I'm familiar with how the Monte Carlo method works but I fail to understand why in many scientific papers, the Monte Carlo method is regarded as the reference value.
In other words, in the absence of a theoretical approach to computing a quantity of interest, one may propose an approximation or a method to evaluate this quantity of interest. Quite often people resort to Monte Carlo to quantify the goodness of the proposed approach.
How can we be certain that the values we are getting from Monte Carlo are "correct" since the method relies on the number of samples used, right?
The Monte Carlo method is a kind of brute force method, but instead of applying all possible inputs (like a typical brute force approach), input data is chosen randomly, and then the results are observed and counted. This is usually much easier than proving a complex mathematical system.
That works to reach an acceptable approximate resulting value if
With today's fast computers, improved random generators, and being able to run the program on multiple computers at the same time, the system is provided with so much random input that the probability that the aggregated result diverges more than an acceptable value is very small.
Some very complex systems/formulas that require human mathematical expertise (that could be error prone) can be comforted by the output of a Monte Carlo algorithm, or questioned if MC diverges too much from the expected results.
The Monty Hall problem solution can be easily "proven" (comforted) thanks to a Monte Carlo approach
repeating that simple algorithm a few million times gives you a good $66.67$% chance to get the car if you chose the other door...