18.1 Bayesian decisions

In Bayesian decision analysis it is possible to include a priori information in the form of a priori distributions, and also a posteriori distributions on the space Θ of possible states of the considered decision situation.

Let Θ = Θ: Θ possible state be the state space, = {d : d possible decision} be the set of possible decisions, and P be a probability distribution on the state space Θ. Moreover let U(.,.) be a utility function, i.e.

where U(θ, d) denotes the utility of the decision d if the system is in state Θ.

Assuming that the corresponding sums or integrals exist, the basis for optimal decisions is the expected utility

Here , is the stochastic quantity describing the random character of the system.

Definition 18.1:

Based on the context above the Bayesian decision dB is the solution of the following optimization problem:

Therefore the Bayesian decision dB is maximizing the expected utility.

Remark 18.1:

If the approach by loss considerations is used, instead ...

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