Small formula. A formula is a mathematical expression, combining specific features (Q-functions of different models) by using standard mathematical operators (Roc-A web addition, subtraction, logarithm, etc.). The discrete E/E strategy space is the set of all formulas which can be built by combining at most n features/operators (such a set is denoted by Fn ). OPPS-DS does not come with any guarantee. However, the UCB1 bandit algorithm used to identify the best E/E strategy within the set of strategies provides statistical guarantees that the best E/E strategies are identified with high probability after a certain budget of experiments. However, it is not clear that the best strategy of the E/E strategy space considered yields any high-performance strategy regardless the problem. Tested values: ?S 2 fF2 ?12? F3 ?43? F4 ?226? F5 ?1210? F6 ?7407 , ? 2 50, 500, 1250, 2500, 5000, 10000, 100000, 1000000. 5.1.5 BAMCP. Bayes-adaptive Monte Carlo Planning (BAMCP) [7] is an evolution of the Upper Confidence Tree (UCT) algorithm [11], where each transition is sampled according to the history of observed transitions. The principle of this algorithm is to adapt the UCT principle for planning in a Bayes-adaptive MDP, also called the belief-augmented MDP, which is an MDP obtained when considering augmented states made of the concatenation of the actualPLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,10 /Benchmarking for Bayesian Reinforcement Learningstate and the posterior. The BAMCP algorithm is made computationally tractable by using a sparse sampling strategy, which avoids sampling a model from the posterior distribution at every node of the planification tree. Note that the BAMCP also comes with theoretical guarantees of convergence towards Bayesian optimality. In practice, the BAMCP relies on two parameters: (i) Parameter K which defines the number of nodes created at each time-step, and (ii) Parameter depth which defines the depth of the tree from the root. Tested values: ?K 2 1, 500, 1250, 2500, 5000, 10000, 25000, ?depth 2 15, 25, 50. 5.1.6 BFS3. The Bayesian Forward Relugolix biological activity search Sparse Sampling (BFS3) [6] is a Bayesian RL algorithm whose principle is to apply the principle of the FSSS (Forward Search Sparse Sampling, see [12] algorithm to belief-augmented MDPs. It first samples one model from the posterior, which is then used to sample transitions. The algorithm then relies on lower and upper bounds on the value of each augmented state to prune the search space. The authors also show that BFS3 converges towards Bayes-optimality as the number of samples increases. In practice, the parameters of BFS3 are used to control how much computational power is allowed. The parameter K defines the number of nodes to develop at each time-step, C defines the branching factor of the tree and depth controls its maximal depth. Tested values: ?K 2 1, 500, 1250, 2500, 5000, 10000, ?C 2 2, 5, 10, 15, ?depth 2 15, 25, 50. 5.1.7 SBOSS. The Smarter Best of Sampled Set (SBOSS) [5] is a Bayesian RL algorithm which relies on the assumption that the model is sampled from a Dirichlet distribution. From this assumption, it derives uncertainty bounds on the value of state action pairs. It then uses those bounds to decide how many models to sample from the posterior, and how often the posterior should be updated in order to reduce the computational cost of Bayesian updates. The sampling technique is then used to build a merged MDP, as in [13], and to derive the.Small formula. A formula is a mathematical expression, combining specific features (Q-functions of different models) by using standard mathematical operators (addition, subtraction, logarithm, etc.). The discrete E/E strategy space is the set of all formulas which can be built by combining at most n features/operators (such a set is denoted by Fn ). OPPS-DS does not come with any guarantee. However, the UCB1 bandit algorithm used to identify the best E/E strategy within the set of strategies provides statistical guarantees that the best E/E strategies are identified with high probability after a certain budget of experiments. However, it is not clear that the best strategy of the E/E strategy space considered yields any high-performance strategy regardless the problem. Tested values: ?S 2 fF2 ?12? F3 ?43? F4 ?226? F5 ?1210? F6 ?7407 , ? 2 50, 500, 1250, 2500, 5000, 10000, 100000, 1000000. 5.1.5 BAMCP. Bayes-adaptive Monte Carlo Planning (BAMCP) [7] is an evolution of the Upper Confidence Tree (UCT) algorithm [11], where each transition is sampled according to the history of observed transitions. The principle of this algorithm is to adapt the UCT principle for planning in a Bayes-adaptive MDP, also called the belief-augmented MDP, which is an MDP obtained when considering augmented states made of the concatenation of the actualPLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,10 /Benchmarking for Bayesian Reinforcement Learningstate and the posterior. The BAMCP algorithm is made computationally tractable by using a sparse sampling strategy, which avoids sampling a model from the posterior distribution at every node of the planification tree. Note that the BAMCP also comes with theoretical guarantees of convergence towards Bayesian optimality. In practice, the BAMCP relies on two parameters: (i) Parameter K which defines the number of nodes created at each time-step, and (ii) Parameter depth which defines the depth of the tree from the root. Tested values: ?K 2 1, 500, 1250, 2500, 5000, 10000, 25000, ?depth 2 15, 25, 50. 5.1.6 BFS3. The Bayesian Forward Search Sparse Sampling (BFS3) [6] is a Bayesian RL algorithm whose principle is to apply the principle of the FSSS (Forward Search Sparse Sampling, see [12] algorithm to belief-augmented MDPs. It first samples one model from the posterior, which is then used to sample transitions. The algorithm then relies on lower and upper bounds on the value of each augmented state to prune the search space. The authors also show that BFS3 converges towards Bayes-optimality as the number of samples increases. In practice, the parameters of BFS3 are used to control how much computational power is allowed. The parameter K defines the number of nodes to develop at each time-step, C defines the branching factor of the tree and depth controls its maximal depth. Tested values: ?K 2 1, 500, 1250, 2500, 5000, 10000, ?C 2 2, 5, 10, 15, ?depth 2 15, 25, 50. 5.1.7 SBOSS. The Smarter Best of Sampled Set (SBOSS) [5] is a Bayesian RL algorithm which relies on the assumption that the model is sampled from a Dirichlet distribution. From this assumption, it derives uncertainty bounds on the value of state action pairs. It then uses those bounds to decide how many models to sample from the posterior, and how often the posterior should be updated in order to reduce the computational cost of Bayesian updates. The sampling technique is then used to build a merged MDP, as in [13], and to derive the.