Monte Carlo Integration

You are probably familiar with Reimann sum integration in which the space is partitioned into intervals and a rectangle is constructed for each interval with height equal to the function value at a specific point within the interval.

What's different in Monte Carlo integration is we have a sampling distribution p(x) from which we draw samples and evaluate our function f(x) at those samples. This is particularly useful for evaluating high-dimensional integrals encountered in computing posterior distributions.

Some benefits of Monte Carlo integration include:
1. General and simple to use method
2. Scales to high-dimensional integrals
3. Versatile approach with a variety of applications
4. Computes uncertainty estimate (error bars) in the final answer

In summary, Monte Carlo integration offers simplicity, flexibility, and robustness, making it a valuable tool for numerical computations.

Link to notebook:

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00:00 Introduction
00:29 Monte Carlo Integration Goal
01:53 Monte Carlo Expectation Formula
02:20 Estimating Pi
03:31 Python Notebook