Bayesian Inference & Inventory Optimization

• Bayesian inference with inventory optimization is a framework that integrates probabilistic demand learning and risk-aware dynamic policies to manage inventory uncertainty.
• It leverages methods such as Poisson-Gamma empirical Bayes, Bayesian dynamic programming, and simulation-based Bayesian optimization across varied inventory settings.
• This approach improves decision-making by reducing cost variance, scaling to high-dimensional systems, and effectively addressing both stationary and nonstationary demand.

Bayesian inference with inventory optimization refers to the class of methodologies that integrate Bayesian learning of uncertain demand and supply parameters with the dynamic or static optimization of inventory policies. This intersection provides a principled probabilistic treatment of parameter uncertainty, allowing for data-driven, risk-aware, and sample-efficient inventory control. The literature spans Poisson-Gamma empirical Bayes models for high-dimensional item pools, finite-horizon Markov decision processes with Bayesian risk functionals, episodic Bayesian optimal control, system-dynamic Monte Carlo simulations with Bayesian optimization, and scalable proxies using Bayesian neural networks. The approaches address single-period (newsvendor), multi-period, and continuous-review inventory settings, both under fully and partially observed, stationary and non-stationary demand.

1. Bayesian Inference in Classical Inventory Models

Inventory systems often assume unknown parameters such as item demand rates or demand curves. Bayesian inference utilizes observed data to construct a posterior distribution over these parameters, allowing decision-makers to update beliefs as more data accumulates.

In Poisson demand models, item-level demands $\{X_i\}$ are aggregations of underlying Poisson processes with unknown rates $\{\theta_i\}$. An empirical Bayes framework posits a hierarchical prior, typically $\theta_i \sim \mathrm{Gamma}(\alpha, \beta)$. Hyperparameters $(\alpha, \beta)$ are estimated via marginal maximum likelihood or the method of moments, exploiting the closed-form negative binomial marginal distribution for observed counts. For each item, the posterior $$\theta_i | X_i \sim \mathrm{Gamma}(\hat{\alpha} + X_i, \hat{\beta} + T)$$ yields the posterior predictive demand, which informs order quantities through newsvendor or $(s, S)$ policies. This empirical Bayes shrinkage effect is particularly robust when item demand is sparse and effectively reduces variance in outlier cases, outperforming plug-in point estimators (Anderson et al., 5 Aug 2025).

The key principles are:

• The posterior mean $$\mathbb{E}[\theta_i|X_i]=( \hat{\alpha} + X_i ) / ( \hat{\beta} + T )$$ balances sample information with population-level shrinkage.
• Full posterior predictive distributions enable calibrated multi-period policies and credible intervals for demand.
• Groupings among items, tested via likelihood-ratio statistics, may be formed to account for heterogeneity, but over-partitioning reduces effective sample size and is generally not beneficial absent strong covariate effects.

2. Bayesian Risk and Dynamic Programming Formulations

Extending to multi-period and dynamic settings, Bayesian inference allows updating of demand models as new observations are sequentially incorporated. In Bayesian Risk Markov Decision Processes (BR-MDP), discrete posteriors over parameters (e.g., Poisson rates on a grid) evolve according to Bayes' rule. The policy optimization objective becomes a nested risk functional—most notably, the conditional value-at-risk (CVaR)—applied recursively at each decision epoch (Lin et al., 2021).

The BR-MDP dynamic programming equation for finite time horizon $T$ is: $$V_t(s_t, \mu_t) = \min_{a_t} \rho_{\mu_t} \Big[ \mathbb{E}_{\xi\sim f(\cdot;\theta)}\left[ \mathcal{C}_t(s_t,a_t,\xi) + V_{t+1}(s_{t+1},\mu_{t+1}) \right]\Big]$$ where $\mu_t$ is the posterior over $\theta$, and $\rho_{\mu_t}$ typically takes the form of CVaR under $\mu_t$. This provides time-consistent, risk-aware policies incorporating parameter uncertainty and learning.

Approximate methods employ “$\alpha$-function” convex approximations and backward induction with gradient descent on auxiliary variables, scaling to larger systems with negligible performance loss compared to exact dynamic programming. Notably, empirical studies show that with moderate $\alpha$, BR-MDPs outperform both nominal plug-in and distributionally robust optimizations in terms of mean cost and cost variance, especially under limited data (Lin et al., 2021).

3. Bayesian Optimization for Inventory Policy Learning

Bayesian optimization (BO), classically developed for global optimization in black-box settings, has been adapted to the inventory domain for both single and multi-product systems, particularly when closed-form models of objective functions are unavailable or prohibitively expensive to evaluate. Typically, BO is coupled with Monte Carlo system-dynamic simulation of inventory flows, using a Gaussian process (GP) surrogate to model expected profit or cost as a function of order policy vectors $Q$ (Maitra, 15 Feb 2024).

The workflow involves:

• Simulation of stochastic inventory processes for candidate policies,
• GP regression to estimate a posterior mean and variance of the profit function,
• Acquisition functions such as Expected Improvement (EI) to sequentially select new candidate policies balancing exploitation and exploration,
• Iterative improvement yielding policies with higher expected profit and approximately optimal service-level constraint satisfaction.

Empirically, BayesOpt achieves 3–5$\times$ improvements in expected profit over brute-force Monte Carlo scans and scales gracefully to higher-dimensional settings. Sensitivity analyses demonstrate that optimal policies identified by BO are robust to moderate perturbations in demand mean and variance (Maitra, 15 Feb 2024).

4. Bayesian Learning in Dynamic and Partially Observed Regimes

Inventory systems with nonstationary or partially observed demand are natural candidates for Bayesian filtering and learning. In Markov-modulated settings, system demand is driven by an unobserved Markov chain $M_t$ which must be inferred online via a posterior $\pi_t$ over regimes. The sufficient statistic $(\pi_t, P_t)$ evolves via explicit stochastic differential equations (SDEs) or jump-diffusion processes, with belief updates at censored demand observations or stock-outs (Bayraktar et al., 2012).

The optimal replenishment policy in this framework is typically $(s(\pi), S(\pi))$-type, parameterized by the inferred regime posterior. Comprehensive numerical illustrations confirm that accounting for partial observability and censoring can materially alter inventory policies and values compared to naively assuming full observability. The explicit Bayesian filtering equations enable calculation of belief-dependent reorder and restocking points (Bayraktar et al., 2012).

In episodic Bayesian optimal control, uncertainty in the demand distribution is managed via sequential policy updates after new demand data are observed at the end of each episode. Each episode solves a “Bayesian average” optimal control problem, integrating over the current posterior on demand parameters, and so policies and value functions converge at an $O(N^{-1/2})$ rate to the true-optimal ones under assumptions of model correctness (Shapiro et al., 2023). Stochastic Dual Dynamic Programming (SDDP), when combined with Bayesian updating and warm-starting of affine cuts, accelerates convergence in high-dimensional, convex-linear inventory settings.

5. Bayesian Optimization and Deep Learning Proxies under Constraints

When the objective is to solve complex, possibly non-convex, constrained inventory optimization problems with limited labeled data and computation, Bayesian neural network (BNN) proxies serve as a flexible approach. In the proposed framework, a BNN is trained via stochastic variational inference to minimize both supervised loss on labeled (input, optimal-policy) pairs and unsupervised loss enforcing feasibility constraints using large banks of unlabeled input vectors (e.g., demand scenarios) (Pareek et al., 4 Oct 2024). Training alternates (“sandwiched”) between these two objectives.

Post-training, predictive mean and variance over decision variables are computed from the approximate posterior. Probabilistic high-confidence bounds on maximum equality and inequality constraint violations (MEG, MIG) are constructed using empirical Bernstein inequalities parameterized by predictive variances (MPV) from the BNN.

Experimentally, “sandwich” BNNs demonstrate significant reduction in cost gap and constraint violations compared to standard DNN baselines, despite limited labeled data. Posterior sampling and selection methods further halve the maximum constraint violations at no additional solver overhead. This suggests that semi-supervised BNN proxies, with uncertainty quantification, are practical tools for constraint-aware inventory optimization in data-constrained regimes (Pareek et al., 4 Oct 2024).

6. Summary and Implementation Considerations

The integration of Bayesian inference with inventory optimization encompasses a broad methodological and modeling spectrum:

Methodology	Demand Model	Policy Structure
Empirical Bayes (Poisson)	Hierarchical Poisson	Newsvendor, (s, S)
BR-MDP w/ CVaR	Discrete parametric	Nested dynamic programming
BO (Monte Carlo GP)	Black-box stochastic	Policy vector search
Episodic Bayes Control	Parametric, unknown	Episode-wise Bayesian DP
BNN Proxy (“Sandwich”)	Arbitrary	DNN surrogate, constraint-aware

• Model specification must be compatible with closed-form posterior updates or permit efficient sampling for scalable BO or deep Bayesian surrogates.
• Overfitting hierarchical or flexible priors when sample sizes per group are small degrades inventory performance; judicious grouping or regularization is necessary.
• Bayesian methods naturally propagate uncertainty into both point policies and credible intervals, crucial for risk-averse or constrained applications.
• Computational strategies include convex analytic approximations for dynamic programming, gradient-based optimization for variational inference, and SDDP for convex-linear dynamics.

Bayesian inference with inventory optimization provides a theoretically grounded, empirically validated framework for decision-making under demand uncertainty, encompassing single and multi-item, stationary and nonstationary, data-rich and data-sparse regimes across the full range of operational research and supply chain contexts.