Test distribution is unknown (p0 ??6?pM ??. M In the inaccurate case, we have no assumption on the transition matrix. We represented this lack of knowledge by a uniform FDM distribution, where each transition has been observed one single time ( = [1, ???, 1]). Sections 5.2.1, 5.2.2 and 5.2.3 describes the three distributions considered for this study.PLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,12 /Benchmarking for Bayesian Reinforcement LearningFig 3. Illustration of the GC distribution. doi:10.1371/journal.pone.0157088.g5.2.1 Generalised Chain distribution r ; . The Generalised Chain (GC) distribution is inspired from the five-state chain problem (5 states, 3 actions) [15]. The agent starts at State 1, and has to go through State 2, 3 and 4 in order to reach the last state (State 5), where the best rewards are. The agent has at its disposal 3 actions. An action can either let the agent move from State x(n) to State x(n+1) or force it to go back to State x(1). The transition matrix is drawn from a FDM U0126-EtOH cost parameterised by GC, and the reward function is denoted by GC. Fig 3 illustrates the distribution and more details can be found in S2 File. GDL GDL 5.2.2 Generalised Double-Loop distribution r ; . The Generalised DoubleLoop (GDL) distribution is inspired from the double-loop problem (9 states, 2 actions) [15]. Two loops of 5 states are crossing at State 1, where the agent starts. One loop is a trap: if the agent enters it, it has no choice to exit but crossing over all the states composing it. Exiting this loop provides a small reward. The other loop is yielding a good reward. However, each action of this loop can either let the agent move to the next state of the loop or force it to return to State 1 with no reward. The transition matrix is drawn from an FDM parameterised by GDL, and the reward function is denoted by GDL. Fig 4 illustrates the distribution and more details can be found in S2 File. Grid Grid 5.2.3 Grid distribution r ; . The Grid distribution is inspired from the Dearden’s maze problem (25 states, 4 actions) [15]. The agent is placed at a corner of a 5×5 grid (the S cell), and has to reach the opposite corner (the G cell). When it succeeds, it returns to its initial state and receives a reward. The agent can perform 4 different actions, corresponding to theGC GCFig 4. Illustration of the GDL distribution. doi:10.1371/journal.pone.0157088.gPLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,13 /Benchmarking for Bayesian Reinforcement LearningFig 5. Illustration of the Grid distribution. doi:10.1371/journal.pone.0157088.g4 directions (up, down, left, right). However, depending on the cell on which the agent is, each action has a certain probability to fail, and can prevent the agent to move in the selected direction. The transition matrix is drawn from an FDM parameterised by Grid, and the reward function is denoted by Grid. Fig 5 illustrates the distribution and more details can be found in S2 File.5.3 Discussion of the results5.3.1 Accurate case. As it can be seen in Fig 6, OPPS is the only algorithm whose offline time cost varies. In the three different settings, OPPS can be launched after a few seconds, but order Bay 41-4109 behaves very poorly. However, its performances increased very quickly when given at least one minute of computation time. Algorithms that do not use offline computation time have a wide range of different scores. This variance represents the different possible configurations for these algorithms, whic.Test distribution is unknown (p0 ??6?pM ??. M In the inaccurate case, we have no assumption on the transition matrix. We represented this lack of knowledge by a uniform FDM distribution, where each transition has been observed one single time ( = [1, ???, 1]). Sections 5.2.1, 5.2.2 and 5.2.3 describes the three distributions considered for this study.PLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,12 /Benchmarking for Bayesian Reinforcement LearningFig 3. Illustration of the GC distribution. doi:10.1371/journal.pone.0157088.g5.2.1 Generalised Chain distribution r ; . The Generalised Chain (GC) distribution is inspired from the five-state chain problem (5 states, 3 actions) [15]. The agent starts at State 1, and has to go through State 2, 3 and 4 in order to reach the last state (State 5), where the best rewards are. The agent has at its disposal 3 actions. An action can either let the agent move from State x(n) to State x(n+1) or force it to go back to State x(1). The transition matrix is drawn from a FDM parameterised by GC, and the reward function is denoted by GC. Fig 3 illustrates the distribution and more details can be found in S2 File. GDL GDL 5.2.2 Generalised Double-Loop distribution r ; . The Generalised DoubleLoop (GDL) distribution is inspired from the double-loop problem (9 states, 2 actions) [15]. Two loops of 5 states are crossing at State 1, where the agent starts. One loop is a trap: if the agent enters it, it has no choice to exit but crossing over all the states composing it. Exiting this loop provides a small reward. The other loop is yielding a good reward. However, each action of this loop can either let the agent move to the next state of the loop or force it to return to State 1 with no reward. The transition matrix is drawn from an FDM parameterised by GDL, and the reward function is denoted by GDL. Fig 4 illustrates the distribution and more details can be found in S2 File. Grid Grid 5.2.3 Grid distribution r ; . The Grid distribution is inspired from the Dearden’s maze problem (25 states, 4 actions) [15]. The agent is placed at a corner of a 5×5 grid (the S cell), and has to reach the opposite corner (the G cell). When it succeeds, it returns to its initial state and receives a reward. The agent can perform 4 different actions, corresponding to theGC GCFig 4. Illustration of the GDL distribution. doi:10.1371/journal.pone.0157088.gPLOS ONE | DOI:10.1371/journal.pone.0157088 June 15,13 /Benchmarking for Bayesian Reinforcement LearningFig 5. Illustration of the Grid distribution. doi:10.1371/journal.pone.0157088.g4 directions (up, down, left, right). However, depending on the cell on which the agent is, each action has a certain probability to fail, and can prevent the agent to move in the selected direction. The transition matrix is drawn from an FDM parameterised by Grid, and the reward function is denoted by Grid. Fig 5 illustrates the distribution and more details can be found in S2 File.5.3 Discussion of the results5.3.1 Accurate case. As it can be seen in Fig 6, OPPS is the only algorithm whose offline time cost varies. In the three different settings, OPPS can be launched after a few seconds, but behaves very poorly. However, its performances increased very quickly when given at least one minute of computation time. Algorithms that do not use offline computation time have a wide range of different scores. This variance represents the different possible configurations for these algorithms, whic.