###### tags: `one-offs` `monte carlo` `non-technical`

# Principles of the Monte Carlo Method

**Overview**: In this note, I describe some rough thoughts on which 'principles' (if any) lie at the core of the Monte Carlo method. No claims are made to originality, insight, or rigour in this discussion.

## Discussion

Some months ago, I offered the following three statements as a motivation for 'the Monte Carlo method', broadly construed:

1. Quantities of interest can be expressed as integrals,
2. Integrals can be expressed as expectations of random variables, and
3. Expectations of random variables can be approximated through simulation,

where each use of 'can' is intended be interpreted in the sense of 'it is sometimes possible to', rather than 'it is always possible to'. The contention is not that these are essential or exhaustive, but that they cover some relevant aspects of the approach.

Having offered these statements, one can be curious about whether any of them are 'unique' to Monte Carlo, at least at some resolution.

I think that one can contend that the third statement certainly is: upon using simulation of random variables to approximate expectations of random variables, you are well and truly in Monte Carlo territory.

The second statement is quite relevant, but doesn't directly implicate the use of randomness. Still, once you make the observation that integrals can be expressed as expectations, it is relatively common to end up implementing some form of Monte Carlo. A potential counter-example would be to use the probabilistic interpretation of expectations to construct some appropriately-adapted quadrature rule.

The first statement seems to be strictly broader than Monte Carlo, and should also apply equally to e.g. classical quadrature.

Still, in isolation, the third statement only suggests a practical solution if your problem has already been cast in the form of evaluating an expectation. Similarly, taking only the second and third statements in isolation, they are only relevant if your problem is already written in the form of an integral. As such, I see a case that the conjunction of all three points may be needed to capture the essence of the Monte Carlo approach.

I think that I feel this way because the broadest version of the principle suggests that one might put in additional work to write your problem in a specific form (i.e. an integral or expectation), in much the same way that others might work to systematically reduce their own problems to linear algebra, convex optimisation, and so on.

In any case, these are not especially formal thoughts. It is thus manifestly possible that either i) there exists a simpler principle which better captures the essence of the approach, ii) there exists a broader principle which concisely captures even more detail, or iii) some combination of the two.