Table 2 Summary of the main notation used throughout the paper

D_y

Dimension of the data.

L

Number of data available.

\(\mathbf {y} \in \mathbb {R}^{LD_{\mathrm {y}}}\)

LD_y-dimensional observations vector, y=vec{y₁,…,y_L} with \(\mathbf {y}_{i} \in \mathbb {R}^{D_{\mathrm {y}}}\) for i=1,…,L.

D_θ

Dimension of the parameter space.

\(\Theta = \Theta _{1} \times \cdots \times \Theta _{D_{\theta }}\phantom {\dot {i}\!}\)

Feature space for the parameter vector θ.

\(\boldsymbol {\theta } \in \mathbb {R}^{D_{\theta }}\)

D_θ-dimensional parameter vector, \(\phantom {\dot {i}\!}\boldsymbol {\theta } = [\theta _{1}, \ldots, \theta _{D_{\theta }}]\) with θ_d∈Θ_d for d=1,…,D_θ.

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.

ℓ(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.

T_b

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 M≥N in MIS.

\(\bar {q}(\boldsymbol {\theta })\), \(\bar {q}_{t}(\boldsymbol {\theta })\), \(\bar {q}_{m,t}(\boldsymbol {\theta })\)

Proposal PDF.

q(θ), q_t(θ), q_m,t(θ)

Proposal function (i.e., non-negative but unnormalized) for t=1,…,T and m=1,…,M.

w_m,t(θ)

Unnormalized weight of the mth particle (m=1,…,M) at the tth iteration (t=1,…,T) for AIS approaches.

w_m,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 \(\mathcal {I}\).