Efficient Uncertainty Propagation in Bayesian Two-Step Procedures

TL;DR

This work tackles efficient uncertainty propagation in Bayesian two-step models by combining Pareto Smoothed Importance Sampling (PSIS) with Importance Weighted Moment Matching (IWMM) to approximate second-step posteriors across many first-step draws $\tau$, avoiding a full MCMC run for every draw. The authors develop an iterative algorithm that uses representative draws and optional mixture proposals to construct accurate posterior approximations while performing only a minority of expensive MCMC fits; they demonstrate large-scale computational gains without sacrificing accuracy. The method is validated in missing-data settings (via MICE imputations) and surrogate-model scenarios, including a real-world NYC taxi case study, showing substantial reductions in log-likelihood and gradient evaluations and close agreement with gold-standard MCMC via diagnostics like PSIS $\hat{k}$ and MMD tests. Overall, the approach provides a scalable, diagnostics-guided framework for uncertainty propagation in Bayesian two-step workflows with broad applicability to modular inference and cut-model contexts.

Abstract

Bayesian inference provides a principled framework for probabilistic reasoning. If inference is performed in two steps, uncertainty propagation plays a crucial role in accounting for all sources of uncertainty and variability. This becomes particularly important when both aleatoric uncertainty, caused by data variability, and epistemic uncertainty, arising from incomplete knowledge or missing data, are present. Examples include surrogate models and missing data problems. In surrogate modeling, the surrogate is used as a simplified approximation of a resource-heavy and costly simulation. The uncertainty from the surrogate-fitting process can be propagated using a two-step procedure. For modeling with missing data, methods like Multivariate Imputation by Chained Equations (MICE) generate multiple datasets to account for imputation uncertainty. These approaches, however, are computationally expensive, as multiple models must be fitted separately to surrogate parameters respectively imputed datasets. To address these challenges, we propose an efficient two-step approach that reduces computational overhead while maintaining accuracy. By selecting a representative subset of draws or imputations, we construct a mixture distribution to approximate the desired posteriors using Pareto smoothed importance sampling. For more complex scenarios, this is further refined with importance weighted moment matching and an iterative procedure that broadens the mixture distribution to better capture diverse posterior distributions.

Figures (28)

• Figure 1: Overview of problem setting: The goal is to propagate the uncertainty induced by multiple draws of $\tau$ into the posterior space. (a) Standard approach: Run costly simulation, i.e. MCMC, for each realization of $\tau$. (b) Proposed method in this paper: Only a few costly simulations, i.e. MCMC, are required; the remaining posterior distributions are approximated using importance sampling.
• Figure 2: Exemplary posterior distributions for one model parameter in the special case of the missing data problem (cf. Section \ref{['sec:special_case:MI']}) from the simulation study in Section \ref{['sec:MI_sim_study']} (dataset with $n=100$ observations, $p=10$ predictors and 15% of rows containing missing values; imputed using MICE). The proposal and target distribution correspond to each one imputed dataset (each approximated with HMC). PSIS alone fails with a $\hat{k} = 0.93 > k_{\rm threshold}$, which implies an unsuccessful approximation of the target distribution as is clearly visible in the figure. In contrast, the combination of PSIS and IWMM is appropriately approximating the target distribution, as indicated by $\hat{k}_{\rm MM} = 0.27 < k_{\rm threshold}$.
• Figure 3: Iterative Algorithm (cf. Algorithm \ref{['algo:method']}) for the special case of the missing data problem. Multiple datasets are generated by the MICE algorithm, representing the uncertainty induced by the missing values in the original dataset. A representative dataset is selected and the posterior draws accessed through MCMC. This posterior is used to approximate the posteriors of the resulting imputed datasets with PSIS and IWMM, until all posteriors can be approximated using resampling and the Monte Carlo estimate can be calculated.
• Figure 4: Posterior distributions $p(\theta_I \mid \tau^{(i)}, D_I)$ for $i=1,\dots,100$ from the logistic-surrogate simulation study in Section \ref{['sec:surrogates_sim_study']}. The $N_\mathrm{mix}=10$ distributions highlighted in red were selected by the proposed method based on log-likelihoods evaluated over prior samples.
• Figure 5: Number of log-probability (blue) and gradient log-probability evaluations (red) required to approximate the posteriors of $m = 100$ imputed datasets ($N = 100$, $p = 10$) with 15% of the rows containing missing values in the original data. Representative datasets were selected using the Medoids method. Error bars indicate the standard deviation across repeated runs of the same setup.