A Metropolis method is used to explore a distribution P(x)that is actually a 1000-dimensional spherical Gaussian distribution of standard deviation 1 in all dimensions. The proposal density Q is...

A Metropolis method is used to explore a distribution P(x)that is actually a 1000-dimensional spherical Gaussian distribution of standard deviation 1 in all dimensions. The proposal density Q is a1000-dimensional spherical Gaussian distribution of standard deviation∈. Roughly what is the step size ∈ if the acceptance rate is 0.5? Assuming this value of ∈,

(a) roughly how long would the method take to traverse the distributionand generate a sample independent of the initial condition?

(b) By how much does long(x) change in a typical step? By how much should ln P(x) vary when x is drawn from P(x)?

(c) What happens if, rather than using a Metropolis method that tries to change all components at once, one instead uses a concatenation of Metropolis updates changing one component at a time?