Gaussian Processes for Machine Learning

C. E. rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of AMathematical Joint, Marginal and Conditional ProbabilityLet then(discrete or continuous) random variablesy1,..,ynhave ajointjoint probabilityprobabilityp(y1,..,yn), orp(y) for , one ought to distin-guish between probabilities (for discrete variables) and probability densities forcontinuous variables. Throughout the book we commonly use the term prob-ability to refer to both. Let us partition the variables inyinto two groups,yAandyB, whereAandBare two disjoint sets whose union is the set{1,..,n},so thatp(y) =p(yA,yB). Each group may contain one or more ofyAis given bymarginal probabilityp(yA) = p(yA,yB)dyB.( )The integral is replaced by a sum if the variables are discrete valued. Noticethat if the setAcontains more than one variable, then the marginal probabilityis itself a joint probability whether it is referred to as one or the other dependson the context.

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, ISBN 026218253X. 2006 Massachusetts Institute of Technology.c www.GaussianProcess.org/gpml

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