# Monte Carlo Methods

The this card describes Monte Carlo methods in the words of David MacKay. A lot of this card's content is drawn from Chapter 29 of MacKay's book.

### Monte Carlo methods. Definition.

*Monte Carlo methods* is a term for something that
utilizes a *random number generator* to solve one or both of the
following problems.

- Generate [something] from a given [what?]
- Estimate expectations of functions under this probability distribution.
For example, for some function \( \phi \), we want to know:
[\[\int {\dd}^N \vb{x} \;\; ? \;\; \phi(\vb{x}) \]]

The notation used above does a lot of heavy lifting. "\( \mathbb{P}(\vb{x}) \)" suggests a set \( L \subset \mathbb{R}^N \) is the co-domain of an implied random variable \( Y : ? \to L \). \( L \) is also the domain of the function \( \mathbb{P} \) which must have a signature of the form \( \mathbb{P} : L \to \mathbb{R} \). The function \( \phi \) seems to have a signature \( \phi : L \to \mathbb{R} \). It is composed with \( Y \) like \( \phi \circ Y \) and thus, \( \phi \) is a random variable. The composition implies that the domain of \( \phi \) is \( L \). The integration formula further implies either that the co-domain of \( \phi \) must be \( \mathbb{R} \) (alternatively, the integration is syntax is allowing for the interpretation of a vector valued integration which seems a stretch).

The reverse side describes Lake plankton analogy to the above problems. Can you remember the details? The reverse side also rewords the above description in the language of probability distributions.