Reinforcement learning in mobile health (mHealth) interventions requires balancing intervention efficacy with user burden, particularly when state measurements (for example, user surveys or feedback) are costly yet essential. The Act-Then-Measure (ATM) heuristic addresses this challenge by decoupling control and measurement actions within the Action-Contingent Noiselessly Observable Markov Decision Process (ACNO-MDP) framework. However, the standard ATM algorithm relies on a temporal-difference-inspired Q-learning method, which is prone to instability in sparse and noisy environments. In this work, we propose a Bayesian extension to ATM that replaces standard Q-learning with a Kalman filter-style Bayesian update, maintaining uncertainty-aware estimates of Q-values and enabling more stable and sample-efficient learning. We evaluate our method in both toy environments and clinically motivated testbeds. In small, tabular environments, Bayesian ATM achieves comparable or improved scalarized returns with substantially lower variance and more stable policy behavior. In contrast, in larger and more complex mHealth settings, both the standard and Bayesian ATM variants perform poorly, suggesting a mismatch between ATM's modeling assumptions and the structural challenges of real-world mHealth domains. These findings highlight the value of uncertainty-aware methods in low-data settings while underscoring the need for new RL algorithms that explicitly model causal structure, continuous states, and delayed feedback under observation cost constraints.

While off-policy temporal difference methods have been broadly used in reinforcement learning due to their efficiency and simple implementation, their Bayesian counterparts have been relatively understudied. This is mainly because the max operator in the Bellman optimality equation brings non-linearity and inconsistent distributions over value function. In this paper, we introduce a new Bayesian approach to off-policy TD methods using Assumed Density Filtering, called ADFQ, which updates beliefs on action-values (Q) through an online Bayesian inference method. Uncertainty measures in the beliefs not only are used in exploration but they provide a natural regularization in the belief updates. We also present a connection between ADFQ and Q-learning. Our empirical results show the proposed ADFQ algorithms outperform comparing algorithms in several task domains. Moreover, our algorithms improve general drawbacks in BRL such as efficiency, usage of uncertainty, and nonlinearity.