 Math and science::INF ML AI

# 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.

1. Generate [something] from a given [what?]
2. 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.