Monte Carlo

Resources

• https://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/The_Monte_Carlo_Estimator

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle.

Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action.

Monte Carlo Simulation solves deterministic problems using a probabilistic analog.

Definition

Monte Carlo is the art of approximating an expectation by the sample mean of a function of simulated random variables.

Explanation

Monte Carlo methods vary, but tend to follow a particular pattern:

1. Define a domain of possible inputs

2. Generate inputs randomly from a probability distribution over the domain

3. Perform a deterministic computation on the inputs

4. Aggregate the results

For example, consider a quadrant (circular sector) inscribed in a unit square. Given that the ratio of their areas is π/4, the value of π can be approximated using a Monte Carlo method:[12]

1. Draw a square, then inscribe a quadrant within it
2. Uniformly scatter a given number of points over the square
3. Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

4. The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, π/4. Multiply the result by 4 to estimate π.

In this procedure the domain of inputs is the square that circumscribes the quadrant. We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of π.

There are two important points:

1. If the points are not uniformly distributed, then the approximation will be poor.

2. There are a large number of points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

Definition

Monte Carlo simulation: a simulation is a fictitious representation of reality, uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Monte Carlo method: technique that can be used to solve a mathematical or statistical problem

Examples:

• Simulation: Drawing one pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

• Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

• Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at a large number of different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.