I am difficulty in the question below,

To measure how accurate our Monte Carlo approximations are, we can use the central limit theorem. If the number of samples drawn mm

m is large, then the Monte Carlo sample mean used to estimate E() approximately follows a normal distribution with mean

E() and variance Var()/m. If we substitute the sample variance for Var(), we can get a rough estimate of our Monte Carlo standard error (or standard deviation).

Suppose we have 100 samples from our posterior distribution for \theta

, called i \theta_i^*

i

, and that the sample variance of these draws is 5.2. A rough estimate of our Monte Carlo standard error would then be 5.2/1000.228 \sqrt{ 5.2 / 100 } \approx 0.228

5.2/100

0.228. So our estimate \bar{\theta^*}

is probably within about 0.456 0.456

0.456 (two standard errors) of the true E() \text{E}(\theta)

E().

What does the standard error of our Monte Carlo estimate become if we increase our sample size to 5,000? Assume that the sample variance of the draws is still 5.2.