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Tutorial 5 Problem Set

1. Consider we have a set of weights w = (w1 , w2 , . . . , wn) generated by the importance sampling.  Suppose wi  ≥ 0 for each i = 1, . . . , n and let ∑i wi  < ∞ . Show the effective sample size (ESS) in the form of

ESS1(w) :=  ( ∑ wi)2

satisfies 1 ≤ ESS1(w) ≤ n.

Then, given the sample mean and sample variance of the weights, n  =  ∑ wi  and n(2)  =  ∑(wi  −

n)2 , respectively. Derive ESS1(w) using n  and n(2) .

2. Suppose we want to compute Ep[h(X)] < ∞ using importance sampling with a importance density g(x). Suppose the second moments Ep[h(X)2] is bounded. The so-called defensive importance sampling uses a mixture density in the form of

gα(x) = αgdef(x) + (1 − α)g(x),     α ∈ (0, 1),

where g(x) is a “good” importance density but not necessarily satisfies Requirement 6.4.1 and gdef(x) is the defensive term which satisfies

p(x)  

for some constant c > 0. Show that the new density gα(x) satisfies both Requirement 6.4.1 and Requirement

6.4.2.

3.  Consider a pair of random variable X and Y . Suppose the random variable X follows a Laplace distribution

with density

fX (x) =  exp(−|x|).

Suppose further the conditional random variable Y | X follows the logistic distribution with density

exp(x2  − y)      

fY |X (y | x) =

• Define the joint probability density function of (X, Y ).

• Derive an inverse transform method that can draw random variables (X, Y ) using a pair of uniform random variables (U1, U2 ).