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# Overview of WebPPL

## Introduction

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 recommend ProbMods. ProbMods is an online tutorial in Church, a language very similar to WebPPL, and its early chapters introduce key ideas in Bayesian generative models that are helpful background.

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:


// Function definition using Javascript's isNaN and log primitives:

var verboseLog = function(x){
if (x<=0 || _.isNaN(x)) {
print("Input " + x + " was not a positive number");
return null;
} else {
return Math.log(x);
}
};

// Array with numbers, object, Boolean types
var inputs = [1, 1.5, -1, {key: 1}, true];

print("Apply verboseLog to elements in array: ");
map(verboseLog, inputs);


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;
// => Assignment is allowed only to fields of globalStore.

var table = {key: 1};
var updatedTable = {key: table.key + 1};
print(updatedTable);

// Or use the library function *update*:

var secondUpdatedTable = update(table, {key:10})
print(secondUpdatedTable);


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


var ar = [1,2,3];

// Don't do this:

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

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


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 number of built-in functions for sampling from random variables from different distributions. Many of these are found in scientific computing libraries. A full list of functions is in the WebPPL library source. Try clicking the “Run” button repeatedly to get different i.i.d. random samples:


print('Fair coins: ' + [flip(0.5), flip(0.5)]);
print('Biased coins: ' + [flip(0.9), flip(0.9)]);

var coinWithSide = function(){
return categorical([.49, .49, .02], ['heads', 'tails', 'side']);
};

print(repeat(5, coinWithSide)); // draw i.i.d samples


There are also continuous random variables:


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

print('A single sample from a 2D Gaussian: ' +
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
};

geometric(0.8);


What makes WebPPL different from conventional programming languages is its ability to represent and manipulate probability distributions. Distribution objects 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 $x \sim P$ where $P(x)$ is the distribution.

2. You can compute the probability (or density) the distribution assigns to a value. That is, to compute $\log(P(x))$, 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 built-in 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: p }));
};

var gaussian = function(mu, sigma) {
return sample(Gaussian({ mu: mu, sigma: 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({ method: 'enumerate' }, 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. Here, we use the enumeration method that enumerates each possible value of each random variables (a, b and c) 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({ method: 'enumerate' }, function(){
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({ method: 'enumerate' }, function(){
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.auto to plot distributions:


var appleOrangeDist = Infer({ method: 'enumerate' }, function(){
return flip(0.9) ? "apple" : "orange";
});

viz.auto(appleOrangeDist);


var fruitTasteDist = Infer({ method: 'enumerate' }, function(){
return {
fruit: categorical([0.3, 0.3, 0.4], ["apple", "banana", "orange"]),
tasty: flip(0.7)
};
});

viz.auto(fruitTasteDist);


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

viz.auto(positionDist);


### Next

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