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I am trying to integrate $e^{-x^2}$ from $0$ to $1$ using Metropolis sampling. While plain Monte Carlo integration gives the right result, Metropolis sampling grossly underestimates it. I have tried different probability densities for the sampling. I cannot figure out where I am making a mistake in my code. My code (in Python) is as follows.

import numpy as np

def func(x):
    return np.exp(-x * x)

def weight(x):
    return np.exp(-x)

def mcmc_int(func, weight, num_points=1000):
    current_point = 0.0
    f_vals = []
    accepted_points = 0
    for point in range(num_points):
        delta = np.random.random()
        new_point = current_point + delta
        acceptance_prob = min(1, weight(new_point) / weight(current_point))
        if acceptance_prob > np.random.random():
            current_point = new_point
            accepted_points += 1
        f_vals.append(func(current_point))
    integral = sum(f_vals) / num_points
    acceptance_rate = accepted_points / num_points
    return integral, acceptance_rate

Any pointers are appreciated.

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  • 1
    $\begingroup$ How many points are you using? What is the acceptance rate with this weight function? If you are trying to integrate from 0 to 1, why is there no restriction on the value of current_point or new_point? Since you only ever add delta and it can be any value from 0 upto 1, current_point will likely increase considerably outside the range 0 to 1 after a small number of iterations. @monstergroup42 $\endgroup$ – Tyberius Jan 24 at 20:05
  • $\begingroup$ @Tyberius You are right, new_point needs to be restricted between 0 and 1. It works now. With 1000 points, the acceptance rate is around 80% for this weight function. However I have to append func(current_point)/weight(current_point) to f_vals instead of just func(current_point) as I wrote in the original code. $\endgroup$ – monstergroup42 Jan 24 at 20:13

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