Analytic methods vs Monte Carlo (terminology)

Question

What's the correct terminology to say "We can calculate the probability exactly using pure math, as opposed to Monte Carlo simulation"?

• Analytically sounds like we need Calculus, which we may not
• Combinatorically sounds like we can only use combinatorics
• exactly sounds like a precision issue and misses the point
• and closed form is no good, because quite often the exact solution is not in closed form - it's expressed as a recurrence or the like

The point is that we solve the probability formally, not needing to rely on simulation - what is the correct term?

Answer 1

I think your interpretation of Analytically is a bit narrow. An analytical solution is as opposed to a numerical solution, regardless of whether you use calculus. I'm sure you've checked this page and this, which highlights a key feature of a purported analytic solution: You can express the exact answer using elementary functions and operations. Anything else is not an analytic solution, for example:

• Truncated infinite series: It's an approximate analytical solution, but not an analytical solution.
• Recursion: Unless you can shortcut the recursion so that you can represent it as a function of N, you will need to calculate the values by chugging through each step.

Just because you can write out the solution formally does not mean it is analytic. For example: $y(x)=\int ix\Gamma(x^{{x^2}}) dx + C$ is a solution to $y'=ix\Gamma(x^{{x^2}})$, but it certainly is not an analytical solution, as you need to approximate the solution numerically. A grey area is infinite series solutions, which are exact and expressed using simple operations and functions, yet getting an exact numerical answer is impossible, since you have to truncate. However, this falls under analytic as well.

Answer 2

I think a good term would be "theoretically" (or "using a theoretical approach"). To me, theory and simulation are opposed in this sense: when you don't know how to calculate something theoretically, you resort to simulation (in this case, Monte Carlo simulation).