by Owain Evans, Andreas Stuhlmüller, John Salvatier, and Daniel Filan

Probabilistic programming in WebPPL


This chapter introduces the probabilistic programming language WebPPL (pronounced “web people”). The models for agents in this tutorial are all implemented in WebPPL and so it’s important to understand how the language works.

We begin with a quick overview of probabilistic programming. If you are new to probabilistic programming, you might want to read an informal introduction (e.g. here or here) or a more technical survey. For a practical introduction to both probabilistic programming and Bayesian modeling, we highly recommend ProbMods, which also uses the WebPPL language.

The only requirement to run the code for this tutorial is a modern browser (e.g. Chrome, Firefox, Safari). If you want to explore the models in detail and to create your own, we recommend running WebPPL from the command line. Installation is simple and is explained here.

WebPPL: a purely functional subset of Javascript

WebPPL includes a subset of Javascript, and follows the syntax of Javascript for this subset.

This example program uses most of the Javascript syntax that is available in WebPPL:

// Define a function using two external primitives:
// 1. Javascript's `JSON.stringify` for converting to strings
// 2. Underscore's _.isFinite for checking if a value is a finite number
var coerceToPositiveNumber = function(x) {
  if (_.isFinite(x) && x > 0) {
    return x;
  } else {
    print('- Input ' + JSON.stringify(x) +
          ' was not a positive number, returning 1 instead');
    return 1;

// Create an array with numbers, an object, an a Boolean
var inputs = [2, 3.5, -1, { key: 1 }, true];

// Map the function over the array
print('Processing elements in array ' + JSON.stringify(inputs) + '...');
var result = map(coerceToPositiveNumber, inputs);
print('Result: ' + JSON.stringify(result));

Language features with side effects are not allowed in WebPPL. The code that has been commented out uses assignment to update a table. This produces an error in WebPPL.

// Don't do this:

// var table = {};
// table.key = 1;
// table.key = table.key + 1;
// => Syntax error: You tried to assign to a field of table, but you can
//                  only assign to fields of globalStore

// Instead do this:

var table = { key: 1 };
var tableTwo = { key: table.key + 1 };

// Or use the library function `extend`:

var tableThree = extend(tableTwo, { key: 3 })

There are no for or while loops. Instead, use higher-order functions like WebPPL’s built-in map, filter and zip:

var xs = [1, 2, 3];

// Don't do this:

// for (var i = 0; i < xs.length; i++){
//   print(xs[i]);
// }

// Instead of for-loop, use `map`:
map(print, xs);


It is possible to use normal Javascript functions (which make internal use of side effects) in WebPPL. See the online book on the implementation of WebPPL for details (section “Using Javascript Libraries”).

WebPPL stochastic primitives

Sampling from random variables

WebPPL has a large library of primitive probability distributions. Try clicking “Run” repeatedly to get different i.i.d. random samples:

print('Fair coins (Bernoulli distribution):');
print([flip(0.5), flip(0.5), flip(0.5)]);

print('Biased coins (Bernoulli distribution):');
print([flip(0.9), flip(0.9), flip(0.9)]);

var coinWithSide = function(){
  return categorical([.45, .45, .1], ['heads', 'tails', 'side']);

print('Coins that can land on their edge:')
print(repeat(5, coinWithSide)); // draw 5 i.i.d samples

There are also continuous random variables:

print('Two samples from standard Gaussian in 1D: ');
print([gaussian(0, 1), gaussian(0, 1)]);

print('A single sample from a 2D Gaussian: ');
print(multivariateGaussian(Vector([0, 0]), Matrix([[1, 0], [0, 10]])));

You can write your own functions to sample from more complex distributions. This example uses recursion to define a sampler for the Geometric distribution:

var geometric = function(p) {
  return flip(p) ? 1 + geometric(p) : 1


What makes WebPPL different from conventional programming languages is its ability to perform inference operations using these primitive probability distributions. Distribution objects in WebPPL have two key features:

  1. You can draw random i.i.d. samples from a distribution using the special function sample. That is, you sample where is the distribution.

  2. You can compute the probability (or density) the distribution assigns to a value. That is, to compute , you use dist.score(x), where dist is the distribution in WebPPL.

The functions above that generate random samples are defined in the WebPPL library in terms of primitive distributions (e.g. Bernoulli for flip and Gaussian for gaussian) and the built-in function sample:

var flip = function(p) {
  var p = (p !== undefined) ? p : 0.5;
  return sample(Bernoulli({ p }));

var gaussian = function(mu, sigma) {
  return sample(Gaussian({ mu, sigma }));

[flip(), gaussian(1, 1)];

To create a new distribution, we pass a (potentially stochastic) function with no arguments—a thunk—to the function Infer that performs marginalization. For example, we can use flip as an ingredient to construct a Binomial distribution using enumeration:

var binomial = function() {
  var a = flip(0.5);
  var b = flip(0.5);
  var c = flip(0.5);
  return a + b + c;

var MyBinomial = Infer({ model: binomial });

[sample(MyBinomial), sample(MyBinomial), sample(MyBinomial)];

Infer is the inference operator that computes (or estimates) the marginal probability of each possible output of the function binomial. If no explicit inference method is specified, Infer defaults to enumerating each possible value of each random variable in the function body.

Bayesian inference by conditioning

The most important use of inference methods is for Bayesian inference. Here, our task is to infer the value of some unknown parameter by observing data that depends on the parameter. For example, if flipping three separate coins produce exactly two Heads, what is the probability that the first coin landed Heads? To solve this in WebPPL, we can use Infer to enumerate all values for the random variables a, b and c. We use condition to constrain the sum of the variables. The result is a distribution representing the posterior distribution on the first variable a having value true (i.e. “Heads”).

var twoHeads = Infer({
  model() {
    var a = flip(0.5);
    var b = flip(0.5);
    var c = flip(0.5);
    condition(a + b + c === 2);
    return a;

print('Probability of first coin being Heads (given exactly two Heads) : ');

var moreThanTwoHeads = Infer({
  model() {
    var a = flip(0.5);
    var b = flip(0.5);
    var c = flip(0.5);
    condition(a + b + c >= 2);
    return a;

print('\Probability of first coin being Heads (given at least two Heads): ');

Codeboxes and Plotting

The codeboxes allow you to modify our examples and to write your own WebPPL code. Code is not shared between boxes. You can use the special function viz to plot distributions:

var appleOrangeDist = Infer({
  model() {
    return flip(0.9) ? 'apple' : 'orange';

var fruitTasteDist = Infer({
  model() {
    return {
      fruit: categorical([0.3, 0.3, 0.4], ['apple', 'banana', 'orange']),
      tasty: flip(0.7)

var positionDist = Infer({
  model() {
    return {
      X: gaussian(0, 1),
      Y: gaussian(0, 1)};
  method: 'forward',
  samples: 1000



In the next chapter, we will implement rational decision-making using inference functions.