Prior-Inherited Bayesian UCB Selection

• The paper introduces a framework that integrates Bayesian priors with UCB exploration to construct high-probability confidence bounds for belief tree planning.
• It derives explicit, statistically rigorous bounds using Monte Carlo estimates and Hoeffding-type concentration inequalities to quantify uncertainty.
• The methodology extends to multi-armed bandits and reinforcement learning, demonstrating reduced regret and efficient targeted exploration by leveraging prior information.

Prior-Inherited Bayesian UCB Selection encompasses a family of algorithms and planning frameworks that combine Bayesian prior modeling of environmental uncertainty with upper confidence bound (UCB) exploration. These methods exploit prior information—explicitly encoded or adaptively learned—to construct high-probability confidence intervals and guide sequential decision processes via optimism in the face of uncertainty. Incorporating the prior in belief updates and value bounds enables targeted exploration and more efficient exploitation, especially in structured settings such as belief trees, bandits, and reinforcement learning.

1. Bayesian Reinforcement Learning and Belief-Augmented MDPs

In Bayesian reinforcement learning (RL), an agent maintains a belief distribution $\xi$ over the space of Markov decision processes (MDPs), typically representing model uncertainty. Rather than operating with a fixed transition model, the agent starts with a prior and updates it based on observed transitions and rewards. This leads to the concept of the belief-augmented MDP (BAMDP), where the "hyperstate" $\omega = (s, \xi)$ denotes the current true state and the posterior belief over MDPs. Value functions are then defined over these augmented states, with the optimal finite horizon value at leaf node $\omega$ given by

$$V^*(\omega) = \sup_{\pi} \int V_{\mu}^{\pi}(s_\omega) \, \xi_\omega(\mu) \, d\mu,$$

where $V_{\mu}^{\pi}$ is the value of policy $\pi$ under model $\mu$. This representation transforms the planning problem into dynamic programming over the "belief tree”—a planning tree in which each node contains both a state and belief.

2. High-Probability Bounds for Value Function Estimation

The paper develops explicit lower and upper bounds on the value in each hyperstate: $$\int V_{\mu}^{\pi^*(\mu)}(s_\omega) \, \xi_\omega(\mu) \, d\mu \geq V^*(\omega) \geq \int V_{\mu}^{\pi^*(\bar{\mu}_\omega)}(s_\omega) \, \xi_\omega(\mu) \, d\mu,$$ where $\pi^*(\mu)$ is the optimal policy for a particular $\mu$, and $\pi^*(\bar{\mu}_\omega)$ is optimal for the mean MDP $\bar{\mu}_\omega$. This tightens classic POMDP-style bounds by directly accounting for the variability induced by the prior and the sampled models.

For practical computation, Monte Carlo estimates approximate the expectations. Hoeffding-type concentration bounds quantify the probability that the estimated upper bound $\hat{v}^*(\omega)$ deviates from the true mean,

$$P\left(|\hat{v}^*(\omega) - \bar{v}^*(\omega)| > \epsilon\right) < 2 \exp\left(-\frac{2c\epsilon^2}{(V_{\max} - V_{\min})^2}\right),$$

enabling rigorous management of uncertainty during tree expansion.

3. Belief Tree Exploration Strategies

Since full expansion of the infinite belief tree is infeasible, multiple expansion heuristics are employed to select which node to develop at each stage. Strategies include:

• Serial expansion: balanced, pre-determined order
• Random expansion: uniform random selection of leaves
• Highest lower bound: expand node with highest discounted lower value, $$U(\omega) = \gamma^{\text{depth}} v_{\text{lower}}(\omega)$$
• Thompson-style: expand according to sampled upper bounds
• High-probability upper bound: expand node maximizing $$\gamma^{\text{depth}} \max\{\hat{v}^*_c(\omega), \bar{v}(\omega)\}$$

These methods balance near-term reward potential and epistemic uncertainty, leveraging both the prior and the statistical properties of sampled transitions. Algorithms that employ high-probability upper bounds or stochastic selection via sampling generally outperform serial and random approaches, achieving reduced computational cost and improved regret.

4. Comparison to Distribution-Free UCB1 Methods

The Bayesian expansion strategies are contrasted with UCB1, a classic frequentist approach for multi-armed bandit (MAB) problems. UCB1 computes an upper confidence bound for each arm, based solely on sample statistics and concentration inequalities, without use of priors. In contrast, the belief-tree Bayesian methods inherit prior information via $\xi$ and use Monte Carlo sampling to estimate bounds, thus adapting exploration to both prior knowledge and observed data.

While UCB1 is efficient and nearly optimal in MAB, the paper's experiments demonstrate that prior-inherited Bayesian exploration strategies—particularly those exploiting stochastic lookahead or upper-bound selection—achieve lower regret and more targeted sampling when prior structure is present or when the state space is large or augmented.

5. Incorporation and Utility of Prior Information

The algorithms developed in the paper naturally facilitate "prior-inherited" UCB selection: rather than generic confidence intervals, bounds are computed via the posterior that incorporates both prior and current evidence. This inherited information enables adaptation of the exploration bonus:

• Regions of the tree where the prior is tight—and where uncertainty is low—can be exploited more aggressively.
• Conversely, areas where the prior is diffuse or the model class is broad receive proportionally higher exploration scores.

Specifically, upper bounds combine belief-conditioned Monte Carlo samples with concentration inequalities, ensuring that the exploration bonus reflects both empirical variance and prior compatibility with observed data.

6. Extension to Multi-Armed Bandits and Generalized Settings

The paper extends the developed tree expansion methods to standard bandit problems, demonstrating that prior-inherited Bayesian UCB selection techniques match or exceed the performance of UCB1 even in classical MAB regimes. Where prior information about the reward structure is available (such as likelihood of arm optimality, correlation structure, or parameter constraints), these Bayesian approaches are particularly efficient.

The methodology generalizes to belief-augmented MDPs, POMDPs, contextual bandits, and combinatorial decision settings, and is applicable wherever prior modeling and dynamic belief updating are computationally feasible.

7. Implications for Theory and Practice

Prior-Inherited Bayesian UCB Selection synthesizes Bayesian reinforcement learning, belief tree planning, and UCB-style optimism. By integrating explicit prior modeling with dynamic, high-probability bounds on state-action values, these methods enable effective targeted exploration, especially in settings with high environmental structure or prior-derived regularization.

This approach also suggests a design principle for computationally adaptive exploration: favoring development of paths that may be suboptimal under current estimates but remain plausible under the prior and posterior uncertainty—thus reducing both sample complexity and worst-case regret when prior information is reliable.

The conceptual framework presented in the paper underpins a class of planning and decision algorithms that combine Bayesian updating, principled uncertainty quantification, and upper bound-based selection. The approach holds particular promise for scalable RL, structured bandits, and belief-model planning in domains where prior knowledge can be fruitfully exploited.