---
title: "Functions Reference"
sidebar_position: 7
---

import { SquiggleEditor } from "../../src/components/SquiggleEditor";

_The source of truth for this document is [this file of code](https://github.com/quantified-uncertainty/squiggle/blob/develop/packages/squiggle-lang/src/rescript/ReducerInterface/ReducerInterface_GenericDistribution.res)_

## Inventory distributions

We provide starter distributions, computed symbolically.

### Normal distribution

The `normal(mean, sd)` function creates a normal distribution with the given mean
and standard deviation.

<SquiggleEditor initialSquiggleString="normal(5, 1)" />

#### Validity

- `sd > 0`

### Uniform distribution

The `uniform(low, high)` function creates a uniform distribution between the
two given numbers.

<SquiggleEditor initialSquiggleString="uniform(3, 7)" />

#### Validity

- `low < high`

### Lognormal distribution

The `lognormal(mu, sigma)` returns the log of a normal distribution with parameters
`mu` and `sigma`. The log of `lognormal(mu, sigma)` is a normal distribution with mean `mu` and standard deviation `sigma`.

<SquiggleEditor initialSquiggleString="lognormal(0, 0.7)" />

An alternative format is also available. The `to` notation creates a lognormal
distribution with a 90% confidence interval between the two numbers. We add
this convenience as lognormal distributions are commonly used in practice.

<SquiggleEditor initialSquiggleString="2 to 10" />

#### Future feature:

Furthermore, it's also possible to create a lognormal from it's actual mean
and standard deviation, using `lognormalFromMeanAndStdDev`.

TODO: interpreter/parser doesn't provide this in current `develop` branch

<SquiggleEditor initialSquiggleString="lognormalFromMeanAndStdDev(20, 10)" />

#### Validity

- `sigma > 0`
- In `x to y` notation, `x < y`

### Beta distribution

The `beta(a, b)` function creates a beta distribution with parameters `a` and `b`:

<SquiggleEditor initialSquiggleString="beta(10, 20)" />

#### Validity

- `a > 0`
- `b > 0`
- Empirically, we have noticed that numerical instability arises when `a < 1` or `b < 1`

### Exponential distribution

The `exponential(rate)` function creates an exponential distribution with the given
rate.

<SquiggleEditor initialSquiggleString="exponential(1.11)" />

#### Validity

- `rate > 0`

### Triangular distribution

The `triangular(a,b,c)` function creates a triangular distribution with lower
bound `a`, mode `b` and upper bound `c`.

#### Validity

- `a < b < c`

<SquiggleEditor initialSquiggleString="triangular(1, 2, 4)" />

### Scalar (constant dist)

Squiggle, when the context is right, automatically casts a float to a constant distribution.

## `fromSamples`

The last distribution constructor takes an array of samples and constructs a sample set distribution.

<SquiggleEditor initialSquiggleString="fromSamples([1,2,3,4,6,5,5,5])" />

#### Validity

For `fromSamples(xs)`,

- `xs.length > 5`
- Strictly every element of `xs` must be a number.

## Operating on distributions

Here are the ways we combine distributions.

### Mixture of distributions

The `mixture` function combines 2 or more other distributions to create a weighted
combination of the two. The first positional arguments represent the distributions
to be combined, and the last argument is how much to weigh every distribution in the
combination.

<SquiggleEditor initialSquiggleString="mixture(uniform(0,1), normal(1,1), [0.5, 0.5])" />

It's possible to create discrete distributions using this method.

<SquiggleEditor initialSquiggleString="mixture(0, 1, [0.2,0.8])" />

As well as mixed distributions:

<SquiggleEditor initialSquiggleString="mixture(3, 8, 1 to 10, [0.2, 0.3, 0.5])" />

An alias of `mixture` is `mx`

#### Validity

Using javascript's variable arguments notation, consider `mx(...dists, weights)`:

- `dists.length == weights.length`

### Addition

A horizontal right shift

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 + dist2`}
/>

### Subtraction

A horizontal left shift

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 - dist2`}
/>

### Multiplication

TODO: provide intuition pump for the semantics

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 * dist2`}
/>

We also provide concatenation of two distributions as a syntax sugar for `*`

<SquiggleEditor initialSquiggleString="(0.1 to 1) triangular(1,2,3)" />

### Division

TODO: provide intuition pump for the semantics

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 / dist2`}
/>

### Exponentiation

TODO: provide intuition pump for the semantics

<SquiggleEditor initialSquiggleString={`(0.1 to 1) ^ beta(2, 3)`} />

### Taking the base `e` exponential

<SquiggleEditor
  initialSquiggleString={`dist = triangular(1,2,3)
exp(dist)`}
/>

### Taking logarithms

<SquiggleEditor
  initialSquiggleString={`dist = triangular(1,2,3)
log(dist)`}
/>

<SquiggleEditor
  initialSquiggleString={`dist = beta(1,2)
log10(dist)`}
/>

Base `x`

<SquiggleEditor
  initialSquiggleString={`x = 2
dist = beta(2,3)
log(dist, x)`}
/>

#### Validity

- `x` must be a scalar
- See [the current discourse](https://github.com/quantified-uncertainty/squiggle/issues/304)

### Pointwise addition

**Pointwise operations are done with `PointSetDist` internals rather than `SampleSetDist` internals**.

TODO: this isn't in the new interpreter/parser yet.

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 .+ dist2`}
/>

### Pointwise subtraction

TODO: this isn't in the new interpreter/parser yet.

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 .- dist2`}
/>

### Pointwise multiplication

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 .* dist2`}
/>

### Pointwise division

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 ./ dist2`}
/>

### Pointwise exponentiation

<SquiggleEditor
  initialSquiggleString={`dist1 = 1 to 10
dist2 = triangular(1,2,3)
dist1 .^ dist2`}
/>

## Standard functions on distributions

### Probability density function

The `pdf(dist, x)` function returns the density of a distribution at the
given point x.

<SquiggleEditor initialSquiggleString="pdf(normal(0,1),0)" />

#### Validity

- `x` must be a scalar
- `dist` must be a distribution

### Cumulative density function

The `cdf(dist, x)` gives the cumulative probability of the distribution
or all values lower than x. It is the inverse of `inv`.

<SquiggleEditor initialSquiggleString="cdf(normal(0,1),0)" />

#### Validity

- `x` must be a scalar
- `dist` must be a distribution

### Inverse CDF

The `inv(dist, prob)` gives the value x or which the probability for all values
lower than x is equal to prob. It is the inverse of `cdf`.

<SquiggleEditor initialSquiggleString="inv(normal(0,1),0.5)" />

#### Validity

- `prob` must be a scalar (please only put it in `(0,1)`)
- `dist` must be a distribution

### Mean

The `mean(distribution)` function gives the mean (expected value) of a distribution.

<SquiggleEditor initialSquiggleString="mean(normal(5, 10))" />

### Sampling a distribution

The `sample(distribution)` samples a given distribution.

<SquiggleEditor initialSquiggleString="sample(normal(0, 10))" />

## Converting between distribution formats

Recall the [three formats of distributions](https://develop--squiggle-documentation.netlify.app/docs/Discussions/Three-Types-Of-Distributions). We can force any distribution into `SampleSet` format

<SquiggleEditor initialSquiggleString="toSampleSet(normal(5, 10))" />

Or `PointSet` format

<SquiggleEditor initialSquiggleString="toPointSet(normal(5, 10))" />

### `toSampleSet` has two signatures

Above, we saw the unary `toSampleSet`, which uses an internal hardcoded number of samples. If you'd like to provide the number of samples, it has a binary signature as well (floored)

<SquiggleEditor initialSquiggleString="toSampleSet(0.1 to 1, 100.1)" />

#### Validity

- Second argument to `toSampleSet` must be a number.

## Normalization

Some distribution operations (like horizontal shift) return an unnormalized distriibution.

We provide a `normalize` function

<SquiggleEditor initialSquiggleString="normalize((0.1 to 1) + triangular(0.1, 1, 10))" />

#### Validity - Input to `normalize` must be a dist

We provide a predicate `isNormalized`, for when we have simple control flow

<SquiggleEditor initialSquiggleString="isNormalized((0.1 to 1) * triangular(0.1, 1, 10))" />

#### Validity

- Input to `isNormalized` must be a dist

## `inspect`

You may like to debug by right clicking your browser and using the _inspect_ functionality on the webpage, and viewing the _console_ tab. Then, wrap your squiggle output with `inspect` to log an internal representation.

<SquiggleEditor initialSquiggleString="inspect(toSampleSet(0.1 to 1, 100))" />

Save for a logging side effect, `inspect` does nothing to input and returns it.

## Truncate

You can cut off from the left

<SquiggleEditor initialSquiggleString="truncateLeft(0.1 to 1, 0.5)" />

You can cut off from the right

<SquiggleEditor initialSquiggleString="truncateRight(0.1 to 1, 10)" />

You can cut off from both sides

<SquiggleEditor initialSquiggleString="truncate(0.1 to 1, 0.5, 1.5)" />