Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem

Technical Summary: Quasi-Bayes Empirical Bayes for the Poisson Compound Decision Problem

1. Problem Statement
The paper addresses the Poisson compound decision problem, a classical statistical challenge involving the estimation of a sequence of means $(\theta_1, \dots, \theta_n)$ for independent Poisson random variables $Y_1, \dots, Y_n$ under squared-error loss. While Empirical Bayes (EB) methods are well-established for this problem in static or batch settings (Robbins, 1956; Zhang, 2003), the authors identify a gap in the literature regarding sequential (streaming) data. Existing EB approaches, particularly those relying on $g$-modeling (estimating the prior distribution $G$), typically require batch optimization or full posterior recomputation, rendering them computationally inefficient for online inference where data arrive sequentially and timely updates are required.

2. Methodology: Sequential Quasi-Bayes Approach
The authors propose a sequential $g$-modeling framework termed Quasi-Bayes Empirical Bayes (QB). The core methodology relies on a recursive learning model that updates the estimate of the mixing distribution $G$ as new observations arrive, avoiding the need for batch re-optimization.

• Recursive Update (Newton's Algorithm): Starting from an initial guess $G_0$, the mixing distribution $G_n$ is updated upon observing $Y_{n+1}$ using a recursive procedure based on Smith and Makov (1978) and Newton et al. (1998):
  $$G_{n+1}(d\theta) = (1 - \alpha_{n+1})G_n(d\theta) + \alpha_{n+1} \frac{\text{Poisson}(Y_{n+1} | \theta)G_n(d\theta)}{\int_\Theta \text{Poisson}(Y_{n+1} | \theta)G_n(d\theta)}$$
  Here, $\alpha_n$ is a learning rate sequence satisfying $\sum \alpha_n = \infty$ and $\sum \alpha_n^2 < \infty$. This update represents a weighted average of the current estimate and the posterior distribution based on the new observation.

• Quasi-Bayesian Interpretation: The model is interpreted as a learning model rather than a data-generating process. It is shown to be "quasi-Bayesian," meaning the sequence of distributions $(G_n)$ converges weakly almost surely to a random limit $\tilde{G}$, and the predictive distribution is asymptotically equivalent to a Bayesian model with a random prior.

• Estimator Construction: The QB estimate for $\theta_i$ given $Y_i=y$ is obtained by plugging the current estimate $G_n$ into the standard Bayes estimator formula:
  $$\hat{\theta}_{G_n}(y) = (y+1) \frac{p_{G_n}(y+1)}{p_{G_n}(y)}$$
  where $p_{G_n}$ is the marginal probability mass function derived from $G_n$.

• Uncertainty Quantification: The authors derive a conditional Gaussian central limit theorem for the estimator. This allows for the construction of asymptotic credible intervals by approximating the variability induced by the substitution of the unknown limit $\tilde{G}$ with the finite-sample estimate $G_n$.

3. Key Contributions and Theoretical Guarantees
The paper establishes several frequentist guarantees for the proposed estimator under the assumption that observations are i.i.d. from a Poisson mixture with a "true" oracle prior $G^*$:

• Consistency and Asymptotic Optimality: Theorem 2.4 demonstrates that the estimator $\hat{\theta}_{G_n}(y)$ converges almost surely to the Bayes estimator corresponding to the Kullback-Leibler projection of $G^*$ onto the class of distributions absolutely continuous with respect to $G_0$. Consequently, the regret (excess Bayes risk) vanishes as $n \to \infty$.
• Computational Efficiency: Unlike batch methods (e.g., Nonparametric Maximum Likelihood or Minimum Distance estimators) that recompute estimates from scratch, the QB method updates the estimate using only the new observation. This results in a constant per-observation computational cost that does not increase with the sample size $n$.
• Asymptotic Normality: Theorem 2.2 provides a conditional central limit theorem, enabling the construction of asymptotic credible intervals that account for both parameter uncertainty and the uncertainty in estimating the prior.

4. Empirical Results
The methodology is evaluated through synthetic and real-data experiments:

• Synthetic Data: The QB estimator is compared against Robbins' method ($f$-modeling), Nonparametric Maximum Likelihood (ML), and Minimum Hellinger Distance (MHD) estimators.

  • Performance: QB achieves performance comparable to ML and MHD, significantly outperforming Robbins' method, which is noted for numerical instability.
  • Efficiency: In terms of CPU time, QB is substantially faster. For $n=400$, QB requires approximately 0.0019 seconds per update, whereas ML and MHD require 1.84 and 2.14 seconds, respectively, to recompute the estimate.
  • Robustness: Sensitivity analyses indicate that QB performance is robust to the choice of the number of quadrature points used for numerical integration.

• Real Data:

  • Twitter Dataset: Applied to retweet counts to estimate instantaneous virality. The QB estimates provide a sequential alternative to batch methods for detecting automated activity or early virality.
  • Accident Dataset: Applied to the benchmark automobile insurance claims data (Efron and Hastie, 2021), demonstrating applicability to standard compound decision problems.

5. Significance and Claims
The paper claims to offer the first sequential methodology for the Poisson compound decision problem that provides both theoretical guarantees (consistency, vanishing regret, asymptotic normality) and convincing empirical support.

The primary significance lies in bridging the gap between Empirical Bayes and sequential inference. By leveraging a quasi-Bayesian learning model based on Newton's algorithm, the authors provide a framework that:

1. Enables Online Inference: Allows for real-time estimation in streaming data environments (e.g., social media, biosurveillance) where batch processing is infeasible.
2. Maintains Theoretical Rigor: Retains the strong frequentist optimality properties (vanishing regret) associated with optimal EB methods.
3. Offers Computational Scalability: Achieves constant computational cost per observation, making it scalable to large, high-frequency data streams where traditional $g$-modeling approaches fail due to computational bottlenecks.

The authors conclude that while $g$-modeling is generally more accurate than $f$-modeling but computationally expensive, their quasi-Bayes approach successfully mitigates the computational cost while preserving accuracy, making it a viable solution for modern streaming data applications.