# Monte Carlo estimation

(1.1 hours to learn)

## Summary

One way to answer queries about a probability distribution is to simulate from the distribution, a procedure known as Monte Carlo estimation. In particular, we estimate the expected value of some function f with respect to a distribution p by generating samples from p and averaging the values of f over those samples.

## Context

This concept has the prerequisites:

- expectation and variance (Monte Carlo methods are a way of estimating expectations.)

## Core resources (read/watch one of the following)

## -Free-

→ Machine learning summer school: Markov chain Monte Carlo (2009)

→ Information Theory, Inference, and Learning Algorithms

A graudate-level textbook on machine learning and information theory.

## -Paid-

→ Machine Learning: a Probabilistic Perspective

A very comprehensive graudate-level machine learning textbook.

Location:
Section 2.7, pages 52-56

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ Coursera: Probabilistic Graphical Models (2013)

An online course on probabilistic graphical models.

Location:
Lecture "Simple sampling"

Other notes:

- Click on "Preview" to see the videos.

## -Paid-

→ Pattern Recognition and Machine Learning

A textbook for a graduate machine learning course, with a focus on Bayesian methods.

Location:
Chapter 11 intro, pages 523-526

## See also

- The law of large numbers justifies the use of Monte Carlo estimation.
- Some commonly used sampling algorithms include:
- Gibbs sampling , a generic and widely applicable sampling algorithm
- particle filters , which are useful for time series modeling
- Metropolis-Hastings algorithm , which is very general

- variational inference , which tries to approximate an intractable posterior distribution with a tractable one
- loopy belief propagation , a strategy which is particular to graphical models