Monte Carlo Integration Overview

Study Notes

Monte Carlo integration uses random sampling to estimate integrals.
It's useful when direct integration is difficult or impossible.
The method leverages the law of large numbers. As the sample size increases, the Monte Carlo estimate converges to the true value of the integral.

The expected value of a function h(X), denoted as E[h(X)], is estimated by averaging the function's values over a large sample of values drawn from the probability distribution of X. This is mathematically expressed as the limit, as the sample size approaches infinity, of the sum of h(x_i) over all sample points (x_i).
The estimate is represented by hn, where hn = (1/n) * (sum of h(xj) for j from 1 to n)
This approach estimates the integral by drawing random values (x_j) from a distribution with probability density function (PDF) f(x). The function is then applied to each sampled value, and the mean of the resulting values estimates the integral of h(x) * f(x).

The Monte Carlo estimator hn is unbiased, meaning its expected value equals the true expected value of the function h(x), E[h(x)].
The variance of hn is inversely proportional to the sample size (n). Larger sample sizes lead to smaller variances, implying better accuracy. This variance is denoted as 1/(n*Var(h(X))).
The variance of the estimator can also be estimated using a Monte Carlo method, denoted var(hn).

If the variance of h(X) is unknown, it can be approximated through Monte Carlo methods.
The method involves calculating g(x) = [h(x) - E[h(x)]]2. This function is then integrated (or approximated though Monte Carlo methods) and used to estimate Var(h(X)).
A Monte Carlo approximation of this is denoted as v_n = (1/n)*sum(g(x_j) for j = 1 to n).
Finally: Var(hn) approximately equals vn.

The central limit theorem states that the sampling distribution of the mean of a large number of independent, identically distributed random variables tends toward a normal distribution.
This allows creating confidence bounds for hn.
This allows knowing that hn is normally distributed, with mean Ef[h(x)] and variance 1/n of Var[h(X)].

Monte Carlo integration can be used to estimate the integral of any function, even if the function is difficult (or impossible) to integrate analytically.
The method is particularly useful for functions defined on complex domains, high dimensions or involve integrals based on complex probability distributions.
This method is used for estimating integrals of functions in different types of problems. These include calculating areas, volumes, and other quantities.

The method involves generating random numbers following a specified probability distribution. For example, using a uniform distribution.
The approximation will get better by drawing progressively more random numbers.
Graphs and plots showcasing the closeness of the observed function and distributions to theoretical expected functions help demonstrate convergence and accuracy.
The examples show use with functions like cos/sin to show an example of using monte carlo for estimating integrals.