Online sequential Monte Carlo smoother for partially observed diffusion processes

where (W _t )_t≥0 is a standard Brownian motion on \(\mathbb {R}^{d}\), \(\alpha : \mathbb {R}^{d}\to \mathbb {R}^{d}\), and \(\Gamma : \mathbb {R}^{d}\to \mathbb {R}^{d\times d}\). The solution to (1) is supposed to be partially observed at times t₀=0,…,t _n through an observation process (Y _k )_0≤k≤n in \(\left (\mathbb {R}^{m}\right)^{n+1}\). In the following, for all 0≤k≤n, the state \(X_{t_{k}}\) at time k is referred to as X _k . For all 0≤k≤n, the distribution of Y _k given X _k has a density with respect to a reference measure λ on \(\mathbb {R}^{m}\) given by g(X _k ,·). For the sake of simplicity, the shorthand notation g _k (X _k ) for g(X _k ,Y _k ) is used. The distribution of X₀ has a density with respect to a reference measure μ on \(\mathbb {R}^{d}\) given by χ. For all 0≤k≤n−1, the conditional distribution of X_k+1 given X _k has a density q _k (X _k ,·) with respect to μ.

when \(\{h_{k}\}_{k=0}^{n-1}\) are given functions on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\). Smoothed additive functionals as (2) are crucial for maximum likelihood inference of latent data models. These quantities appear naturally when computing the Fisher score in hidden Markov models or the intermediate quantity of the expectation maximization algorithm (see Section 5). They are also pivotal to design online expectation maximization-based algorithms which motivates the method introduced in this paper that does not require growing storage and can process observations online.

The algorithm proposed in this paper is based on sequential Monte Carlo methods which offer a flexible framework to approximate such distributions with weighted empirical measures associated with random samples. At each time step, the samples are moved randomly in \(\mathbb {R}^{d}\) and associated with importance weights. In general situations, the computation of these importance weights involve the unknown transition density of the process (1). The solution introduced in Section 3 requires an unbiased estimator of these unknown transition densities. Moreover, this estimator must be almost surely positive and upper bounded. Statistical inference of stochastic differential equations is an active area of research, and several solutions have been proposed to design unbiased estimates of these transition densities. Those estimators require different assumptions on the model (1), we provide below several solutions that can be investigated.

Assumption (i) is somewhat restrictive as it requires α to derive from a scalar potential, however, it has natural applications in many fields such as movement ecology, see [15]. Assumption (ii) is a technical condition which ensures that exact sampling of processes solution to (1) using acceptance rejection methods, see for instance [4, 5, 13]. In addition to provide an unbiased estimate of the transition density, the GPE ensure that this estimate is almost surely positive. Moreover, as detailed below, under additional conditions, a GPE that is almost surely upper bounded can be defined.

Continuous importance sampling-based estimators In the case the previous assumptions are not fulfilled, in particular assumption (i), alternatives to GPEs are given by continuous importance sampling procedures for SDE. In [34], for each 0≤k≤n−1, the transition density between t _k and t_k+1 is expressed as an infinite expansion obtained using the Kolmogorov backward operator associated with (1). This analytical expression of the transition density is not tractable and is estimated by updating random samples at random times between t _k and t_k+1 using tractable proposal distributions (for instance, based on an Euler discretization of the original SDE). Then, these samples are associated with random weights to ensure that the proposed estimator is unbiased. More recently, [14] extended the discrete time importance sampling estimator by introducing updates at random times associated with a renewal process. The random samples are weighted using the Kolmogorov forward operator associated with the SDE which relies on the first two order derivatives of the drift and diffusion coefficients (and is therefore tractable).

The unbiasedness of these procedures and the controls of the variability of the estimates require moments assumptions and Holder type conditions on the parameters of the SDE (1). Their efficiency require a fair amount of tuning as they highly depend on the proposal densities used to obtain the Monte Carlo samples and the point processes generating the underlying random times. In addition to unbiasedness, the proposed algorithm in this work requires that the estimator of the transition density is almost surely positive and upper bounded. This implies additional assumptions on the SDE depending on the chosen estimate and could lead to interesting perspectives.

The approximation of (5) requires first to approximate the sequence of filtering distributions. Sequential Monte Carlo methods provide an efficient and simple solution to obtain these approximations using sets of particles \(\left \{\xi ^{\ell }_{k}\right \}_{\ell =1}^{N}\) associated with weights \(\left \{\omega ^{\ell }_{k}\right \}_{\ell =1}^{N}\), 0≤k≤n.

The most simple choice for p_k−1 and 𝜗 _k is the bootstrap filter proposed by [17] which sets p_k−1=q_k−1 and for all \(x\in \mathbb {R}^{d}\), 𝜗(x)=1. In the case of POD processes, q_k−1 is unknown but it can be replaced by any approximation to sample the particles as any choice of p_k−1 can be made. The approximation can be obtained using a discretization scheme such as Euler method or a Poisson-based approximation as detailed below. A more appealing choice is the fully adapted particle filter which sets for all \(x,x'\in \mathbb {R}^{d}\), p_k−1(x,x^′)∝q_k−1(x,x^′)g _k (x^′) and for all \(x\in \mathbb {R}^{d}\), \(\vartheta (x) = \int q_{k-1}\left (x,x'\right)g_{k}\left (x'\right)\mu \left (\mathrm {d} x'\right)\). Here, again q_k−1 has to be replaced by an approximation. In Section 5, it is replaced by the Gaussian approximation provided by a Euler scheme which leads to a Gaussian proposal density p_k−1 as the observation model is linear and Gaussian.

The PaRIS algorithm uses the same decomposition as the FFBS algorithm introduced in [12] and the FFBSi algorithm proposed by [16] to approximate smoothing distributions. It combines both the forward-only version of the FFBS algorithm with the sampling mechanism of the FFBSi algorithm. It does not produce an approximation of the smoothing distributions but of the smoothed expectation of a fixed additive functional and thus may be used to approximate (2). Its crucial property is that it does not require a backward pass, the smoothed expectation is computed on-the-fly with the particle filter and no storage of the particles or weights is needed.

It is clear from steps (i) to (iii) that each time a new observation Y_n+1 is received, the quantities \(\left (\tau _{n+1}^{i}\right)_{1\le i \le N}\) can be updated only using Y_n+1, \(\left (\tau _{n}^{i}\right)_{1\le i \le N}\) and the particle filter at time n. This means that storage requirements do not increase when processing additional data.

As proved in [28], the algorithm is asymptotically consistent (as N goes to infinity) for any precision parameter \(\tilde N\). However, there is a significant qualitative difference between the cases \(\tilde {N} = 1\) and \(\tilde {N} \geq 2\). As for the FFBSi algorithm, when there exists σ₊ such that 0<q _k <σ₊, PaRIS algorithm may be implemented with \(\mathcal {O}(N)\) complexity using the accept-reject mechanism of [10].

In Algorithm 1, M independent copies \(\left (\zeta ^{m}_{k-1}\right)_{1\le m \le M}\) of ζ_k−1 are sampled and the empirical mean of the associated estimates of the transition density are used to compute \(\widehat {\omega }^{\ell }_{k}\) instead of a single realization. Therefore, to obtain a generalized random version of PaRIS algorithm, we only need to be able to sample from the discrete probability distribution \(\Lambda _{k}^{N}(i,\cdot)\) in the case of POD processes.

Lemma 1

Assume that (10) holds for some 0≤k≤n−1. For all 1≤i≤N, define the random variable \(J_{k}^{i}\) as follows:

Then, the conditional probability distribution given \(\mathcal {G}_{k+1}^{N}\) of \(J_{k}^{i}\) is \(\Lambda _{k}^{N}(i,\cdot)\).

If only assumption (12) holds, the algorithm has a quadratic complexity. The bound of (10) is uniform (it does not depend on the particles) and can be used for every particle 1≤i≤N. However, this bound can be large (with respect to the simulated set of particles) for the algorithm of Lemma 1. The bound of (12) requires N computations per particle (therefore, N² computations). However, it is clear that this second bound is sharper that the one of (10) for the acceptance rejection procedure and may lead to a computationally more efficient algorithm.

Assume that there exist random variables L _w and U _w such that for all 0≤s≤Δ _k , L _w ≤ϕ(w _s )≤U _w . The performance of the estimator depends on the choice of L _w and U _w which is specific to the SDE. In the case of the models analyzed in Section 5, these bounds are discussed in [13] for the SINE model and in [27] for the log-growth model. Note that in the case where ϕ is not upper bounded, [5] proposed the EA3 algorithm. This layered Brownian bridge construction first samples random variables to determine in which layer the Brownian bridge lies before simulating the bridge conditional on the event that it belongs to the layer. By continuity of ϕ, L _w , and U _w can be computed easily.

is upper bounded almost surely by \(\hat {\sigma }^{k}_{+}\). In particular, if L _w is bounded below almost surely, (14) always satisfies assumption (12) and Algorithm 1 can be used. This condition is always satisfied for models in the domains required for the applications of exact algorithms EA1, EA2, and EA3 defined in [6].

When (10) or (11) holds, it can be nonetheless of practical interest to choose the bounds \(\hat {\sigma }^{k,i}_{+}\), 1≤i≤N, corresponding to (12). Indeed, this might increase significantly the acceptance rate of the algorithm, and therefore reduce the number of draws of the random variable ζ _k , which has a much higher cost than the computation of \(\rho _{\Delta _{k}}\), as it requires simulations of Brownian bridges. Moreover, this option allows to avoid numerical optimization if no analytical expression of \(\hat {\sigma }_{+}^{k}\) is available. In practice, this seems more efficient in terms of computational time when N has moderate values.

Assumption H2 depends on the algorithm used to estimate the transition densities and on the tuning parameters of the SMC filter. The most common choice is 𝜗 _k =1 so that under H1, the only requirement is to control \(\widehat {q}_{k-1}\) and p_k−1. For instance, in the case of the GPE-1, as explained in Section 3, H2 is satisfied if ϕ is upper bounded (as for the EA1).