_{t}\), the update step for \(\mathcal {S}_{t}\) can be split in two parts: choose a “bad” point in \(\mathcal {Z}\in \{z_{1},\dots,z_{M}\}\) and then test whether it should be added or not. Thus, first a z^{′}=z_{ i } is selected among the samples in \(\mathcal {Z}\) with probability proportional to
for i=1,…,M.^{Footnote 8} This step selects (with high probability) a sample where the proposal value is far from the target. Then, the point z^{′} is included in \(\mathcal {S}_{t}\) with probability
exactly as in Eq. (7). Therefore, the probability of adding a point z_{ i } to \(\mathcal {S}_{t}\) is
that is a probability mass function defined over M+1 elements: z_{1},…, z_{ M } and the event {no addition} that, for simplicity, we denote with the empty set symbol ∅. Thus, the update rule in step 3 of Table 4 can be rewritten as a unique step,
where we have used \(1\sum _{i=1}^{(r)} P_{\mathcal {Z}}(z_{i})=\frac {M}{\sum _{j=1}^{M}\varphi _{t}\left (z_{j}\right)}\).
8 Range of applicability and multivariate generation
The range of applicability of the sticky MCMC methods is briefly discussed below. On the one hand, sticky MCMC methods can be employed as standalone algorithms. Indeed, in many applications, it is necessary to draw samples from complicated univariate target pdf (as example in signal processing, see [38]). In this case, the sticky schemes provide virtually independent samples (i.e., with correlation close to zero) very efficiently. It is also important to remark that AISM and AISMTM also provide automatically an estimation of the normalizing constant of the target (a.k.a. marginal likelihood or Bayesian evidence) (since, with a suitable choice of the update test, the proposal approaches the target pdf almost everywhere). This is usually a hard task using MCMC methods [1, 2, 11].
AISM and AIMTM can be also applied directly to draw from a multivariate distribution if a suitable construction procedure of the multivariate sticky proposal is designed (e.g, see [30, 31, 39, 40] and ([21], Chapter 11)). However, devising and implementing such procedures in high dimensional state spaces are not easy tasks. Therefore, in this paper, we focus on the use of the sticky schemes within other Monte Carlo techniques (such as Gibbs sampling or the hit and run algorithm) to draw from multivariate densities. More generally, Bayesian inference often requires drawing samples from complicated multivariate posterior pdfs, \(\widetilde {\pi }(\mathbf {x}\mathbf {y})\) with
For instance, this happens in blind equalization and source separation, or spectral analysis [3, 4]. For simplicity, in the following we denote the target pdf as \(\widetilde {\pi }(\mathbf {x})\). When direct sampling from \(\widetilde {\pi }(\mathbf {x})\) in the space \(\mathbb {R}^{L}\) is unfeasible, a common approach is the use of Gibbstype samplers [2]. This type of methods split the complex sampling problem into simpler univariate cases. Below we briefly summarize some wellknown Gibbstype algorithms.
Gibbs sampling. Let us denote as x^{(0)} a randomly chosen starting point. At iteration k≥1, a Gibbs sampler obtains the ℓth component (ℓ=1,…,L) of x, x_{ ℓ }, drawing from the full conditional \(\widetilde {\pi }_{\ell }\left (x\mathbf {x}_{1:\ell 1}^{(k)}, \mathbf {x}_{\ell +1:L}^{(k1)}\right)\) given all the information available, namely:

1.
Draw \(x_{\ell }^{(k)} \sim \widetilde {\pi }_{\ell }\left (x\mathbf {x}_{1:\ell 1}^{(k)}, \mathbf {x}_{\ell +1:L}^{(k1)}\right)\) for ℓ=1,…,L.

2.
Set \(\mathbf {x}^{(k)}=\left [x_{1}^{(k)},\ldots,x_{L}^{(k)}\right ]^{\top }\).
The steps above are repeated for k=1,…,N_{ G }, where N_{ G } is the total number of Gibbs iterations. However, even sampling from \(\widetilde {\pi }_{\ell }\) can often be complicated. In some specific situations, rejection samplers [41–45] and their adaptive versions, adaptive rejection sampling (ARS) algorithms, are employed to generate (one) sample from \(\widetilde {\pi }_{\ell }\) [12, 19, 25, 27–29, 40, 46, 47]. The ARS algorithms are very appealing techniques since they construct a nonparametric proposal in order to mimic the shape of the target pdf, yielding in general excellent performance (i.e., independent samples from \(\widetilde {\pi }_{\ell }\) with an high acceptance rate). However, their range of application is limited to some specific classes of densities [19, 47].
More generally, it is impossible to draw from a fullconditional pdf \(\widetilde {\pi }_{\ell }\) (neither a rejection sampler can be applied), an additional MCMC sampler is required in order to draw from \(\widetilde {\pi }_{\ell }\) [33]. Thus, in many practical scenarios, we have an MCMC (e.g., an MH sampler) inside another MCMC scheme (i.e., the Gibbs sampler). In the socalled MHwithinGibbs approach, only one MH step is often performed within each Gibbs iteration, in order to draw from each complicated fullconditionals. This hybrid approach preserves the ergodicity of the Gibbs sampler and provides good performance in many cases. On the other hand, several authors have noticed that using a single MH step for the internal MCMC is not always the best solution in terms of performance (cf. [48]). Other approximated approaches have been also proposed, considering the application of the importance sampling within the Gibbs sampler [32].
Using a larger number of iterations for the MH algorithm, there is more probability of avoiding the “burnin” period so that the last sample be distributed as the fullconditional [33–35]. Thus, this case is closer to the ideal situation, i.e., sampling directly from the fullconditional pdf. However, unless the proposal is very well tailored to the target, a properly designed adaptive MCMC algorithm should provide less correlated samples than a standard MH algorithm. Several more sophisticated (adaptive or not) MH schemes for the application “withinGibbs” have been proposed in literature [12, 13, 16, 18, 20, 23, 49, 50]. In general, these techniques employ a nonparametric proposal pdf in the same fashion of the ARS schemes (and as the sticky MCMC methods). It is important to remark that performing more steps of a standard or adaptive MH within a Gibbs sampler can provide better performance than performing a longer Gibbs chain applying only one MH step (see, e.g., [12, 13, 16, 17]).
Recycling Gibbs sampling. Recently, an alternative Gibbs scheme, called Recycling Gibbs (RG) sampler, has been proposed in literature [51]. The combined use of RG with a sticky algorithm is particularly interesting since RG recycles and employs all the samples drawn from each fullconditional pdfs in the final estimators. Clearly, this scheme fits specially well for the use of a adaptive sticky MCMC algorithm where different MCMC steps are performed for each fullconditional pdfs.
Hit and Run. The Gibbs sampler only allows movements along the axes. In certain scenarios, e.g., when the variables x_{ ℓ } are highly correlated, this can be an important limitation that slows down the convergence of the chain to the stationary distribution. The Hit and Run sampler is a valid alternative. Starting from x^{(0)}, at the kth iteration, it applies the following steps:

1.
Choose uniformly a direction d^{(k)} in \(\mathbb {R}^{L}\). For instance, it can be done drawing L samples v_{ ℓ } from a standard Gaussian \(\mathcal {N}(0,1)\), and setting
$$ \mathbf{d}^{(k)}=\frac{\mathbf{v}}{\sqrt{\mathbf{v}\mathbf{v}^{\top}} }, $$where v=[v_{1},…,v_{ L }].

2.
Set x^{(k)}=x^{(k−1)}+λ^{(k)}d^{(k)} where λ^{(k)} is drawn from the univariate pdf
$$p(\lambda)\propto \widetilde{\pi}\left(\mathbf{x}^{\left(k1\right)}+\lambda \mathbf{d}^{(k)}\right), $$where \(\widetilde {\pi }\left (\mathbf {x}_{\ell }^{(k1)}+\lambda \mathbf {d}^{(k)}\right)\) is a slice of the target pdf along the direction d^{(k)}.
Also in this case, we need to be able to draw from the univariate pdf p(λ) using either some direct sampling technique or another Monte Carlo method (e.g., see [50]).
There are several methods similar to the Hit and Run where drawing from a univariate pdf is required; for instance, the most popular one is the Adaptive Direction Sampling [52].
Sampling from univariate pdfs is also required inside other types of MCMC methods. For instance, this is the case of exchangetype MCMC algorithms [53] for handling models with intractable partition functions. In this case, efficient techniques for generating artificial observations are needed. Techniques which generalize the ARS method, using nonparametric proposals, have been applied for this purpose (see [54]).
9 Numerical simulations
In this section, we provide several numerical results comparing the sticky methods with several wellknown MCMC schemes, such as the ARMS technique [12], the adaptive MH method in [10], and the slice sampler [55].^{Footnote 9} The first two experiments (which can be easily reproduced by interested users) correspond to bimodal onedimensional and twodimensional targets, respectively, and are used as benchmarks to compare different variants of the AISM and AISMTM methods with other techniques. They allow us to show the benefits of the nonparametric proposal construction, even in these two simple experiments. Then, in the third example, we approximate the hyperparameters of a Gaussian process (GP) [56], which is often used for regression purposes in machine learning for signal processing applications.
9.1 Multimodal target distribution
We study the ability of different algorithms to simulate multimodal densities (which are clearly nonlogconcave). As an example, we consider a mixture of Gaussians as target density,
where \(\mathcal {N}\left (x;\mu,\sigma ^{2}\right)\) denotes the normal distribution with mean μ and variance σ^{2}. The two modes are so separated that ordinary MCMC methods fail to visit one of the modes or remains indefinitely trapped in one of them. The goal is to approximate the expected value of the target (E[X]=0 with \(X\sim \widetilde {\pi }(x)\)) via Monte Carlo. We test the ARMS method [12] and the proposed AISM and AISMTM algorithms. For AISM and AISMTM, we consider different construction procedures for the proposal pdf:

P1: the construction given in [12] formed by exponential pieces, specifically designed for ARMS.

P2: alternative construction formed by exponential pieces obtained by a linear interpolation in the logpdf domain, given in [13].

P3: the construction using uniform pieces in Eq. (3).

P4: the construction using linear pieces in Eq. (4).
Furthermore, for AISM and AISMTM, we consider the Update Rule 1 (R1) with different values of the parameter β, the Update Rule 2 (R2) with different value of the parameter ε, and the Update Rule 3 (R3) for the inclusion of a new node in the set \(\mathcal {S}_{t}\) (see Section 4). More specifically, we first test AISM and AISMTM with all the construction procedures P1, P2, P3, and P4 jointly with the rule R3. Then, we test AISM with the construction P4 and the update test R2 with ε∈{0.005,0.01,0.1,0.2}. For Rule 1 we consider β∈{0.3,0.5,0.7,2,3,4}. All the algorithms start with \(\mathcal {S}_{0}=\{10,8,5,10\}\) and initial state x_{0}=−6.6. For AISMTM, we have set M∈{10,50}. For each independent run, we perform T=5000 iterations of the chain.
The results given in Table 5 are the averages over 2000 runs, without removing any sample to account for the initial burnin period. Table 5 shows the Mean Square Error (MSE) in the estimation E[X], the autocorrelation function ρ(τ) at different lags, τ∈{1,10,50} (normalized, i.e., ρ(0)=1), the approximated effective sample size (ESS) of the produced chain ([57], Chapter 4)
(clearly, ESS≤T), the final number of support points m_{ T } and the computing time normalized with respect to the time spent by ARMS [12]. For simplicity, in Table 5, we have reported only the case of R2 with ε∈{0.005,0.01}; however, other results are shown in Fig. 5.
AISM and AIMTM outperform ARMS, providing a smaller MSE and correlation (both close to zero). This is because ARMS does not allow a complete adaptation of the proposal pdf as highlighted in [13]. The adaptation in AISM and AIMTM provides a better approximation of the target than ARMS, as also indicated by the ESS which is substantially higher in the proposed methods. ARMS is in general slower than AISM for two main reasons. Firstly, the construction P1 (used by ARMS) is more costly since it requires the computation of several intersection points [12]. It is not required for the procedures P2, P3, and P4. Secondly, the effective number of iterations in ARMS is higher than T=5000 (the averaged value is ≈5057.83) due to the discarded samples in the rejection step (in this case, the chain is not moved forward).
Figure 6a–d depicts the averaged autocorrelation function ρ(τ) for τ=1,…,100 for the different techniques and constructions. Figure 6e–h shows the average acceptance probability (AAP; the value of α of the MHtype techniques) of accepting a new state as function of the iterations t. We can see that, with AISM and AIMTM, AAP approaches 1 since q_{ t } becomes closer and closer to π. Figure 7 shows the evolutions of the number of support points, m_{ t }, as function of t=1,…,T=5000, again for the different techniques and constructions. Note that, with AIMTM and P3P4, AAP approaches 1 so quickly and the correlation is so small (virtually zero) that it is difficult to recognize the corresponding curves which are almost constant close to one or zero, respectively. The constructions P3 and P4 provide the better results. In this experiment, P4 seems to provide the best compromise between performance and computational cost. We also test AISM with update R2 for different values of ε (and different constructions). The number of nodes m_{ t } and AAP as function of t for these cases are shown in Fig. 5. These figures and the results given in Table 5 show that AISMP4R2 provides extremely good performance with a small computational cost (e.g., the final number of points is only m_{ T }≈43 with ε=0.005). This shows that the update rule R2 is a very promising choice given the obtained results. Moreover, we can observe that the update rule R1 is very parsimonious in adding new points even considering a great range of values of β, from 0.3 to 4. The results are good also in this case with R1, so that this rule seems to be a more robust interesting alternative to R2 (which seems more dependent on the choice of β). Finally, Fig. 8 shows the histograms of the 5000 samples obtained by one run of AISMP3R1 with β=0.1 and β=3. The target pdf is depicted in solid line and the final construction proposal pdf is shown in dashed line.
9.2 Missing mode experiment
Let us consider again the previous bimodal target pdf,
shown in Fig. 8. Here, we consider a bad choice of the initial support points, such as \(\mathcal {S}_{0}=\{5,6,10\}\) cutting out one of the two modes (we consider that no information about the range of the target pdf is provided). We test the robust implementation described in Appendix E.1 Mixture of proposal densities, i.e., we employ the proposal density defined
where \( \widetilde {q}_{1}(x)=\mathcal {N}\left (x;0,\sigma _{p}^{2}\right)\) and \(\widetilde {q}_{2}(x\mathcal {S}_{t})\) is a sticky proposal constructed using the procedure P3 in Eq. (3) (we use the update rule 1 with β=0.1). We consider the most defensive strategy defining α_{ t }=α_{0}=0.5 for all t. We test σ_{ p }∈{2,3,8,10}. We compute the mean absolute error (MAE), in estimating the variance Var[X]=49.55 where \(X\sim \widetilde {\pi }(x)\), with different MCMC methods generating chains of length T=10^{4}. We compare this Robust AISMP3R1 scheme with a standard MH method using \(\widetilde {q}_{1}(x)\) as proposal pdf and the Adaptive MH technique where the scale parameter \(\sigma _{p}^{(t)}\) is adapted online [10] (starting with \(\sigma _{p}^{(0)}\in \{2,3,8,10\}\)). The results, averaged over 10^{3} independent runs, are given in Table 6.
9.3 Heavytailed target distribution
In this section, we test the AISM method from drawing with a target heavy tails. We show that the sticky MCMC schemes can be applied in this scenario, even by using a proposal pdf with exponential (i.e., “light”) tails. However, we recall that an alternative construction of the tails is always possible, as suggested in Appendix E.2 Heavy tails using Pareto tails, for instance. More specifically, we consider the Lévy density, i.e.
∀x≥λ. Given a random variable \(X \sim {\bar \pi }(x)\), we have that E[X]=∞ and Var[X]=∞ due to the heavytail of the Lévy distribution. However, the normalizing constant, \(\frac {1}{c_{\pi }}\), such that \({\bar \pi }(x) = \frac {1}{c_{\pi }} \pi (x)\) integrates to one, can be determined analytically, and is given by \(\frac {1}{c_{\pi }} = \sqrt {\frac {\nu }{2\pi }}\).
Our goal is estimating the normalizing constant \(\frac {1}{c_{\pi }}\) via Monte Carlo simulation, when λ=0 and ν=2. In general, it is difficult to estimate a normalizing constant using MCMC outputs [2, 58, 59]. However, in the sticky MCMC algorithms (with update rules as R1 and R3 in Table 2), the normalizing constant of the adaptive nonparametric proposal approaches the normalizing constant of the target. We compare AISMP4R3 and different Multipletry Metropolis (MTM) schemes. For the MTM schemes, we use the following procedure: given the MTM outputs obtained in one run, we use these samples as nodes, then construct the approximated function using the construction P4 (considering these nodes), and finally compute the normalizing constant of this approximated function. Note that we use the same construction procedure P4, for a fair comparison.
For AISM, we start with only m_{0}=3 support points, \(\mathcal {S}_{0}=\{s_{1}=0,s_{2},s_{3}\}\), where two nodes are randomly chosen at each run, i.e., \(s_{2},s_{3} \sim \mathcal {U}([1,10])\) with s_{2}<s_{3}. We also test three different MTM techniques, two of them using an independent proposal pdf (MTMind) and the last one a random walk proposal pdf (MTMrw). For the MTM schemes, we set M=1000 tries and importance weights designed again to choose the best candidate in each step [37]. We set T=5000 for all the methods. Note that, the total number of target valuation E of AISM is only E=T=5000 whereas we E=MT=5·10^{6} for the MTMind schemes and E=2MT=10^{7} for the MTMrw algorithm (see [37] for further details). For the MTMind methods, we use an independent proposal \(\widetilde {q}(x)\propto \exp ((x\mu)^{2}/(2\sigma ^{2}))\) with μ∈{10,100} and σ^{2}=2500. In MTMrw, we have a random walk proposal \(\widetilde {q}(xx_{t1})\propto \exp \left ((xx_{t1})^{2}\left /\left (2\sigma ^{2}\right)\right.\!\right)\) with σ^{2}=2500. Note that we need to choose huge values of σ^{2} due to the heavytailed feature of the target.
The results, averaged over 2000 runs, are summarized in Table 7. Note that the real value of \(\frac {1}{c_{p}}\) when ν=2 is \(\frac {1}{\sqrt {\pi }}=0.5642\). The AISMP4R3 provides better results than all of the MTM approaches tested with only a fraction of their computational cost. Furthermore, AISMP4R3 avoids the critical issue of parameter selection (selecting a small value of σ^{2} in this case can easily lead to very poor performance).
9.4 Sticky MCMC methods within Gibbs sampling
9.4.1 Example 1: comparing different MCMCwithinGibbs schemes
In this example we show that, even in a simple bivariate scenario, AISM schemes can be useful within a Gibbs sampler. Let us consider the bimodal target density
with A=16, B=10^{−2}, and \(\sigma _{1}^{2}=\sigma _{2}^{2}=\frac {10^{4}}{2}\). Densities with this nonlinear analytic form have been used in the literature (cf. [10]) to compare the performance of different Monte Carlo algorithms. We apply N_{ G } steps of a Gibbs sampler to draw from \(\widetilde {\pi }(x_{1},x_{2})\), using ARMS [12], AISMP4R3, and AISMTMP4R3 within of the Gibbs sampler to generate samples from the fullconditionals, starting always with the initial support set \(\mathcal {S}_{0}=\{10, 6, 4.3, 0, 3.2, 3.8, 4.3, 7, 10\}\). From each fullconditional pdf, we draw T samples and take the last one as the output from the Gibbs sampler. We also apply a standard MH algorithm with a random walk proposal \(q\left (x_{\ell,t}x_{\ell,t1}\right) \propto \exp \left ((x_{\ell,t}x_{\ell,t1})^{2}\left /\left (2\sigma _{p}^{2}\right)\right.\!\right)\) for ℓ∈{1,2}, σ_{ p }∈{1,2,10}, 1≤t≤T. Furthermore, we test an adaptive parametric approach (as suggested in [8]). Specifically, we apply the adaptive MH method in [10] where the scale parameter of q(x_{ℓ,t}x_{ℓ,t−1}) is adapted online, i.e., σ_{p,t} varies with t (we set σ_{p,0}=3). We also consider the application of the slice sampler [55] and the Hamiltonian Monte Carlo (HMC) method [60]. For the standard MH and the slice samplers we have used the function mhsample.m and slicesample.m directly provided by MATLAB. For HMC, we consider the code provided in [61] with ε_{ d }=0.01 as discretization parameter and L=1 as length of the trajectory.^{Footnote 10} We recall that a preliminary code of AISM is also available at MatlabFileExchange webpage.
We consider two initializations for all the methodswithinGibbs: (In1) \(x_{\ell,0}^{(k)}=1\); (In2) \(x_{\ell,0}^{(k)}=1\) and \(x_{\ell,0}^{(k)}=x_{\ell,T}^{(k1)}\) for k=1,…,N_{ G }. We use all the samples to estimate four statistics that involve the first four moments of the target: mean, variance, skewness, and kurtosis. Table 8 provides the mean absolute error (MAE; averaged over 500 independent runs) for each of the four statistics estimated, and the time required by the Gibbs sampler (normalized by considering 1.0 to be the time required by ARMS with T=50).
The results are provided in Table 8. First of all, we notice that AISM outperforms ARMS and the slice sampler for all values of T and N_{ G }, in terms of performance and computational time. Regarding the use of the MH algorithm within Gibbs, the results depend largely on the choice of the variance of the proposal, \(\sigma _{p}^{2}\), and the initialization, showing the need for adaptive MCMC strategies. For a fixed value of T×N_{ G }, the AISM schemes provide results close to the smallest averaged MAE for In1 and the best results for In2 with a slight increase in the computing time, w.r.t. the standard MH algorithm. Finally, Table 8 shows the advantage of the nonparametric adaptive independent sticky approach w.r.t. the parametric adaptive approach [8, 10].
9.4.2 Example 2: comparison with an ideal Gibbs sampler
The ideal scenario for the Gibbs sampling scheme is that we are able to draw samples from the fullconditional pdfs (using a transformation or a direct method). In this section, we compare the performance of MH and AISMwithinGibbs schemes with the ideal case. Let us consider two Gaussian fullconditional densities,
with ξ_{1}=1 and ξ_{2}=0.2. The joint pdf is a bivariate Gaussian pdf with mean vector μ=[0,0]^{⊤} and covariance matrix Σ=[1.08 0.54; 0.54 0.31]. We apply a Gibbs sampler with N_{ G } iterations to estimate both the mean and the covariance of the joint pdf. Then, we calculate the average MSE in the estimation of all the elements in μ and Σ, averaged over 2000 independent runs. We use this simple case, where we can draw directly from the fullconditionals, to check the performance of MH and AISMP3R3 within Gibbs as a function of T and N_{ G }. For the MH scheme, we use a Gaussian random walk proposal, \(\widetilde {q}\left (x_{\ell,t}^{(k)}\left x_{\ell,t1}^{(k)}\right.\right) \propto \exp \left (\left.\left (x_{\ell,t}^{(k)}0.5x_{\ell,t1}^{(k)}\right)^{2}\right /\left (2\sigma _{p}^{2}\right)\right)\) for ℓ∈{1,2}, 1≤t≤T and 1≤k≤N_{ G }. For AISMP3R3, we start with \(\mathcal {S}_{0}=\{2,0,2\}\).
We set N_{ G }=10^{3} and \(x_{\ell,0}^{(i)}=1\) (both for MH and AISMP3R3), and increase the value of T. The results can be seen in Fig. 9. AISMwithinGibbs easily reaches the same performance as the ideal case (sampling directly from the full conditionals) even for small values of T, whereas the MHwithinGibbs needs a substantially larger value of T (up to T=500 for σ_{ p }=0.1) to attain a similar performance. Note the importance of using a proper parameter σ_{ p } for attaining good performance. This observation shows the importance of employing an adaptive technique withinGibbs.
9.5 Sticky MCMC methods within Recycling Gibbs sampling
In this section, we test the sticky MCMC methods within the Recycling Gibbs (RG) sampling scheme where the intermediate samples drawn from each fullconditional pdf are sued in the final estimator [51]. We consider a simple numerical simulation (easily reproducible by any practitioner) involving a bidimensional target pdf
where μ_{1}=4, μ_{2}=1, \(\delta _{1}=\sqrt {\frac {5}{2}}\) and δ_{2}=1. Note that \({\widetilde \pi }(x_{1},x_{2})\) is bimodal and is not Gaussian. The goal is to approximate via Monte Carlo the expected value, \(\mathbb {E}[\mathbf {X}]\) where \(\mathbf {X}=\left [X_{1},X_{2}\right ] \sim {\widetilde \pi }(x_{1},x_{2})\).
We test different Gibbs techniques: the MH [2] and AISMP3R3 algorithm (with update rule 3 and proposal construction in Eq. (3)), within the Standard Gibbs (SG) and within the RG sampling schemes. For the MH method, we use a Gaussian random walk proposal,
for σ>0, ℓ∈{1,2}, 1≤k≤N_{ G } and 1≤t≤T. We set \(x_{\ell,0}^{(k)}=1\) and \(x_{\ell,0}^{(k)}=x_{\ell,T}^{(k1)}\) for k=1,…,N_{ G }, for all schemes.
9.5.1 Optimal scale parameter for MH
First of all, we obtain the MSE in estimation of E[ X] for different values of the σ parameter for MHwithinSG (with T=1 and N_{ G }=1000). Figure 10a shows the results averaged over 10^{5} independent runs. The performance of the Standard Gibbs (SG) sampler depends strongly on the choice of σ of the internal MH method. We can observe that there exists an optimal value σ^{∗}≈3. This shows the need of using an adaptive scheme for drawing from the fullconditional pdfs. In the following, we compare the performance of AISM with the performance of this optimized MH using the optimal scale parameter σ^{∗}=3, in order to show the capability of the nonparametric adaptation employed in AISM, with respect to a standard adaptation procedure [10].
9.5.2 Comparison among different schemes
For AISMP3R3, we start with the set of support points \(\mathcal {S}_{0}=\{ 10,6,2,2,6,10\}\). We have averaged the MSE values over 10^{5} independent runs for each Gibbs scheme.
In Fig. 10b (represented in logscale), we fix N_{ G }=1000 and vary T. As T grows, when a standard Gibbs (SG) sampler is used, the curves show an horizontal asymptote since the internal chains converge after some value T≥T^{∗}. Considering an RG scheme, the increase of T yield lower MSE since now we recycle the internal samples. Figure 10b shows the advantage of using AISMR3P3 even when compared with the optimized MH method. The advantage of AISMR3P3 is clearer with small T values (10<T<30; recall that in this experiment N_{ G }=1000 is kept fixed). The performance of AISMR3P3 and optimized MH (within Gibbs) becomes more similar as T increases. This is due to the fact that, in this case, with a high enough value of T, the MH chain is able to exceed its burnin period and eventually converges.
9.6 Tuning of the hyperparameters of a Gaussian process (GP)
9.6.1 Exponential Power kernel function
Let assume to observe the pairs of data \(\{y_{j},\mathbf {z}_{j}\}_{j=1}^{P}\), with \(y_{j} \in \mathbb {R}\) and \(\mathbf {z}_{j} \in \mathbb {R}^{d_{Z}}\), and denote the corresponding vectors y=[y_{1},…,y_{ P }] and Z=[z_{1},…,z_{ P }]. We address the regression problem of inferring the hidden function y=f(z), linking the variable y and z. For this goal, we assume the model
where e∼N(e;0,σ^{2}). For simplicity, we set d_{ Z }=1. We consider the f is a Gaussian process (GP) [56], i.e., we assume a GP prior over f, so f∼GP(μ(z),κ(z,r)) where μ(z)=0, and the kernel function is
Therefore, the vector f=[f(z_{1}),…,f(z_{ P })] is distributed as \(p(\mathbf {f}\mathbf {Z},\kappa,\beta,\delta)=\mathcal {N}(\mathbf {f};\mathbf {0},\mathbf {K})\) where 0 is a 1×P vector, K:=κ(z_{ i },z_{ j }), for all i,j=1,…,P is a P×P matrix, and we have expressed explicitly the dependence on the choice of the kernel family κ in Eq. (22). Moreover, we denote the hyperparameters of the model as θ=[θ_{1}=σ,θ_{2}=β,θ_{3}=δ], i.e., the standard deviation of the observation noise and the two parameters of the kernel κ(z,r). We assume a prior with independent truncated positive Gaussian components for the hyperparameters \(p(\boldsymbol {\theta })=p(\sigma,\beta,\delta)=\mathcal {N}(\sigma ;0,5) \mathcal {N}(\beta ;0,5) \mathcal {N}(\delta ;0,5) \mathbb {I}_{\sigma }\mathbb {I}_{\beta }\mathbb {I}_{\gamma }\) where \(\mathbb {I}_{v}=1\) if v>0, and \(\mathbb {I}_{v}=0\) if v≤0. To simplify the expression of the posterior pdf, let us focus on the filtering problem and the tune of the parameters, namely we desire to infer f and θ. Hence, the posterior pdf is given by
with \(p(\mathbf {y}\mathbf {f},\mathbf {Z},{\boldsymbol {\theta }}, \kappa)=\mathcal {N}\left (\mathbf {y};\mathbf {0},\sigma ^{2} \mathbf {I}\right)\) and \(p(\mathbf {f}\mathbf {y}, \mathbf {Z},{\boldsymbol {\theta }}, \kappa) =\mathcal {N}(\mathbf {f};{\boldsymbol {\mu }}_{p}, {\boldsymbol {\Sigma }}_{p})\), with mean μ_{ p }=K(K+σ^{2}I)^{−1}y^{⊤} and covariance matrix Σ_{ p }=K−K(K+σ^{2}I)^{−1}K^{⊤}, representing the solution of the GP given the specific choice of the hyperparameters θ. The marginal posterior of the hyperparameters [56] is
where
Hence, the logmarginal posterior is
for θ_{1},θ_{2},θ_{3}>0, where clearly K depends on θ_{1}=σ, θ_{2}=β and θ_{3}=δ.^{Footnote 11} We apply a Gibbs sampler from drawing from p(θy,Z,κ). We fix Z=[−10:0.1:10] (i.e., a grid between −10 and 10 with step 0.1); hence, P=201, and the data y are artificially generated according to the model (21) considering the values θ^{∗}=[σ^{∗}=1,β^{∗}=0.5,δ^{∗}=3]. We average the results using 10^{3} independent runs. At each run, we generate new data y according to the model with θ^{∗}, and run the Gibbs sampler in order to approximate p(θy,Z,κ) considering N_{ G }=2000 samples (without removing any burnin period). We approximate the expected value of the posterior \( \widehat {{\boldsymbol {\theta }}}\approx E_{p}[{\boldsymbol {\theta }}]\) using these N_{ G } samples and compare with θ^{∗} (with enough number of data, it can be considered the groundtruth). For drawing from the fullconditional pdfs, we set T=10, we employ a standard MH with Gaussian random proposal a \(q(x_{\ell,t}x_{\ell,t1}) \propto \exp \left ((x_{\ell,t}x_{\ell,t1})^{2}/\left (2\sigma _{p}^{2}\right)\right)\) for ℓ∈{1,2,3}, and we test different values of σ_{ p }∈{1,2,3}. Moreover, we apply AISMP4R3 with T=10 and the initial support points \(\mathcal {S}_{0}=\{0.01, 0.2, 0.5,1,2,4,7,10\}\). We also test the IA^{2}RMS method [13] which is a special case of AISM technique (see Section 6.1). For IA^{2}RMS, we use the construction procedure P4 as in AISM (both methods employ the update rule R3). The initializations for all techniques is set \(x_{\ell,0}^{(k)}=1\) and \(x_{\ell,0}^{(k)}=x_{\ell,T}^{(k1)}\) for ℓ=1,2,3 and k=1,…,N_{ G }. The mean square error (MSE) in the estimation of θ^{∗}, averaged over 10^{3} runs, is shown in Table 9. AISM outperforms the MH methods. IA^{2}RMS provides better results w.r.t. AISM since it uses a better equivalent proposal p_{ t }(x)∝ min{q_{ t }(x),π(x)}. However, IA^{2}RMS is slower than AISM due to its rejection step (necessary in order to produce samples from the equivalent proposal p_{ t }(x)∝ min{q_{ t }(x),π(x)}). We recall that IA^{2}RMS is a special case of AISM technique. Finally, Table 9 shows the MSE in the estimation of the hyperparameters θ^{∗} employing a Riemann quadrature, i.e., using a grid approximation [ 0,A]^{3} with A=100 and with step ε_{ g }∈{0.1,0.2,0.5,1,2} (note this method excludes the possibility that the hyperparameters are greater than A). The computing times are normalized w.r.t. the time spent by MH in Tables 9 and 10.
9.6.2 Automatic Relevant Determination kernel function
Here we consider the estimation of the hyperparameters of the Automatic Relevance Determination (ARD) covariance ([62], Chapter 6). Let us assume again the P observed data pairs \(\{y_{j},\mathbf {z}_{j}\}_{j=1}^{P}\), with \(y_{j}\in \mathbb {R}\) and
where d_{ Z } is the dimension of the input features. We also denote the corresponding P×1 output vector as y=[y_{1},…,y_{ P }]^{⊤} and the d_{ Z }×P input matrix Z=[z_{1},…,z_{ P }]. We again address the regression problem of inferring the unknown function f which links the variable y and z. Thus, the assumed model is y=f(z)+e, where e∼N(e;0,σ^{2}), and that f(z) is a realization of a Gaussian process (GP) [56]. Hence \(f(\mathbf {z}) \sim \mathcal {GP}(\mu (\mathbf {z}),\kappa (\mathbf {z},\mathbf {r}))\) where μ(z)=0, \(\mathbf {z},\mathbf {r} \in \mathbb {R}^{d_{Z}}\), and we consider the ARD kernel function
for ℓ=1,…,d_{ Z }. Note that we have a different hyperparameter δ_{ ℓ } for each input component z_{ ℓ }; hence, we also define \({\boldsymbol {\delta }}=\delta _{1:d_{Z}}=[\delta _{1},\ldots,\delta _{d_{Z}}]\). Unlike in the previous section, note that here β is assumed known (β=2). This type of kernel function is often employed to perform an automatic relevance determination (ARD) of the input components with respect the output variable ([62], Chapter 6). Namely, using ARD allows us to infer the relative importance of different components of inputs: a small value of δ_{ ℓ } means that a variation of the ℓcomponent z_{ ℓ } impacts the output more, while a high value of δ_{ ℓ } shows virtually independence between the ℓcomponent and the output. Therefore, the complete vector containing all the hyperparameters of the model is
i.e., all the parameters of the kernel function in Eq. (22) and standard deviation σ of the observation noise. We assume \(p({\boldsymbol {\theta }})=\prod _{\ell =1}^{d_{Z}+1}\frac {1}{\theta _{\ell }^{\alpha }}\mathbb {I}_{\theta _{\ell }}\) where α=1.3, \(\mathbb {I}_{v}=1\) if v>0, and \(\mathbb {I}_{v}=0\) if v≤0. We desire to compute the expected value \({\mathbb E}[{\boldsymbol {\Theta }}]\) with Θ∼p(θy,Z,κ), via Monte Carlo quadrature.
More specifically, we apply a AISMP4R3 withinGibbs (with \(\mathcal {S}_{0}=\{0.01,0.5,1,2,5,8,10,15\}\)) and the Single Component Adaptive Metropolis (SCAM) algorithm [63] withinGibbs to draw from π(θ)∝p(θy,Z,κ). Note that dimension of the problem is D=d_{ X }+1 since \({\boldsymbol {\theta }}\in \mathbb {R}^{D}\). For SCAM, we use the Gaussian random walk proposal \(q(x_{\ell,t}x_{\ell,t1}) \propto \exp \left ((x_{\ell,t}x_{\ell,t1})^{2}/\left (2\gamma _{\ell,t}^{2}\right)\right)\). In SCAM, the scale parameters γ_{ℓ,t} are adapted (one for each component) considering all the previous corresponding samples (starting with γ_{ℓ,0}=1).
We generated the P=500 pairs of data, \(\{y_{j},\mathbf {z}_{j}\}_{j=1}^{P}\), drawing \(\mathbf {z}_{j}\sim \mathcal {U}\left ([0,10]^{d_{Z}}\right)\) and y_{ j } according to the model in Eq. (21), considered d_{ Z }∈{1,3,5,7,9} so that D∈{2,4,6,8,10}, and set \(\sigma ^{*}=\frac {1}{2}\) and \(\delta _{\ell }^{*}=2\), ∀ℓ, for all the experiments (recall that θ^{∗}=[δ^{∗},σ^{∗}]). We consider θ^{∗} as ground truth and compute the MSE obtained by the different Monte Carlo techniques.
We have averaged the results using 10^{3} independent runs. We consider N_{ G }=1000 and T=20 for both schemes, AISMwithinGibbs and SCAMwithinGibbs. The results are provided in Table 11. We can see that AISMP4R3 provides the better performance and the difference increases with the dimension D=d_{ Z }+1 of the problem.
10 Conclusions
In this work, we have introduced a new class of adaptive MCMC algorithms for anypurpose stochastic simulation. We have discussed the general features of the novel family, describing the different parts which form a generic sticky adaptive MCMC algorithm. The proposal density used in the new class is adapted online, constructed by employing nonparametric procedures. The name “sticky” remarks that the proposal pdf becomes progressively more and more similar to the target. Namely, a complete adaptation of the shape of the proposal is obtained (unlike using parametric proposals). The role of the update control test for the inclusion of new support points has been investigated. The design of this test is extremely important, since it controls the tradeoff between computational cost and the efficiency of the resulting algorithm. Moreover, we have discussed how the combined design of a suitable proposal construction and a proper update test ensures the ergodicity of the generated chain.
Two specific sticky schemes, AISM and ASMTM, have been proposed and tested exhaustively in different numerical simulations. The numerical results show the efficiency of the proposed algorithms with respect to other stateoftheart adaptive MCMC methods. Furthermore, we have showed that other wellknown algorithms already introduced in the literature are encompassed by the novel class of methods proposed. A detailed description of the related works in the literature and their range of applicability are also provided, which is particularly useful for the interested practitioners and researchers. The novel methods can be applied both as a standalone algorithm or within any Monte Carlo approach that requires sampling from univariate densities (e.g., the Gibbs sampler, the hitandrun algorithm or adaptive direction sampling). A promising future line is designing suitable constructions of the proposal density in order to allow the direct sampling from multivariate target distributions (similarly as [21, 30, 31, 39, 40]). However, we remark that the structure of the novel class of methods is valid regardless of the dimension of the target.
11 Appendix A: Proof of Theorem 1
Note that Eq. (9) in Theorem 1 is a direct consequence of Theorem 2 in [14], which requires \(x_{t} \sim q(x\mathcal {S}_{t})\) to be independent of the current state, x_{t−1}, and the satisfaction of the strong Doeblin condition. Regarding the first issue, x_{ t } is independent of x_{t−1} by construction of the algorithm, so we only need to focus on the second issue. The strong Doeblin condition is satisfied if, given a proposal pdf, \(\widetilde {q}_{t}(x\mathcal {S}_{t}) = \frac {1}{c_{t}} q_{t}(x\mathcal {S}_{t})\), and a target, \(\widetilde {\pi }(x) = \frac {1}{c_{\pi }} \pi (x)\) with support \(\mathcal {X} \subseteq \mathbb {R}\), there exists some a_{ t }∈(0,1] such that, for all \(x \in \mathcal {X}\) and \(t \in \mathbb {N}\),
First of all, note that Eq. (28) can be rewritten as
Then, note also that
where the last inequality is due to the fact that min{1,x}≤x. Therefore, a possible value of a_{ t } that allows us to satisfy Eq. (29) is
From Eq. (30) it is clear that a_{ t }≤1, so all that remains to be shown is that a_{ t }>0. Let us recall that \(\mathcal {I}_{t} = (s_{1},s_{m_{t}}]\), where s_{1} and \(s_{m_{t}}\) are the smallest and largest support points in \(\mathcal {S}_{t} = \{s_{1}, \ldots, s_{m_{t}}\}\), respectively. Then, since \(q_{t}(x\mathcal {S}_{t}) > 0\) for all \(x \in \mathcal {X}\) (condition 1 in Definition 1) and \(t \in \mathbb {N}\), and π(x) is assumed to be bounded, we have
And regarding the tails, note that \(q_{t}(x\mathcal {S}_{t})\) must be uniformly heavier tailed by construction (condition 4 in Definition 1),^{Footnote 12} so \(q_{t}(x\mathcal {S}_{t}) \ge \pi (x)\) for all \(x \in \mathcal {I}_{t}^{c} = (\infty,s_{1}] \cup (s_{m_{t}},\infty)\) and we also have
Therefore, we conclude that 0<a_{ t }≤1, the strong Doeblin condition is satisfied and thus all the conditions for Theorem 2 in [14] are fulfilled.
12 Appendix B: Argumentation for Conjecture 1
Let us define \(\mathcal {I}_{t} = (s_{1}, s_{m_{t}}]\) and \(\mathcal {I}_{t}^{c} = (\infty, s_{1}] \cup (s_{m_{t}}, \infty)\), where s_{1} and \(s_{m_{t}}\) are the smallest and largest points of the set of support points at time step t, \(\phantom {\dot {i}\!}\mathcal {S}_{t}=\{s_{1},\ldots,s_{m_{t}}\}\) with \(\phantom {\dot {i}\!}s_{1}<\ldots <s_{m_{t}}\). Then, the L_{1} distance between the target and the proposal can be expressed as \(D_{1}(\pi,q_{t}) = D_{\mathcal {I}_{t}}(\pi,q_{t}) + D_{\mathcal {I}_{t}^{c}}(\pi,q_{t})\), where \(D_{\mathcal {I}_{t}}(\pi,q_{t}) = \int _{\mathcal {I}_{t}}{d_{t}(x)\ dx}\) and \(D_{\mathcal {I}_{t}^{c}}(\pi,q_{t}) = \int _{\mathcal {I}_{t}^{c}}{d_{t}(x)\ dx}\) with d_{ t }(x)=π(x)−q_{ t }(x). Let us focus first on \(D_{\mathcal {I}_{t}}(\pi,q_{t})\). Since q_{ t }(x) is constructed as a piecewise polynomial approximation on the intervals \(\mathcal {I}_{t,i} = (s_{i},s_{i+1}]\),
where
is the L_{1} distance between the target and the proposal in the ith interval. Now, using Theorem 3.1.1 in [65] we can easily bound d_{ t }(x) for the ℓth order interpolation polynomial (with ℓ∈{0,1} in this case) used within the ith interval. For ℓ=0 and assuming that π(s_{ i })≥π(s_{i+1}) (and thus \(q_{t}(x)=\pi (s_{i})\ \forall x \in \mathcal {I}_{t,i}\)) without loss of generality,^{Footnote 13}
where \(\dot {\pi }(\xi)\) denotes the first derivative of π(x) evaluated at x=ξ, ξ∈(s_{ i },s_{i+1}] is some point inside the interval whose value depends on x, x_{ i } and π(x), and this bound is finite since we assume that the first derivative of π(x) is bounded. Therefore, for the PWC approximation we have
Similarly, for ℓ=1 we have
where \(\ddot {\pi }(\xi)\) denotes the second derivative of π(x) evaluated at x=ξ, ξ∈(s_{ i },s_{i+1}] is some point inside the interval, and this bound is again finite since we assume that the second derivative of π(x) is also bounded. And the L_{1} distance for the PWL approximation can thus be bounded as
Note that the two cases can be summarized in a single expression:
where
with \(C_{t}^{(0)} = \max _{x \in \mathcal {I}_{t}}\dot {\pi }(x)\) and \(C_{t}^{(1)} = \frac {1}{2} \max _{x \in \mathcal {I}_{t}}\ddot {\pi }(x)\).
Now, let us assume that a new point, \(s' \in \mathcal {I}_{t,k} = \left [s_{k},s_{k+1}\right ]\) for 1≤k≤m_{ t }−1, is added at some iteration t^{′}>t using the mechanism described in the AISM algorithm (see Table 1) and that no other points have been incorporated to the support set for t+1,…,t^{′}−1. In this case, the construction of the proposal function changes only inside the interval \(\mathcal {I}_{t,k}\), which splits now into \(\mathcal {I}_{t',k}=[s_{k},s']\) and \(\mathcal {I}_{t',k+1}=[s',s_{k+1}]\). Then, the new bound for the distance inside \(\mathcal {I}_{t'} = \mathcal {I}_{t}\) is \(D_{\mathcal {I}_{t'}}(\pi,q_{t'}) \le L_{t'}^{(\ell)}\), with
where the last inequality is obtained by applying Newton’s binomial theorem, which states that A^{ℓ+1}+B^{ℓ+1}<(A+B)^{ℓ+1} for any A,B>0, using A=s^{′}−s_{ k }>0 and B=s_{k+1}−s^{′}>0. Hence, the bound in Eq. (36) can never increase when a new support point is incorporated and indeed tends to decrease as new points are added to the support set.
Note that we could still have \(L_{t}^{(\ell)} \to K > 0\) as t→∞. However, the conditions of Definition 1 ensure that the support of the proposal always contains the support of the target (i.e., \(q_{t}(x\mathcal {S}_{t})>0\) whenever π(x)>0 for any t and \(\mathcal {S}_{t}\)) and it has uniformly heavier tails (implying that \(q_{t}(x\mathcal {S}_{t}) \to 0\) slower than π(x) as x→±∞). Consequently, support points can be added anywhere inside the support of the target, \(\mathcal {X} \subseteq \mathbb {R}\). This implies that \(L_{t}^{(\ell)} \to 0\) as t→∞, since (s_{i+1}−s_{ i })→0 as more points are added inside \(\mathcal {I}_{t}\), and thus also \(D_{\mathcal {I}_{t}}(\pi,q_{t}) \to 0\) as t→∞. Let us focus now on \(D_{\mathcal {I}_{t}^{c}}(\pi,q_{t})\). Let us assume, without loss of generality, that a new point, s^{′}∈(−∞,s_{1}],^{Footnote 14} is added at some iteration t^{′}>t using the mechanism described in the AISM algorithm (see Table 1) and that no other points have been incorporated to the support set for t+1,…,t^{′}−1. In this case, it is clear that the distance in the tails decreases (i.e., \(D_{\mathcal {I}_{t'}^{c}}(\pi,q_{t}) < D_{\mathcal {I}_{t}^{c}}(\pi,q_{t})\)) at the expense of increasing the distance in the central part of the target (i.e., \(D_{\mathcal {I}_{t'}}(\pi,q_{t}) > D_{\mathcal {I}_{t}}(\pi,q_{t})\)). However, even if this leads to a momentary increase in the overall distance, note that we still have \(D_{\mathcal {I}_{t'}}(\pi,q_{t}) \to 0\) as t^{′}→∞ as long as new support points can be added inside \(\mathcal {I}_{t'}\), something which is guaranteed by the AISM algorithm. Finally, since there is always a nonnull probability of incorporating points in the tails,^{Footnote 15} thus implying that \(D_{\mathcal {I}_{t}^{c}}(\pi,q_{t}) \to 0\) as t→∞, since \(\mathcal {I}_{t}^{c}\) becomes smaller and smaller as t increases.
Therefore, we can guarantee that using the AISM algorithm in Table 1, with a valid proposal that fulfills Definition 1 and an acceptance rule according to Definition 3, we obtain a sticky proposal that fulfills Definition 2.
13 Appendix C: Support points
In this appendix we provide the proofs of Theorem 3 and Corollary 4, which bound the expected growth of the number of support points.
13.1 C.1 Proof of Theorem 3
Given the support set \(\mathcal {S}_{t}\) and the state x_{t−1}, the expected probability of adding a new point to \(\mathcal {S}_{t}\) at the tth iteration is given by
where \(d_{t}(z)=\left \pi (z)q_{t}(z\mathcal {S}_{t})\right \) and
represents the kernel function of AISM given x_{t−1} and \(\mathcal {S}_{t}\). Since candidate points \(x' \in \mathcal {X}\) are directly drawn from the proposal pdf, we have \(p_{t}\left (x'x_{t1},\mathcal {S}_{t}\right) = \widetilde {q}_{t}\left (x'\mathcal {S}_{t}\right)\), and from the structure of the AISM in Table 1 it is straightforward to see that
where \(\alpha (x_{t1},x') =\min \left [1,\frac {\pi (x')q_{t}(x_{t1}\mathcal {S}_{t})}{\pi (x_{t1})q_{t}(x'\mathcal {S}_{t})}\right ]\). Inserting these two expressions in Eq. (38), the kernel function of AISM becomes
Let us recall now the integral form of Jensen’s inequality for a concave function φ(x) with support \(\mathcal {X} \subseteq \mathbb {R}\) [66]:
which is valid for any nonnegative function f(x) such that \(\int _{\mathcal {X}}{f(x)\ dx}=1\). Then, since we assume that η_{ t }(z,d)=η_{ t }(d), η_{ t }(d) is a concave function of d by condition 4 of Definition 3, and \(\int _{\mathcal {X}} p_{t}(zx_{t1},\mathcal {S}_{t}) dz=1\), we have
with
where we have used (39) to obtain the final expression in (41). Now, for the first term in the right hand side of (41), note that \(\left [\int _{\mathcal {X}} \alpha (x_{t1},x') \ \widetilde {q}_{t}(x'\mathcal {S}_{t})\ dx' \right ] \le 1\), since 0≤α(x_{t−1},x^{′})≤1 and \(\int _{\mathcal {X}}{\widetilde {q}_{t}(x'\mathcal {S}_{t})\ dx'} = 1\). And for the second term, we have
where we recall that \(D_{1}(\pi,q_{t}) = \int _{\mathcal {X}}{d_{t}(z)\ dz} = \int _{\mathcal {X}}{\pi (z)q_{t}(z\mathcal {S}_{t})\ dz}\) and \(C = \max _{z\in \mathcal {X}} \widetilde {q}_{t}(z\mathcal {S}_{t}) < \infty \), since we have assumed that π(x) is bounded and thus, by condition 4 in Definition 1, \(\widetilde {q}_{t}(z\mathcal {S}_{t})\) is also bounded. Therefore, we obtain
and inserting (42) into (40) we have the following bound for the expected probability of adding a support point at the tthe iteration,
Finally, noting C<∞, that both d_{ t }(x_{t−1})→0 and D_{1}(q_{ t },π)→0 as t→∞ by Conjecture 1, and that η_{ t }(0)=0 by condition 2 in Definition 3, we have \(E[P_{a}(z)x_{t1}, \mathcal {S}_{t}]\to 0\) as t→∞.
13.2 C.2 Proof of Corollary 4
First of all, recall that a semimetric fulfills all the properties of a metric except for the triangle inequality. Therefore, we have \(\widetilde {d}_{t}(\pi (z),q_{t}(z)) \ge 0\), \(\widetilde {d}_{t}(\pi (z),q_{t}(z)) = 0 \iff \pi (z) = q_{t}(z)\) and \(\widetilde {d}_{t}(\pi (z),q_{t}(z)) = \widetilde {d}_{t}(q_{t}(z),\pi (z))\). Now, from the proof of Theorem 3 (see Appendix C.1) we can see that η_{ t } is not used until Eq. (40). Since \(\eta _{t}(\widetilde {d}_{t}(z))\) is a concave function of \(\widetilde {d}_{t}(z)\), we can still use Jensen’s inequality and this equation becomes
where, following the same procedure as in Appendix C.1 (which is still valid due to the fact that \(\widetilde {d}_{t}(\pi (z),q_{t}(z))\) is a semimetric), the term inside η_{ t } can be now bounded by
with \(\widetilde {D}_{t}(\pi, q_{t}) = \int _{\mathcal {X}}{\widetilde {d}_{t}(z)\ dz}\). Therefore, we have
with \(E[P_{a}(z)x_{t1},\mathcal {S}_{t}] \to 0\) as t→∞ under the conditions of Conjecture 1.
14 Appendix D: Variate generation
The proposal density \(\widetilde {q}_{t}(x\mathcal {S}_{t}) \propto q_{t}(x\mathcal {S}_{t})\), built using one of the interpolation procedures in Section 3.1, is composed of m_{ t }+1 pieces (including the two tails). More specifically, the function \(q_{t}(x\mathcal {S}_{t})\) can be seen as a finite mixture
with \(\sum _{i=0}^{m_{t}} \eta _{i}=1\), whereas ϕ_{ i }(x) is a linear pdf or a uniform pdf (depending on the employed construction; see Eqs. (3)(4)) defined in the interval \(\mathcal {I}_{i}\), and ϕ_{ i }(x)=0 for \(x \notin \mathcal {I}_{i}\). The tails, ϕ_{0}(x) and \(\phi _{m_{t}}(x)\), are truncated exponential pdfs (or Pareto tails see Appendix E.2 Heavy tails). Hence, in order to draw a sample from \(\widetilde {q}_{t}(x\mathcal {S}_{t}) \propto q_{t}(x\mathcal {S}_{t})\), it is necessary to perform the following steps:

1.
Compute the area A_{ i } below each piece composing \(q_{t}(x\mathcal {S}_{t})\), i=0,…,m_{ t }. This is straightforward for the construction procedures in Eqs. (3)(4) since the function \(q_{t}(x\mathcal {S}_{t})\) is formed by linear or constant pieces, so that it can be easily done analytically. Moreover, since the tails are exponential functions also in this case we compute the areas below A_{0} and \(A_{m_{t}}\) analytically. Then, we need to normalize them,
$$ \eta_{i} = \frac{A_{i}}{\sum_{j=1}^{m} A_{j}}, \quad \text{for} \quad i=0,\ldots, m. $$ 
2.
Choose a piece (i.e., an index j^{∗}∈{0,…,m_{ t }}) according to the weights η_{ i } for i=0,…,m_{ t }.

3.
Given the index j^{∗}, draw a sample x^{′} in the interval \(\phantom {\dot {i}\!}\mathcal {I}_{j^{*}}\) with pdf \(\phantom {\dot {i}\!}\phi _{j^{*}}(x)\), i.e., \(\phantom {\dot {i}\!}x' \sim \phi _{j^{*}}(x)\).
15 Appendix E: Robust algorithms
In this appendix, we briefly discuss how to increase the robustness of the method, both with respect to a bad choice of the initial set \(\mathcal {S}_{0}\) (e.g., when information about the range of the target pdf is not available) and w.r.t. the heavy tails that appear in many target pdfs.
15.1 E.1 Mixture of proposal densities
Let us define a proposal density as
where \(\widetilde {q}_{2}(x\mathcal {S}_{t})\) is a sticky proposal pdf built as described in Section 3. The density \(\widetilde {q}_{1}(x)\) is a generic proposal function with an explorative task. The explorative behavior of \(\widetilde {q}_{1}\) can be controlled by its scale parameter. The weight α_{ t } can be kept constant α_{ t }=α_{0}=0.5 for all t (this is the most defensive strategy), or it can be decreased with the iteration t, i.e., α_{ t }→0 as t→∞. The joint adaptation of the weight α_{ t }, the scale parameter of \(\widetilde {q}_{1}\) and \(\widetilde {q}_{2}\) using a sticky procedure needs and deserves additional studies.
15.2 E.2 Heavy tails
The choice of the tails for the proposal is important for two reasons: (a) to accelerate the convergence of the chain to the target (especially for heavytailed target distributions) and (b) to increase the robustness of the method w.r.t. the initial choice of the set \(\mathcal {S}_{0}\). Indeed, often the construction of tails with a bigger area below them can reduce the dependence on a specific choice of the set of initial support points. For heavy tailed constructions, there are several possibilities. For instance, here we propose to use Pareto pieces, which have the following analytic form
with γ_{ j }>1, j∈{0,m_{ t }}. In the logdomain, this results in
i.e., \(q_{t}(x\mathcal {S}_{t})=\exp \left (w_{i}(x)\right)\) with i∈{0,m_{ t }}. Let us denote V(x)= log[π(x)]. Fixing the parameters μ_{ j }, j∈{0,m_{ t }}, the remaining parameters, ρ_{ j } and γ_{ j }, are set in order to satisfy the passing conditions through the points (s_{1},V(s_{1})) and (s_{2},V(s_{2})), and through the points \((s_{m_{t}1},V(s_{m_{t}1}))\) and \((s_{m_{t}},V(s_{m_{t}}))\), respectively. The parameters μ_{ j } can be arbitrarily chosen by the user, as long as they fulfill the following inequalities:
Values of μ_{ j } such that μ_{0}≈s_{2} and \(\mu _{m_{t}}\approx s_{m_{t}1}\) yield small values of γ_{ j } (close to 1) and, as a consequence, fatter tails. Larger differences in μ_{0}−s_{2} and \(\mu _{m_{t}} s_{m_{t}1}\) yield γ_{ j }→+∞, i.e., lighter tails. Note that we can compute analytically the integral of q_{ t }(x) in \(\mathcal {I}_{0}\) and \(\mathcal {I}_{m_{t}}\):
Moreover, we can also draw samples easily from each Pareto tail using the inversion method [2].
Notes
The adjective “sticky” highlights the ability of the proposed schemes to generate a sequence of proposal densities that progressively “stick” to the target.
The purpose of this work is to provide a family of methods applicable to a wide range of signal processing problems. A generic Matlab code (not focusing on any specific application) is provided at http://www.lucamartino.altervista.org/STICKY.zip.
A preliminary version of this work has been published in [64]. With respect to that paper, the following major changes have been performed: we discuss exhaustively the general structure of the new family (not just a particular algorithm); we perform a complete theoretical analysis of the AISM algorithm; we extend substantially the discussion about related works; we introduce the AISMTM algorithm; we show how sticky methods can be used to sample from multivariate pdfs by embedding them within a Gibbs sampler or the hit and run algorithm; and we provide additional numerical simulations (including comparisons with other benchmark sampling algorithms and the estimation of the hyper parameters of a Gaussian processes).
For simplicity, we assume that π(x) is bounded. However, the case of unbounded target pdfs can also be tackled by designing a suitable proposal construction that takes into account the vertical asymptotes of the target function. Similarly, we consider a target function defined in a continuous space \(\mathcal {X}\) for the sake of simplicity, although the support domain could also be discrete.
Note that any other MCMC technique could be used.
Note that \(d_{t}(z) \le \max \{\pi (z), q_{t}(z\mathcal {S}_{t})\} \le M_{\pi }\), since \(M_{t}=\max \limits _{z\in \mathcal {X}} q_{t}(z\mathcal {S}_{t})\le M_{\pi }\) for all of the constructions described in Section 3 for the proposal function. Therefore, all the ε_{ t }≥M_{ π } lead to equivalent update rules.
Regarding the definition of ε_{ t }, this threshold should decrease over time (to guarantee that new support points can always be added), but not too fast (to avoid adding too many points and thus increasing the computational cost). Selecting the optimum threshold can be very challenging, but many of the rules that have been used in the area of stochastic filtering for the update parameter could be used here. For instance, good update rules could be ε_{ t }=κM_{ π }·e^{−γt} or \(\varepsilon _{t} = \frac {\kappa M_{\pi }}{t+1}\) for some appropriate values of 0<κ<1 and γ>0. Exploring this issue is out of the scope of this paper, but we plan to address this in future works.
We have used the equality \(d_{t}(z_{i})=\pi (z_{i})q_{t}(z_{i}\mathcal {S}_{t})=\max \{\pi (z_{i}),q_{t}(z_{i}\mathcal {S}_{t})\}\min \{\pi (z_{i}),q_{t}(z_{i}\mathcal {S}_{t})\}\).
Preliminary Matlab code for the AISM algorithm, with the constructions described in Section 3.1 and the update control rule R3, is provided at https://www.mathworks.com/matlabcentral/fileexchange/54701adaptiveindependentstickymetropolis–aism–algorithm.
Other related codes can be also found at http://mcstan.org.
Recall that if θ_{1}θ_{2}θ_{3}≤0 then p(θy,Z,κ)=0.
Note that we can always guarantee that \(q_{t}(x\mathcal {S}_{t})\) is heavier tailed than π(x) by using an appropriate construction for the tails of the proposal, as discussed in Section 3 and Appendix E.2 Heavy tails.
If we consider the complementary case (i.e., π(s_{i+1})≥π(s_{ i }) and thus \(q_{t}(x)=\pi (s_{i+1})\ \forall x \in \mathcal {I}_{t,i}\)) we obtain exactly the same bound following an identical procedure.
The same conclusion is obtained if we consider a point \(s' \in (s_{m_{t}},\infty)\).
Note that the proposals are assumed to be uniformly heavier tailed than the target by Condition 4 of Definition 1. Therefore, we can guarantee that enough candidate samples are generated in the tails.
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Funding
This work has been supported by the Spanish Ministry of Economy and Competitiveness (MINECO) through the MIMODPLC (TEC201564835C33R) and KERMES (TEC201681900REDT/AEI) projects; by the Italian Ministry of Education, University and Research (MIUR); by PRIN 201011 grant; and by the European Union (Seventh Framework Programme FP7/20072013) under grant agreement no:630677.
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Martino, L., Casarin, R., Leisen, F. et al. Adaptive independent sticky MCMC algorithms. EURASIP J. Adv. Signal Process. 2018, 5 (2018). https://doi.org/10.1186/s1363401705246
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DOI: https://doi.org/10.1186/s1363401705246