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Table 2 Summary of the main notation used throughout the paper

From: A survey of Monte Carlo methods for parameter estimation

Notation Description
Dy Dimension of the data.
L Number of data available.
\(\mathbf {y} \in \mathbb {R}^{LD_{\mathrm {y}}}\) LDy-dimensional observations vector, y=vec{y1,…,yL} with \(\mathbf {y}_{i} \in \mathbb {R}^{D_{\mathrm {y}}}\) for i=1,…,L.
Dθ Dimension of the parameter space.
\(\Theta = \Theta _{1} \times \cdots \times \Theta _{D_{\theta }}\phantom {\dot {i}\!}\) Feature space for the parameter vector θ.
\(\boldsymbol {\theta } \in \mathbb {R}^{D_{\theta }}\) Dθ-dimensional parameter vector, \(\phantom {\dot {i}\!}\boldsymbol {\theta } = [\theta _{1}, \ldots, \theta _{D_{\theta }}]\) with θdΘd for d=1,…,Dθ.
θ(m) mth sample of the parameter vector in MC and RS.
θ(t) Sample of the parameter vector at the tth iteration in MCMC methods.
\(\bar {\pi }(\boldsymbol {\theta }|\mathbf {y}) \equiv \bar {\pi }(\boldsymbol {\theta })\) Target (i.e., posterior) PDF.
π(θ|y)≡π(θ) Target function (i.e., non-negative but unnormalized).
p0(θ) Prior probability density function.
(y|θ) Likelihood.
Z(y) Normalizing constant of the target (a.k.a. partition function, marginal likelihood, or model evidence).
\(\bar {\pi }(\theta _{d}|\boldsymbol {\theta }_{\neg d})\) Full conditional PDF for the dth parameter given all the other parameters (used in the Gibbs sampler).
T Number of Monte Carlo iterations performed.
Tb Number of iterations for the burn-in period in MCMC.
N Number of proposals used in multiple IS approaches.
M Number of samples drawn in the MC algorithm, RS and IS approaches. Usually MN in MIS.
\(\bar {q}(\boldsymbol {\theta })\), \(\bar {q}_{t}(\boldsymbol {\theta })\), \(\bar {q}_{m,t}(\boldsymbol {\theta })\) Proposal PDF.
q(θ), qt(θ), qm,t(θ) Proposal function (i.e., non-negative but unnormalized) for t=1,…,T and m=1,…,M.
wm,t(θ) Unnormalized weight of the mth particle (m=1,…,M) at the tth iteration (t=1,…,T) for AIS approaches.
wm,t(θ) Normalized weight of the mth particle (m=1,…,M) at the tth iteration (t=1,…,T) for AIS approaches.
\(\hat {\pi }(\boldsymbol {\theta })\) Random measure used to approximate the target at the tth iteration.
\(\boldsymbol {\mathcal {N}}(\mu,\mathbf {C})\), \(\boldsymbol {\mathcal {N}}(\cdot |\mu,\mathbf {C})\) Gaussian PDF with mean μ and covariance C.
\(\boldsymbol {\mathcal {U}}(\mathcal {I})\) Uniform PDF within the interval \(\mathcal {I}\).