---
sidebar_position: 3
title: Distribution
---

import Admonition from "@theme/Admonition";
import TOCInline from "@theme/TOCInline";

Distributions are the flagship data type in Squiggle. The distribution type is a generic data type that contains one of three different formats of distributions.
These subtypes are [point set](/docs/Api/DistPointSet), [sample set](/docs/Api/DistSampleSet), and symbolic. The first two of these have a few custom functions that only work on them. You can read more about the differences between these formats [here](/docs/Discussions/Three-Formats-Of-Distributions).

Several functions below only can work on particular distribution formats.
For example, scoring and pointwise math requires the point set format. When this happens, the types are automatically converted to the correct format. These conversions are lossy.

<TOCInline toc={toc} />

## Distribution Creation

These are functions for creating primative distributions. Many of these could optionally take in distributions as inputs. In these cases, Monte Carlo Sampling will be used to generate the greater distribution. This can be used for simple hierarchical models.

See a longer tutorial on creating distributions [here](/docs/Guides/DistributionCreation).

### normal

```
normal: (distribution|number, distribution|number) => distribution
normal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution
normal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```js
normal(5, 1)
normal({ p5: 4, p95: 10 })
normal({ mean: 5, stdev: 2 })
normal(5 to 10, normal(3, 2))
normal({ mean: uniform(5, 9), stdev: 3 })
```

### lognormal

```
lognormal: (distribution|number, distribution|number) => distribution
lognormal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution
lognormal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```javascript
lognormal(0.5, 0.8);
lognormal({ p5: 4, p95: 10 });
lognormal({ mean: 5, stdev: 2 });
```

### uniform

```
uniform: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
uniform(10, 12);
```

### beta

```
beta: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
beta(20, 25);
```

### cauchy

```
cauchy: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
cauchy(5, 1);
```

### gamma

```javascript
gamma: (distribution|number, distribution|number) => distribution
```

**Examples**

```js
gamma(5, 1);
```

### logistic

```
logistic: (distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
gamma(5, 1);
```

### exponential

```
exponential: (distribution|number) => distribution
```

**Examples**

```javascript
exponential(2);
```

### bernoulli

```
bernoulli: (distribution|number) => distribution
```

**Examples**

```javascript
bernoulli(0.5);
```

### triangular

```javascript
triangular: (number, number, number) => distribution;
```

**Examples**

```javascript
triangular(5, 10, 20);
```

### to / credibleIntervalToDistribution

The `to` function is an easy way to generate simple distributions using predicted _5th_ and _95th_ percentiles.

If both values are above zero, a `lognormal` distribution is used. If not, a `normal` distribution is used.

`To` is an alias for `credibleIntervalToDistribution`. However, because of its frequent use, it is recommended to use the shorter name.

```
to: (distribution|number, distribution|number) => distribution
credibleIntervalToDistribution(distribution|number, distribution|number) => distribution
```

**Examples**

```javascript
5 to 10
to(5,10)
-5 to 5
```

### mixture

```
mixture: (...distributionLike, weights?:list<float>) => distribution
mixture: (list<distributionLike>, weights?:list<float>) => distribution
```

**Examples**

```javascript
mixture(normal(5, 1), normal(10, 1), 8);
mx(normal(5, 1), normal(10, 1), [0.3, 0.7]);
mx([normal(5, 1), normal(10, 1)], [0.3, 0.7]);
```

## Functions

### sample

One random sample from the distribution

```
sample: (distribution) => number
```

**Examples**

```javascript
sample(normal(5, 2));
```

### sampleN

N random samples from the distribution

```
sampleN: (distribution, number) => list<number>
```

**Examples**

```javascript
sampleN(normal(5, 2), 100);
```

### mean

The distribution mean

```
mean: (distribution) => number
```

**Examples**

```javascript
mean(normal(5, 2));
```

### stdev

Standard deviation. Only works now on sample set distributions (so converts other distributions into sample set in order to calculate.)

```
stdev: (distribution) => number
```

### variance

Variance. Similar to stdev, only works now on sample set distributions.

```
variance: (distribution) => number
```

### mode

```
mode: (distribution) => number
```

### cdf

```
cdf: (distribution, number) => number
```

**Examples**

```javascript
cdf(normal(5, 2), 3);
```

### pdf

```
pdf: (distribution, number) => number
```

**Examples**

```javascript
pdf(normal(5, 2), 3);
```

### quantile

```
quantile: (distribution, number) => number
```

**Examples**

```javascript
quantile(normal(5, 2), 0.5);
```

### toPointSet

**TODO: Will soon be called "PointSet.make"**

Converts a distribution to the pointSet format.

```
toPointSet: (distribution) => pointSetDistribution
```

**Examples**

```javascript
toPointSet(normal(5, 2));
```

### toSampleSet

**TODO: Will soon be called "SampleSet.make"**

Converts a distribution to the sampleSet format, with n samples.

```
toSampleSet: (distribution, number) => sampleSetDistribution
```

**Examples**

```javascript
toSampleSet(normal(5, 2), 1000);
```

### truncateLeft

Truncates the left side of a distribution. Returns either a pointSet distribution or a symbolic distribution.

```
truncateLeft: (distribution, l => number) => distribution
```

**Examples**

```javascript
truncateLeft(normal(5, 2), 3);
```

### truncateRight

Truncates the right side of a distribution. Returns either a pointSet distribution or a symbolic distribution.

```
truncateRight: (distribution, r => number) => distribution
```

**Examples**

```javascript
truncateLeft(normal(5, 2), 6);
```

### klDivergence

[Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between two distributions.

```
klDivergence: (distribution, distribution) => number
```

**Examples**

```javascript
klDivergence(normal(5, 2), normal(5, 4)); // returns 0.57
```

## Display

### toString

```
toString: (distribution) => string
```

**Examples**

```javascript
toString(normal(5, 2));
```

### sparkline

Produce a sparkline of length n. For example, `▁▁▁▁▁▂▄▆▇██▇▆▄▂▁▁▁▁▁`. These can be useful for testing or quick text visualizations.

```
sparkline: (distribution, n = 20) => string
```

**Examples**

```javascript
toSparkline(truncateLeft(normal(5, 2), 3), 20); // produces ▁▇█████▇▅▄▃▂▂▁▁▁▁▁▁▁
```

### inspect

Prints the value of the distribution to the Javascript console, then returns the distribution. Useful for debugging.

```
inspect: (distribution) => distribution
```

**Examples**

```javascript
inspect(normal(5, 2)); // logs "normal(5, 2)" to the javascript console and returns the distribution.
```

## Normalization

There are some situations where computation will return unnormalized distributions. This means that their cumulative sums are not equal to 1.0. Unnormalized distributions are not valid for many relevant functions; for example, klDivergence and scoring.

The only functions that do not return normalized distributions are the pointwise arithmetic operations and the scalewise arithmetic operations. If you use these functions, it is recommended that you consider normalizing the resulting distributions.

### normalize

Normalize a distribution. This means scaling it appropriately so that it's cumulative sum is equal to 1.

```
normalize: (distribution) => distribution
```

**Examples**

```javascript
normalize(normal(5, 2));
```

### isNormalized

Check of a distribution is normalized. Most distributions are typically normalized, but there are some commands that could produce non-normalized distributions.

```
isNormalized: (distribution) => bool
```

**Examples**

```javascript
isNormalized(normal(5, 2)); // returns true
```

### integralSum

**Note: If you have suggestions for better names for this, please let us know.**

Get the sum of the integral of a distribution. If the distribution is normalized, this will be 1.0. This is useful for understanding unnormalized distributions.

```
integralSum: (distribution) => number
```

**Examples**

```javascript
integralSum(normal(5, 2));
```

## Regular Arithmetic Operations

Regular arithmetic operations cover the basic mathematical operations on distributions. They work much like their equivalent operations on numbers.

The infixes `+`,`-`, `*`, `/`, `^` are supported for addition, subtraction, multiplication, division, power, and unaryMinus.

```javascript
pointMass(5 + 10) == pointMass(5) + pointMass(10);
```

### add

```
add: (distributionLike, distributionLike) => distribution
```

**Examples**

```javascript
normal(0, 1) + normal(1, 3); // returns normal(1, 3.16...)
add(normal(0, 1), normal(1, 3)); // returns normal(1, 3.16...)
```

### sum

**Todo: Not yet implemented**

```
sum: (list<distributionLike>) => distribution
```

**Examples**

```javascript
sum([normal(0, 1), normal(1, 3), uniform(10, 1)]);
```

### multiply

```
multiply: (distributionLike, distributionLike) => distribution
```

### product

```
product: (list<distributionLike>) => distribution
```

### subtract

```
subtract: (distributionLike, distributionLike) => distribution
```

### divide

```
divide: (distributionLike, distributionLike) => distribution
```

### pow

```
pow: (distributionLike, distributionLike) => distribution
```

### exp

```
exp: (distributionLike, distributionLike) => distribution
```

### log

```
log: (distributionLike, distributionLike) => distribution
```

### log10

```
log10: (distributionLike, distributionLike) => distribution
```

### unaryMinus

```
unaryMinus: (distribution) => distribution
```

**Examples**

```javascript
-normal(5, 2); // same as normal(-5, 2)
unaryMinus(normal(5, 2)); // same as normal(-5, 2)
```

## Pointwise Arithmetic Operations

<Admonition type="caution" title="Unnormalized Results">
  <p>
    Pointwise arithmetic operations typically return unnormalized or completely
    invalid distributions. For example, the operation{" "}
    <code>normal(5,2) .- uniform(10,12)</code> results in a distribution-like
    object with negative probability mass.
  </p>
</Admonition>

Pointwise arithmetic operations cover the standard arithmetic operations, but work in a different way than the regular operations. These operate on the y-values of the distributions instead of the x-values. A pointwise addition would add the y-values of two distributions.

The infixes `.+`,`.-`, `.*`, `./`, `.^` are supported for their respective operations.

The `mixture` methods works with pointwise addition.

### dotAdd

```
dotAdd: (distributionLike, distributionLike) => distribution
```

### dotMultiply

```
dotMultiply: (distributionLike, distributionLike) => distribution
```

### dotSubtract

```
dotSubtract: (distributionLike, distributionLike) => distribution
```

### dotDivide

```
dotDivide: (distributionLike, distributionLike) => distribution
```

### dotPow

```
dotPow: (distributionLike, distributionLike) => distribution
```

### dotExp

```
dotExp: (distributionLike, distributionLike) => distribution
```

## Scale Arithmetic Operations

<Admonition type="caution" title="Likely to change">
  <p>
    We're planning on removing scale operations in favor of more general
    functions soon.
  </p>
</Admonition>

Scale operations are similar to pointwise operations, but operate on a constant y-value instead of y-values coming from a distribution. You can think about this as scaling a distribution vertically by a constant.

The following items would be equivalent.

```js
scalePow(normal(5,2), 2)
mapY(normal(5,2), {|y| y ^ 2}) // Not yet available
```

### scalePow

```
scalePow: (distributionLike, number) => distribution
```

### scaleExp

```
scaleExp: (distributionLike, number) => distribution
```

### scaleLog

```
scaleLog: (distributionLike, number) => distribution
```

### scaleLog10

```
scaleLog10: (distributionLike, number) => distribution
```

## Special

### Declaration (Continuous Functions)

Adds metadata to a function of the input ranges. Works now for numeric and date inputs. This is useful when making formal predictions. It allows you to limit the domain that your prediction will be used and scored within.

Declarations are currently experimental and will likely be removed or changed in the future.

```
declareFn: (dict<{fn: lambda, inputs: array<dict<{min: number, max: number}>>}>) => declaration
```

**Examples**

```javascript
declareFn({
  fn: {|a,b| a },
  inputs: [
    {min: 0, max: 100},
    {min: 30, max: 50}
  ]
})
```
