What is Monte Carlo integration?

What’s the average distance that someone can jump? Let’s estimate it! I have a silly function that gives me the distance that someone can jump, based on their height h in centimeters:

function jumpDistance(h: number): number {
  return h < 0 || h > 360 ? 0 : 300 * Math.sin(h * (Math.PI / 400));
}

Next we’ll describe the heights of people in the population. To start, let’s say the population is just two people: Jane is 160 cm tall, and Peter is 180 cm tall. Then we can estimate the average jump distance:

const janeJump = jumpDistance(160);
const peterJump = jumpDistance(180);
const averageJump = (janeJump + peterJump) / 2;

With a bigger population, we can describe the population with an array of heights:

const populationHeights = [160, 180, 170, 190, 150];
let totalJump = 0;
for (const h of populationHeights) {
  totalJump += jumpDistance(h);
}
const averageJump = totalJump / populationHeights.length;

For a yet larger population, we can describe the population by keeping a tally for each height:

const populationHeights = new Map([
  [160, 23],
  [170, 45],
  [180, 32],
  [190, 12],
  [200, 8],
]);
let totalJump = 0;
let totalPeople = 0;
for (const [h, count] of populationHeights) {
  totalJump += count * jumpDistance(h);
  totalPeople += count;
}
const averageJump = totalJump / totalPeople;

For an even larger population, we can describe the population by a probability distribution, like $\text{Normal}(\mu=170, \sigma=20)$. Then the precise average jump distance is:

\[ \int_{h = -\infty}^{\infty} \text{jumpDistance}(h) \, \text{Normal}(h; \mu=170, \sigma=20) \, dh \]

Yuck! Suddenly we can’t solve the problem by just running the code, because there are infinitely many heights to consider. Our nice finite loop turned into an infinite integral. And it’s horrible to solve analytically, especially because jumpDistance is a piecewise function.

What can we do instead? If we have access to a function that gives us the population count at each height, we can do what we were doing before, but with some chosen heights:

function popCountForHeight(h: number) {
  return 1000 * Math.exp(-0.5 * ((h - 170) / 20) ** 2);
}

let totalJump = 0;
let totalPeople = 0;
for (let h = 0; h < 360; h += 1) {
  const count = popCountForHeight(h);
  totalJump += count * jumpDistance(h);
  totalPeople += count;
}
const averageJump = totalJump / totalPeople;

This approach could be called a Riemann sum. One problem with this approach is that we need to know a range of heights to consider that will cover most of the population, but not too much that we’re wasting time on heights that are very unlikely. Above, we chose to consider heights from 0 to 360 cm.

Another approach is to sample from the distribution:

const populationHeights = new NormalDistribution({ mean: 170, stdDev: 20 });

let totalJump = 0;
let totalPeople = 0;
for (let i = 0; i < 1000; i++) {
  const h = populationHeights.sample();
  totalJump += jumpDistance(h);
  totalPeople++;
}
const averageJump = totalJump / totalPeople;

Surprise, this is Monte Carlo integration!

Tagged #monte-carlo-integration, #monte-carlo, #integration, #numerical-methods, #probability, #statistics, #math, #programming.

More by Jim

What does the dot do in JavaScript?

foo.bar, foo.bar(), or foo.bar = baz - what do they mean? A deep dive into prototypical inheritance and getters/setters. 2020-11-01

Smear phishing: a new Android vulnerability

Trick Android to display an SMS as coming from any contact. Convincing phishing vuln, but still unpatched. 2020-08-06

A probabilistic pub quiz for nerds

A “true or false” quiz where you respond with your confidence level, and the optimal strategy is to report your true belief. 2020-04-26

Time is running out to catch COVID-19

Simulation shows it’s rational to deliberately infect yourself with COVID-19 early on to get treatment, but after healthcare capacity is exceeded, it’s better to avoid infection. Includes interactive parameters and visualizations. 2020-03-14

The inception bar: a new phishing method

A new phishing technique that displays a fake URL bar in Chrome for mobile. A key innovation is the “scroll jail” that traps the user in a fake browser. 2019-04-27

The hacker hype cycle

I got started with simple web development, but because enamored with increasingly esoteric programming concepts, leading to a “trough of hipster technologies” before returning to more productive work. 2019-03-23

Project C-43: the lost origins of asymmetric crypto

Bob invents asymmetric cryptography by playing loud white noise to obscure Alice’s message, which he can cancel out but an eavesdropper cannot. This idea, published in 1944 by Walter Koenig Jr., is the forgotten origin of asymmetric crypto. 2019-02-16

How Hacker News stays interesting

Hacker News buried my post on conspiracy theories in my family due to overheated discussion, not censorship. Moderation keeps the site focused on interesting technical content. 2019-01-26

My parents are Flat-Earthers

For decades, my parents have been working up to Flat-Earther beliefs. From Egyptology to Jehovah’s Witnesses to theories that human built the Moon billions of years in the future. Surprisingly, it doesn’t affect their successful lives very much. For me, it’s a fun family pastime. 2019-01-20

The dots do matter: how to scam a Gmail user

Gmail’s “dots don’t matter” feature lets scammers create an account on, say, Netflix, with your email address but different dots. Results in convincing phishing emails. 2018-04-07

The sorry state of OpenSSL usability

OpenSSL’s inadequate documentation, confusing key formats, and deprecated interfaces make it difficult to use, despite its importance. 2017-12-02

I hate telephones

I hate telephones. Some rational reasons: lack of authentication, no spam filtering, forced synchronous communication. But also just a visceral fear. 2017-11-08

The Three Ts of Time, Thought and Typing: measuring cost on the web

Businesses often tout “free” services, but the real costs come in terms of time, thought, and typing required from users. Reducing these “Three Ts” is key to improving sign-up flows and increasing conversions. 2017-10-26

Granddad died today

Granddad died. The unspoken practice of death-by-dehydration in the NHS. The Liverpool Care Pathway. Assisted dying in the UK. The importance of planning in end-of-life care. 2017-05-19

How do I call a program in C, setting up standard pipes?

A C function to create a new process, set up its standard input/output/error pipes, and return a struct containing the process ID and pipe file descriptors. 2017-02-17

Your syntax highlighter is wrong

Syntax highlighters make value judgments about code. Most highlighters judge that comments are cruft, and try to hide them. Most diff viewers judge that code deletions are bad. 2014-05-11

Want to build a fantastic product using LLMs? I work at Granola where we're building the future IDE for knowledge work. Come and work with us! Read more or get in touch!

This page copyright James Fisher 2024. Content is not associated with my employer. Found an error? Edit this page.

What is Monte Carlo integration?

Similar posts

More by Jim