Optimal Sequential Procedures with Bayes Decision Rules

In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. Let X1,X2,... be a discrete-time stochastic process, whose distribution depends on an unknown parameter θ, θ ∈ Θ. We consider a problem of optimal sequen-tial decision-making in the following framework. Let wn(θ, d;x1,..., xn), θ ∈ Θ, d ∈ D, be a loss function representing losses from making a de-cision d at stage n of a statistical experiment, when the true parameter value is θ, and the data observed up to this stage are x1,..., xn. Let Knθ (x1,..., xn) be the cost of the observations when θ is the true value of the parameter. The decision is supposed to be taken through a se-quential decision-making procedure (τ, δ), where τ is a stopping time with respect to the sequence of σ-algebras Fn = σ(X1,X2,...,Xn), n = 1, 2,..., and δ is an Fτ-measurable decision function with values in D. For any sequential decision procedure (τ, δ) let us define the average loss due to incorrect decision W (θ; τ, δ) = Eθwτ (θ, δ;X1,... Xτ), and the average cost of observations as C(θ; τ) = EθKτθ (X1,...,Xτ). Let, finally, the “risk function ” be defined as R(τ, δ) =