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mathmachine-learningprobabilitybayesianpattern-recognition

Bayesian curve fitting model

With respect to bayesian curve fitting, eq 1.68 of Bishop - Pattern recognition

How is the following result derived :

p(t|x, x, t) = Integration{ p(t|x, w)p(w|x, t) } dw

Solution

  • Lets just consider a simpler case using the Law of total probability. If w1, w2 are disjoint events then

    p(A) = p(A|w1) p(w1) + p(A|w2) p(w2)

    we can extend this to any number of items

    p(A) = sum_{wi} p(A|wi) p(wi)

    or indeed take the limit

    p(A) = int_{w} p(A|w) p(w) dw

    We can make A depend on another independent event B that the w's might depend on

    p(A|B) = int_{w} p(A|w) p(w|B) dw

    or an event C which the w's do not depend on

    p(A|B,C) = = int_{w} p(A|w,C) p(w|B) dw

    which is just your formula with different variables.