In the city of Monaco, France, lies a district of 15,000 people famous for its nightlife, known as Monte Carlo. Here lies a famous casino where the element of chance inspired the naming of a mathematical method that relies heavily on randomness.
The Monte Carlo Method was created by mathematicians John von Neumann and Stanislaw Ulam during World War II as a way to improve decision making under uncertain conditions. The premise of their idea is that solutions to mathematical problems can be derived through the random evolution of sampling. Said differently, when creating mathematical models, one of the key input variables is random since that input variable (condition) is uncertain). This is akin to running surveys on a large population by choosing samples that are truly random in order to avoid introducing bias to the data, but deriving a clear picture of the population — only getting clearer with the greater the population that is surveyed.
To estimate the possible outcomes of an uncertain event, this doesn’t use standard forecasting methods, but instead builds a model of possible results by leveraging a probability distribution, such as a uniform or normal distribution, for any variable that has inherent uncertainty. Because the input is a probability distribution, we can assume that the output in this case will also be a probability distribution.
The results are then recalculated in the simulation repeatedly with each instance using a different set of random numbers between the minimum and maximum values. The more simulations that are run, the more accurate and confident we can be in our results thanks to the Law of Large Numbers. In this law, an average comes closer to its expected value the more samples we have.
With an exhaustive list of possible outcomes, a person can make a better-informed decision around risks under these uncertain conditions. While most statistics problems would estimate random quantities in a deterministic manner, Monte Carlo methods are the inverse where random quantities are used to generate estimates of deterministic quantities.
Usage
Monte Carlo methods are used in various fields, including machine learning & predicting stock prices. They’re also useful in making long-term predictions due to their increasing accuracy as the number of samples increases. The end result of a Monte Carlo simulation would yield a range of possible outcomes with the probability of each result occurring.
An example is calculating the probability of values when rolling two dice. There are 36 combinations of dice rolls, so based on this we can manually compute the probability of a particular outcome. Using a Monte Carlo Simulation, you can simulate rolling the dice 10,000 times (or more) to achieve more accurate predictions.
In a world where Big Data continues to deepen its footprint, improved computational processes will imply greater accuracy in the output of simulations, primarily in machine learning models.