Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

A AppendixDefinition A.6 (Independent Random Variables).The countably valued random variables X 1 ,...,X n are said to be independent if for any sequenceof real numbers x 1 ,...,x n we haveP(X 1 = x 1 ,...,X n = x n ) = P(X 1 = x 1 )P(X 2 = x 2 )···P(X n = x n ).A.4 Definitions for OptimizationThe following definitions are all taken from [46].Definition A.7 (Objective Function).An objective function f : X ↦→ Y with Y ⊆ R is a mathematical function, which is subjectto optimization.Definition A.8 (Problem Space).The problem space X (phenome) of an optimization problem is the set containing all elementsx which could be the solution of the optimization problem.Definition A.9 (Local Maximum).A (local) maximum ˆx l ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˆx l ) ≥ f (x) for all x neighboring ˆx l .Definition A.10 (Local Minimum).A (local) minimum ˇx l ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˇx l ) ≤ f (x) for all x neighboring ˇx l .104