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Machine Learning by Sergios Theodoridis

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Problems

17.1 Let

μ:=E[f(x)]=f(x)p(x)dx

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and q(x) be the proposal distribution. Show that if

w(x):=p(x)q(x),

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and

μ^=1Ni=1Nw(xi)f(xi),

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then the variance

σf2=Eμ^Eμ^2=1Nf2(x)p2(x)q(x)dxμ2.

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Observe that if f2(x)p2(x) goes to zero slower than q(x), then for fixed N, σf2.

17.2 In importance sampling, with weights defined as

w(x)=ϕ(

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