Probabilistic GEM Clause Samples

Probabilistic GEM. ‌ The model presented so far could be sufficient to deal with deterministic and non-deterministic sys- tems. However, for many practical purposes, simply knowing the possible trajectories of a system is not enough; instead, we need to know the probability for taking particular trajectories to evaluate the quality of the system. Therefore we need to turn to stochastic models. This would be needed, for example, to capture the situation where a robot receives sensor input with measurement errors. ⊗ Thus we can assume that a probability measure P(X) is given for each set of trajectories X.1 P(X) describes the probability that a trajectory in X is taken by the system. If the system is gener- ated from an ensemble Sens and an environment Senv , then we assume that probability distributions over their respective trajectory spaces are given, and that the combination operator computes the distribution of S from these. Given a probability measure P and a utility function u for a system S, we define the evaluation of a system S as the expected utility, i.e., evalu(S) = EP[S, u] = ∫ ξ∈S p(ξ)u(ξ)dξ. where p is the probability density of P. The evaluation gives us an easy criterion to compare different systems: a system S1 has a better utility than a system S2 if its evaluation is higher.