Multiple Importance Sampling

• Multiple importance sampling is a Monte Carlo method that integrates samples from various proposal distributions to approximate complex target distributions.
• The sequential MIS algorithm uses softly truncated priors and a balance heuristic to adapt proposals and achieve lower bias and variance in evidence estimation.
• Empirical studies demonstrate that SeMIS improves effective sample sizes and mode mixing, ensuring robust uncertainty quantification in high-dimensional Bayesian inference.

Multiple importance sampling (MIS) is a Monte Carlo methodology in which samples are drawn from several proposal distributions, each potentially targeting different aspects of a complicated target distribution, and these samples are then appropriately weighted and combined to approximate expectations under the target. MIS is widely employed for evidence estimation and efficient Bayesian computation, especially in scenarios characterized by high-dimensionality and multimodality where standard importance sampling is ineffective. The sequential multiple importance sampling (SeMIS) algorithm represents a development within this framework, explicitly addressing high-dimensional Bayesian inference by leveraging adaptive, sequentially constructed proposals and a mode-bridging mechanism through softly truncated priors.

1. Sequential Multiple Importance Sampling Algorithm Structure

SeMIS is designed as an adaptive, iterative procedure. At each stage of the sequence, the proposal distribution is defined as a "softly truncated prior," smoothly interpolating between the prior and posterior through an intermediate parameter $\alpha_i$. Formally, each proposal $q_i(\theta; \alpha_i)$ is constructed as:

$$q_i(\theta; \alpha_i) = \pi(\theta) \cdot L_i(\theta; \alpha_i)$$

where $\pi(\theta)$ is the prior and $$L_i(\theta; \alpha_i) = \min\{L(\theta), \alpha_i L_\text{sup}\}$$, with $L(\theta)$ the likelihood and $L_\text{sup}$ its maximum value. The parameter sequence $\{\alpha_i\}$ is monotonic, with $\alpha_0 = 0$ (pure prior) and $\alpha_{I-1} = 1$ (posterior).

At each stage, $N_i$ samples are drawn from $q_i(\theta; \alpha_i)$ using parallel Markov chain Monte Carlo (MCMC), specifically an elliptical slice sampler which is particularly effective in high dimensions. The proposals are sequenced adaptively: an acceptance rate $p$ between successive proposals is estimated using

$$p = \mathbb{E}_{q_{i-1}}[B_i(\theta)], \quad \text{where}\quad B_i(\theta) = \frac{\min\{L(\theta)/(\alpha_{i-1} L_\text{sup}), 1\}}{\min\{L(\theta)/(\alpha_i L_\text{sup}), 1\}}$$

and the next $\alpha_i$ is chosen by a root-finding procedure to target $p \approx 0.1$. This adaptive schedule allows for more effective mixing, particularly across well-separated posterior modes.

2. Evidence Estimation via MIS and Balance Heuristic

A central objective is the estimation of the Bayesian evidence (marginal likelihood):

$Z = \int \pi(\theta) L(\theta) \, d\theta$

SeMIS utilizes all samples from the sequential proposals, combining them using the balance heuristic—a well-established optimal weighting strategy in MIS. For a total of $I$ proposals with $N_i$ samples each, a weight for each sample $\theta_{i,k} \sim q_i(\theta; \alpha_i)$ is computed:

$$a_i(\theta) = \frac{N_i q_i(\theta; \alpha_i)}{\sum_{j=0}^{I-1} N_j q_j(\theta; \alpha_j)}$$

The full MIS evidence estimator is then

$$\widehat{Z}_\mathrm{MIS} = \sum_{i=0}^{I-1} \frac{1}{N_i} \sum_{k=1}^{N_i} L(\theta_{i,k}) a_i(\theta_{i,k})$$

This approach ensures that all generated samples contribute, with weights reflecting their relevance under all proposals, leading to both lower bias and variance in the evidence estimate relative to classical subset simulation (SuS) and alternative Bayesian updating methods.

3. Posterior Sampling and Effective Sample Size

Posterior samples are drawn using an importance-resampling (or sampling-importance-resampling, SIR) step based on the weighted sample pool. Given the normalized weights $\{a_i(\theta_{i,k}) L(\theta_{i,k})\}$, resampled values $\theta^*_{j}$ are drawn with probabilities proportional to those weights, approximating the true posterior as:

$$p(\theta|D) \approx \sum_{i, k} \omega_{i,k} \delta_{\theta_{i,k}}(\theta), \quad \omega_{i,k} \propto a_i(\theta_{i,k}) L(\theta_{i,k})$$

The effective sample size (ESS),

$\mathrm{ESS} = \frac{1}{\sum_{i,k} \omega_{i,k}^2}$

is used to assess the representational quality of the sample set. Empirical results demonstrate that SeMIS attains a higher ESS compared to SuS and aBUS, especially for challenging (e.g., multimodal) posteriors.

4. Mode-Bridging via Softly Truncated Proposals

A distinctive feature of SeMIS is the use of "softly truncated" priors, as opposed to the hard truncation employed in SuS. At each step, the proposal $$q_i(\theta; \alpha_i) = \pi(\theta) \cdot L_i(\theta; \alpha_i)$$ only gently down-weights areas with likelihood below a fraction of $L_\text{sup}$, rather than excluding them completely. This allows for accepted samples to traverse the parameter space, crossing low-probability zones and moving between isolated modes. As a result, SeMIS exhibits improved mode mixing and ergodicity, which is crucial for accurate uncertainty quantification and posterior exploration in high-dimensional, multimodal settings.

5. Empirical Performance in High-dimensional Bayesian Inference

Benchmark comparisons, including canonical multi-modal problems and real-world high-dimensional finite element model (FEM) updating, demonstrate the strengths of SeMIS. In evidence estimation, SeMIS achieves lower bias and coefficient of variation in $\log$-evidence than SuS and aBUS. For posterior sampling, SeMIS attains higher ESS and obtains empirical distributions closer (in Kolmogorov–Smirnov distance) to the true posterior.

In a 120-degree-of-freedom FE model of a benchmark building, SeMIS is used to localize and quantify damage through inference over stiffness parameters. The algorithm robustly identifies reductions in stiffness corresponding to removed structural elements, and the resulting posterior distributions provide not only point estimates but also precise uncertainty quantification, even under partial observation regimes where the posterior becomes strongly multimodal. The ability of SeMIS to resolve such structure is attributed to its enhanced mode mixing.

6. Theoretical and Algorithmic Implications

SeMIS advances Bayesian computation by providing a general and principled mechanism for bridging between the prior and posterior with smoothly adapting proposals. The balance heuristic enables efficient sample recycling across the sequential steps, maximizing the utility of every generated sample. The softly truncated prior strategy avoids sample impoverishment and improves the exploration of complex posterior topologies. Parallel MCMC sampling further enhances scalability in high dimensions.

7. Future Directions and Open Problems

Several avenues for further development are noted:

• Variance quantification in evidence estimation: Deriving analytical forms for the uncertainty of the MIS-based evidence estimator would enable more rigorous error bounds.
• Improved ESS estimation: A more precise, theoretically grounded expression for the effective sample size in the sequential-MIS context could better characterize sampling efficiency.
• Scalability and real-time monitoring: Applications to real-time structural health monitoring and other large-scale inverse problems may require further algorithmic refinements, such as alternative MCMC kernels or distributed computation strategies.
• Broader applicability: The core methods in SeMIS—sequential adaptive bridging, soft truncation proposals, and full sample recycling under the balance heuristic—are applicable to a range of scientific and engineering problems involving high-dimensional, multimodal, or ill-posed Bayesian inference.

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SeMIS thus represents a robust, theoretically grounded, and practically validated methodology for multiple importance sampling in high-dimensional Bayesian inference, particularly excelling in evidence estimation and complex posterior sampling in multimodal settings (Binbin et al., 7 Jul 2025).