## Categorical Distributions Are Useful

The categorical distribution is a simple but powerful idea that answers the following question:

How do I pick randomly from many different options when the options have different probabilities for being chosen?

When would you ever use this? A few ideas:

• Showing a random post based on number of votes, so more popular posts show up more.
• Simulating random events in a game, where some events are more likely than others.
• Controlling movement of a robot by randomly picking direction based on probabilities of collision

It is a valuable tool in your developer toolkit, and is pretty simple to boot. With the magic of words and the also magic of JavaScript, I’ll walk you through it. We will also touch on some basic topics in math and probability theory, but you don’t need any background to understand.

## Fruitroulette

I am a humble fruit vendor, vending humble fruit. I have decided that I want to offer my customers a chance to live on the edge, and provide a service I call Fruitroulette. The customer gives me some money, and I give them a random fruit. Basically they will be on the Highway to the Danger Zone because the experience will be so intense.

Lets imagine what this would look like in JavaScript.

``````var fruitTypes = ['banana', 'apple', 'peach', 'watermellon', 'mango'];

function pickRandomFruit() {
var randomIndex = Math.ceil(Math.random() * fruitTypes.length) - 1;
console.log("randomIndex is", randomIndex);
return fruitTypes[randomIndex];
}

console.log("pick random fruit:", pickRandomFruit());
``````

Here we create an array to draw randomly from, and just pick a random number from 0 to the length of the array (or `[0,length]` in math). We use that random number as an index into our array to grab a random fruit.

## Fruitroulette, Categorical Style

However, being a shrewd businessman as well as a humble fruit dealer, I realize that some fruits are more expensive than others. I don’t want to be giving out expensive mangos just as often as cheap bananas (Cheap Bananas is also the name of my ska band). I need some way of drawing randomly from a bunch of fruit, but weighted so the cheaper a fruit is, the more likely it is to be chosen.

If you are jumping up and down in your chair, shouting “The categorical distribution!”, you either read the title of the blog post, know this already, or are internet-clairvoyant. Either way, you are rad. Let’s see some example code.

``````var fruits = [
{ fruit: 'banana', cost: 0.1 },
{ fruit: 'apple', cost: 0.15 },
{ fruit: 'peach', cost: 0.3 },
{ fruit: 'watermellon', cost: 2 },
{ fruit: 'mango', cost: 1.1 }
];

function fruitScore(fruit) {
return 1 / fruit.cost;
}
``````

We need a cost for each fruit. We also want the fruit to be picked more often the lower the cost, so we have a little function that inverts the cost to return a higher score for cheaper fruit.

We are getting closer, but we still need a way of organizing the fruit so we can pick randomly from them. What if we arrange them in buckets so that the larger the score, the larger their bucket, and thus the more likely a randomly drawn number is to fall into their bucket? Something like this:

`|______fruit1_|_fruit2_|____________fruit3_________|fruit4|`

``````function createBuckets(fruits, scoreFunction) {
var scoreAccumulator = 0;
fruits.forEach(function(fruit) {
fruit.startScore = scoreAccumulator;
fruit.endScore = scoreAccumulator + scoreFunction(fruit);
scoreAccumulator += scoreFunction(fruit);
});
}
``````

This code will create an interval proportional to the score of the fruit. Now we need one last piece: a function to draw a random fruit.

``````function calculateScoreTotal(items, scoreFunction) {
// Figure out score range
return items.reduce(function(prevScoreTotal, item) {
return prevScoreTotal + scoreFunction(item);
}, 0);
}

function drawFruit(fruits, scoreFunction) {
var scoreTotal,
randomScore,
i;

scoreTotal = calculateScoreTotal(fruits, scoreFunction);

// Draw a random score within the range
randomScore = Math.random() * scoreTotal;

// Figure out what fruit's bucket the score falls in
for (i = 0; i < fruits.length; i++) {
if (fruits[i].startScore < randomScore && fruits[i].endScore > randomScore) {
return fruits[i];
}
}
}

createBuckets(fruits, fruitScore);
console.log("my random fruit is", drawFruit(fruits, fruitScore));
``````

This code may look hairy, but it is doing simple things. We need to create a random number guaranteed to fall within one of the buckets we created. To do that, we need to know what the range of the buckets is so we can scale the number returned by `Math.random()`.

Next, we draw a random number within the score range. Then, we just find the bucket the number falls within, and return the corresponding fruit.

Holy cow, a categorical distribution! We’ve done it!

## Some Mathy Stuff

Now that we have seen an example in code, I’ll explain some of the theory behind the categorical distribution. If you aren’t interested, skip ahead to the next section for a module that lets you use the categorical in your code. If you want to know more, let’s start with the definition of a categorical distribution, from Wikipedia:

[A] categorical distribution […] is a probability distribution that describes the result of a random event that can take on one of K possible outcomes, with the probability of each outcome separately specified.

The first part of the definition tells us that a categorical distribution must be a probability distribution. Roughly, this means that if you add up the probabilities of drawing each item from the distribution, they will equal 1.

The second part says that each possible outcome (or fruit, in our case), has a specific probability of being selected.

Lets confirm that is the case with our distribution.

``````var scoreTotal = calculateScoreTotal(fruits, fruitScore);
var probabilitiesForFruits = fruits.map(function(fruit) {
var scoreRange,
probability;

scoreRange = fruit.endScore - fruit.startScore;
probability = scoreRange / scoreTotal;
console.log("probability for ", fruit.fruit, " is ", probability);
return probability;
});

probabilityTotal = probabilitiesForFruits.reduce(function(probTotalSoFar, prob) {
return prob + probTotalSoFar;
}, 0);

console.log("probability total for all fruits is", probabilityTotal);
``````

## Find It On GitHub

If this looks interesting to you, I wrote a little library for working with categorical distributions in JavaScript. You can find it here. It does some handy optimizations to make creating and drawing from the distribution fast.

Also, if math/computer science stuff is cool to you, you should check out the Papers in CS group. We pick a cool, accessible and foundational paper in Computer Science, and meet via Google Hangout to discuss it. It happens about once a month, and anyone, of any experience level, is welcome. You should join us!