# Distribution

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, sample set, 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.

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.

Distributions are created as sample sets by default. To create a symbolic distribution, use `Sym.`

namespace: `Sym.normal`

, `Sym.beta`

and so on.

## Distribution Creation

These are functions for creating primitive 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.

### normal

```
normal: (distribution|number, distribution|number) => distribution
normal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution
normal: (dict<{p10: distribution|number, p90: distribution|number}>) => distribution
normal: (dict<{p25: distribution|number, p75: distribution|number}>) => distribution
normal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```
normal(5, 1)
normal({ p5: 4, p95: 10 })
normal({ p10: 5, p95: 9 })
normal({ p25: 5, p75: 9 })
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<{p10: distribution|number, p90: distribution|number}>) => distribution
lognormal: (dict<{p25: distribution|number, p75: distribution|number}>) => distribution
lognormal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
```

**Examples**

```
lognormal(0.5, 0.8)
lognormal({ p5: 4, p95: 10 })
lognormal({ p10: 5, p95: 9 })
lognormal({ p25: 5, p75: 9 })
lognormal({ mean: 5, stdev: 2 })
```

### to

The `to`

function is an easy way to generate simple distributions using predicted *5th* and *95th* percentiles.

`To`

is simply an alias for `lognormal({p5:low, p95:high})`

. It does not accept values of 0 or less, as those are not valid for lognormal distributions.

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

**Examples**

```
5 to 10
to(5,10)
```

### uniform

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

**Examples**

`uniform(10, 12)`

### beta

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

**Examples**

```
beta(20, 25)
beta({ mean: 0.39, stdev: 0.1 })
```

### cauchy

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

**Examples**

`cauchy(5, 1)`

### gamma

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

**Examples**

`gamma(5, 1)`

### logistic

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

**Examples**

`gamma(5, 1)`

### exponential

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

**Examples**

`exponential(2)`

### bernoulli

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

**Examples**

`bernoulli(0.5)`

### triangular

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

**Examples**

`triangular(5, 10, 20)`

### mixture

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

Note: If you want to pass in over 5 distributions, you must use the list syntax.

**Examples**

```
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

### make

Make a distribution, starting with either a number of a distribution.

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

### sample

One random sample from the distribution

`sample: (distribution) => number`

**Examples**

`sample(normal(5, 2))`

### sampleN

N random samples from the distribution

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

**Examples**

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

### mean

The distribution mean

`mean: (distribution) => number`

**Examples**

`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`

### Median

Median. The quantile at 0.5.

`median: (distribution) => number`

### cdf

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

**Examples**

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

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

**Examples**

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

### quantile

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

**Examples**

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

### truncate

Truncates both the left side and the right side of a distribution.

`truncate: (distribution, left: number, right: number) => distribution`

**Implementation Details**

Sample set distributions are truncated by filtering samples, but point set
distributions are truncated using direct geometric manipulation. Uniform
distributions are truncated symbolically. Symbolic but non-uniform
distributions get converted to Point Set distributions.

### truncateLeft

Truncates the left side of a distribution.

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

**Examples**

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

### truncateRight

Truncates the right side of a distribution.

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

**Examples**

`truncateRight(normal(5, 2), 6)`

### klDivergence

Kullback–Leibler divergence (opens in a new tab) between two distributions.

Note that this can be very brittle. If the second distribution has probability mass at areas where the first doesn't, then the result will be infinite. Due to numeric approximations, some probability mass in point set distributions is rounded to zero, leading to infinite results with klDivergence.

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

**Examples**

`Dist.klDivergence(normal(5, 2), normal(5, 4)) // returns 0.57`

### logScore

A log loss score. Often that often acts as a scoring rule (opens in a new tab). Useful when evaluating the accuracy of a forecast.

Note that it is fairly slow.

`Dist.logScore: ({estimate: distribution, ?prior: distribution, answer: distribution|number}) => number`

**Examples**

```
Dist.logScore({
estimate: normal(5, 2),
answer: normal(4.5, 1.2),
prior: normal(6, 4),
}) // returns -0.597.57
```

## Display

### toString

`toString: (distribution) => string`

**Examples**

`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**

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

## 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. This only impacts Point Set distributions, because those are the only ones that can be non-normlized.

`normalize: (distribution) => distribution`

**Examples**

`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**

`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**

`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.

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

### add

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

**Examples**

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

### sum

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

### cumulative sum

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

### multiply

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

### product

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

### cumulative product

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

### diff

`diff: (list<distributionLike>) => list<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**

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

## Pointwise Arithmetic Operations

**Unnormalized Results**

Pointwise arithmetic operations typically return unnormalized or completely
invalid distributions. For example, the operation
`normal(5,2) .- uniform(10,12)`

results in a distribution-like
object with negative probability mass.

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`