Policy inference plays an essential role in the contextual bandit problem. In this paper, we use empirical likelihood to develop a Bayesian inference method for the joint analysis of multiple contextual bandit policies in finite sample regimes. The proposed inference method is robust to small sample sizes and is able to provide accurate uncertainty measurements for policy value evaluation. In addition, it allows for flexible inferences on policy comparison with full uncertainty quantification. We demonstrate the effectiveness of the proposed inference method using Monte Carlo simulations and its application to an adolescent body mass index data set.

We propose an estimator and confidence interval for computing the value of a policy from off-policy data in the contextual bandit setting. To this end we apply empirical likelihood techniques to formulate our estimator and confidence interval as simple convex optimization problems. Using the lower bound of our confidence interval, we then propose an off-policy policy optimization algorithm that searches for policies with large reward lower bound. We empirically find that both our estimator and confidence interval improve over previous proposals in finite sample regimes. Finally, the policy optimization algorithm we propose outperforms a strong baseline system for learning from off-policy data.

We develop an empirical likelihood (EL) framework for random forests and related ensemble methods, providing a likelihood-based approach to quantify their statistical uncertainty. Exploiting the incomplete $U$-statistic structure inherent in ensemble predictions, we construct an EL statistic that is asymptotically chi-squared when subsampling induced by incompleteness is not overly sparse. Under sparser subsampling regimes, the EL statistic tends to over-cover due to loss of pivotality; we therefore propose a modified EL that restores pivotality through a simple adjustment. Our method retains key properties of EL while remaining computationally efficient. Theory for honest random forests and simulations demonstrate that modified EL achieves accurate coverage and practical reliability relative to existing inference methods.

We propose a doubly robust estimator for the average treatment effect in high dimensional low sample size observational studies, where contamination and model misspecification pose serious inferential challenges. The estimator combines bounded influence estimating equations for outcome modeling with covariate balancing propensity scores for treatment assignment, embedded within a penalized empirical likelihood framework using nonconvex regularization. It satisfies the oracle property by jointly achieving consistency under partial model correct ness, selection consistency, robustness to contamination, and asymptotic normality. For uncertainty quantification, we derive a finite sample confidence interval using cumulant generating functions and influence function corrections, avoiding reliance on asymptotic approximations. Simulation studies and applications to gene expression datasets (Golub and Khan) demonstrate superior performance in bias, error metrics, and interval calibration, highlighting the method robustness and inferential validity in HDLSS regimes. One notable aspect is that even in the absence of contamination, the proposed estimator and its confidence interval remain efficient compared to those of competing models.

How can we find a tight confidence interval (CI) on counterfactual estimates?

How can we find a tight confidence interval (CI) on counterfactual estimates?

The proof is an adaptation from the standard LP for state-value functions to the case of Q -LP ( De Farias and V an Roy, 2003). The effect of full-rank basis embedding in the example in Section 3.1 can be justified straightforwardly. The algorithm can be generalized to undiscounted MDPs with =1 and finite-horizon MDPs. A similar argument of Section 3.3 for discounted MDPs can be applied to MDPs are strictly more general than multi-armed and contextual bandits. Karampatziakis et al. ( 2019) considers The estimator in Karampatziakis et al. ( 2019) is derived from empirical likelihood with reverse Computationally, the estimator in Karampatziakis et al. ( 2019) requires an extra statistics, i.e., ( max Unfortunately the reverse KL-divergence does not satisfy the conditions in Assumption 1 .

In this study, we introduce a novel methodological framework called Bayesian Penalized Empirical Likelihood (BPEL), designed to address the computational challenges inherent in empirical likelihood (EL) approaches. Our approach has two primary objectives: (i) to enhance the inherent flexibility of EL in accommodating diverse model conditions, and (ii) to facilitate the use of well-established Markov Chain Monte Carlo (MCMC) sampling schemes as a convenient alternative to the complex optimization typically required for statistical inference using EL. To achieve the first objective, we propose a penalized approach that regularizes the Lagrange multipliers, significantly reducing the dimensionality of the problem while accommodating a comprehensive set of model conditions. For the second objective, our study designs and thoroughly investigates two popular sampling schemes within the BPEL context. We demonstrate that the BPEL framework is highly flexible and efficient, enhancing the adaptability and practicality of EL methods. Our study highlights the practical advantages of using sampling techniques over traditional optimization methods for EL problems, showing rapid convergence to the global optima of posterior distributions and ensuring the effective resolution of complex statistical inference challenges.