 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2021/1/c/solution

E_{q} [w (Z)] = \int p (z) d z = 1

. From the ratio in (a),

E_{q} [w (Z)^{2}] = s^{d} \prod_{j = 1}^{d} E_{Z_{j} \sim N (0, s)} [e^{- (1 - s^{- 1}) Z_{j}^{2}}] .

The supplied normal distribution transform applies with quadratic coefficient

- (1 - s^{- 1})

, giving each factor

(2 s - 1)^{- 1/2}

. Thus

Var_{q} (w (Z)) = (\frac{s}{2 s - 1})^{d} - 1.

For

s > 1

, the base

A := s / 2 s - 1

exceeds one because

s^{2} - (2 s - 1) = (s - 1)^{2} > 0

: importance weight variance increases exponentially with dimension. For

s = 1

, the importance weight is identically one and its variance is zero. This calculation concerns the importance weight, rather than the weighted integrand.

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