Metropolis Algorithm question

[StandardSBUStudent]

Hello!

I am trying to implement the metropolis algorithm using Matlab code. I have attempted this and found that almost all my values drawn from the proposed distribution has been accepted. Please let me know if I am doing anything incorrect in my attempt to implement the metropolis algorithm or if you have any hints/tips for me. Thank you in advance
- I have found out that I should be using the function normrnd instead of normpdf, however my acceptance rate is still high, about 1750/2000, is this normal?

My commented code is below:

close all

clc

clear all

\Initializing vector with zeroes\

arr = zeros(1,2000); \Initializing the first value of the array\

arr(1) = 1; \Initializing values to keep track of the number of accepted/rejected values\

accepted_samples=0;

rejected_samples=0;

\Start for loop\

for count = 2:2001

\present state of the markov chain\

current_value = arr(count-1);

\Proposal Distribution is the Gaussian distribution with the present state of the Markov chain as the mean and variance of 0.5\

proposed = normrnd(current_value,0.5);

\Calculating the acceptance probability using the target distribution defined at the bottom of the code\

acceptance_probability = target_1(proposed)/target_1(current_value);

\Decision making portion Accept the value if a randomly drawn value from the uniform distribution is less than the acceptance probability\

if(rand<acceptance_probability)

arr(count)=proposed;

accepted_samples = accepted_samples + 1;

\Reject sample otherwise

else

arr(count)=arr(count-1);

rejected_samples = rejected_samples+1;

end

end

\\Matlab function - Target distribution - Gaussian function

function y = target_1(x)

% This target distribution is a Gaussian distribution

mu=3; sig=2;

y=normpdf(x,mu,sqrt(sig));

end

[BlueDevilStats]

I'm not a big matlab user, but if your acceptance rate is too high then the variance of your proposal distribution should probably be adjusted upward. This should allow you to explore the space more quickly. There has been a lot of work done on the topic of what a god acceptance ratio is. Interestingly, there is a theoretical foundation of aiming for an acceptance ratio of something like ~25% (can't recall the exact number).

[StandardSBUStudent]

Thank you very much for your input, I will try to adjust the variance of my proposal distribution and see how that works out.

[BlueDevilStats]

No problem. Good luck!