Robust Surrogate-Based Bayesian Inference via Sampling-Based Adaptive Active Learning (SALE)

Technical Summary: Robust Surrogate-Based Bayesian Inference via Sampling-Based Adaptive Active Learning (SALE)

Problem Statement
Standard Bayesian inference methods, such as Markov chain Monte Carlo (MCMC), become computationally intractable when likelihood or log-posterior evaluations are expensive. This expense arises in scenarios involving simulators, parameter-dependent normalising constants, repeated integration over latent structures, or numerical solutions to inverse problems. Under limited evaluation budgets, existing surrogate-based approaches face a fundamental tension: they must simultaneously localise regions of high posterior mass and calibrate the surrogate model's accuracy within those regions.

Existing methods often rely exclusively on Bayesian Optimisation (BO) to find high-probability regions or Uncertainty Reduction (UR) to refine the surrogate. However, pure BO strategies often treat the log-posterior as a black-box objective, failing to exploit log-density structures and potentially exploring irrelevant regions. Conversely, posterior-weighted UR strategies can be misled by high uncertainty in the early stages of learning, wasting evaluations in regions with negligible true posterior mass. Furthermore, theoretical guarantees for sequential posterior-weighted UR designs are generally lacking, particularly regarding explicit budget-dependent bounds on the approximation error of the final surrogate posterior.

Methodology: Sampling-Based Adaptive Active Learning (SALE)
The author proposes SALE, a Gaussian Process (GP)-based active learning framework that unifies localisation and calibration through a common sequential design measure: the Expected Posterior (EP), denoted as $\bar{\pi}_t$.

1. The Expected Posterior (EP):
   Instead of using the GP posterior mean ($\mu_t$) as the sole surrogate, SALE utilises the EP, defined as the average of posterior laws induced by GP sample paths: $\bar{\pi}_t(A) = \int \pi(A|g) P_t(dg)$. Theoretically, $\bar{\pi}_t$ minimises the conditional expected Kullback–Leibler (KL) loss among estimators given the current information. Crucially, SALE uses $\bar{\pi}_t$ not just as an inferential output, but as the guiding law for sequential design.

2. State-Dependent Allocation:
   SALE dynamically allocates the evaluation budget between two branches based on a latent state $S_t$:

   • Localisation (BO): When the design resolution around the current maximiser is low, the algorithm prioritises BO to locate high-posterior regions.
   • Calibration (UR): As resolution improves, the algorithm shifts toward UR to ensure approximation accuracy in inference-relevant regions.
     A proxy for the probability of unresolved value discovery is used to determine the allocation, ensuring UR receives a positive asymptotic fraction of evaluations.

3. Annealed Objective (AO) for BO:
   For the localisation branch, SALE introduces the Annealed Objective, which averages $\tau$-tempered posterior laws induced by GP sample paths.

   • At $\tau=1$, AO equals the EP ($\bar{\pi}_t$).
   • As $\tau \to 0$, AO approaches Thompson Sampling (TS).
   • Positive $\tau$-tempering regularises the query law against surrogate-path perturbations, providing stability that TS lacks when the surrogate is highly uncertain (e.g., containing narrow, artificial spikes).
   • Sampling from AO is achieved via Exchange Simulated Annealing (ESA), which handles the intractable normalising constants by introducing auxiliary draws.

4. EP-Weighted Calibration (UR):
   For the calibration branch, SALE defines an ideal UR rule that maximises the expected reduction in a variance functional controlling the Total Variation (TV) distance between the EP and the true posterior.

   • The ideal rule is computationally expensive. SALE implements a practical proxy that restricts candidates to the current EP sample, uses a state-dependent tempering of weights (tilting toward high GP means during localisation), and employs a local ranking proxy based on GP uncertainty and posterior mean.

Key Contributions

1. Unified Design Measure: SALE is the first framework to use the Expected Posterior ($\bar{\pi}_t$) as a common sequential design measure for both region restriction (localisation) and calibration.
2. Stability-Optimisation Trade-off: The paper characterises the trade-off of the Annealed Objective (AO). It proves that positive $\tau$-tempering stabilises the query distribution against surrogate perturbations (via a $W_{1,\infty}$-to-TV continuity bound) while quantifying the associated optimisation penalty through Bayesian regret bounds.
3. Theoretical Guarantees for UR: The author derives an explicit budget-dependent bound on the expected TV error of the final EP under an ideal sequential posterior-weighted UR design. They further establish an expected-TV rate for the implemented UR proxy, showing it matches the polynomial order of the ideal rule up to logarithmic factors.
4. Robustness: The framework is designed to avoid the severe failures seen in external baselines by adaptively switching between localisation and calibration.

Results and Empirical Performance
The author evaluates SALE across analytic benchmarks (e.g., Quartic-tail Gaussian, Neal's Funnel, Rosenbrock, Bimodal Mixture), simulated likelihoods (Spatial Matérn GRF, Lotka–Volterra inverse problem), and two real-world applications (Dynamic Discrete-Choice models for bus-engine replacement and Ultra-Diffuse Galaxy detection).

• Robustness: While no single method dominates all problems in terms of median convergence rate, SALE demonstrates superior robustness. It consistently reduces Total Variation (TV) error across all settings and avoids the severe failures (high TV errors) observed in several external baselines (e.g., GP-UCB, log-GP entropy).
• Localisation vs. Calibration: Ablation studies confirm that the combination of $\bar{\pi}_t$-guided restriction, state-dependent allocation, and the Annealed Objective is critical. Removing the localisation phase (UR-Only) or the restriction (No Restriction) leads to significantly higher error and variability.
• Applications:
  • In the Bus-Engine Replacement model, SALE achieved a posterior approximation comparable to a reference MCMC chain (initialized at the SALE mode) using only 520 likelihood evaluations versus 40,000 for the reference, a roughly 77-fold reduction.
  • In Ultra-Diffuse Galaxy (UDG) detection, SALE produced more reliable posterior marginals for nonlinear parameters and a more accurate residual intensity map compared to a GP-UCB surrogate and an iterative linearisation approach (inlabru).

Significance and Claims
The paper claims that SALE provides a principled, robust framework for Bayesian inference when likelihood evaluations are expensive. By leveraging the Expected Posterior as a unifying design law, SALE effectively bridges the gap between finding high-probability regions and calibrating the surrogate within them. The theoretical contributions provide the first explicit budget-dependent bounds for sequential posterior-weighted UR designs, addressing a gap between theory and finite-budget practice. The empirical results suggest that sequential design, when coupled with a principled exploitation of posterior structure (via the EP and AO), can support reliable Bayesian inference where traditional methods fail or require prohibitive computational resources.