squiggle/packages/website/docs/Features/Distributions.mdx

---
title: "Distribution Creation"
sidebar_position: 8
---

import TOCInline from "@theme/TOCInline";
import { SquiggleEditor } from "../../src/components/SquiggleEditor";
import Admonition from "@theme/Admonition";
import Tabs from "@theme/Tabs";
import TabItem from "@theme/TabItem";

<TOCInline toc={toc} maxHeadingLevel={2} />

## To

`(5thPercentile: number) to (95thPercentile: number)`  
`to(5thPercentile: number, 95thPercentile: number)`

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.

<Tabs>
  <TabItem value="ex1" label="5 to 10" default>
    When <code>5 to 10</code> is entered, both numbers are positive, so it
    generates a lognormal distribution with 5th and 95th percentiles at 5 and
    10.
    <SquiggleEditor initialSquiggleString="5 to 10" />
  </TabItem>
  <TabItem value="ex3" label="to(5,10)">
    <code>5 to 10</code> does the same thing as <code>to(5,10)</code>.
    <SquiggleEditor initialSquiggleString="to(5,10)" />
  </TabItem>
  <TabItem value="ex2" label="-5 to 5">
    When <code>-5 to 5</code> is entered, there's negative values, so it
    generates a normal distribution. This has 5th and 95th percentiles at 5 and
    10.
    <SquiggleEditor initialSquiggleString="-5 to -3" />
  </TabItem>
  <TabItem value="ex4" label="1 to 10000">
    It's very easy to generate distributions with very long tails. If this
    happens, you can click the "log x scale" box to view this using a log scale.
    <SquiggleEditor initialSquiggleString="1 to 10000" />
  </TabItem>
</Tabs>

### Arguments

- `5thPercentile`: number
- `95thPercentile`: number, greater than `5thPercentile`

<Admonition type="tip" title="Tip">
  <p>
    "<bold>To</bold>" is a great way to generate probability distributions very
    quickly from your intuitions. It's easy to write and easy to read. It's
    often a good place to begin an estimate.
  </p>
</Admonition>

<Admonition type="caution" title="Caution">
  <p>
    If you haven't tried{" "}
    <a href="https://www.lesswrong.com/posts/LdFbx9oqtKAAwtKF3/list-of-probability-calibration-exercises">
      calibration training
    </a>
    , you're likely to be overconfident. We recommend doing calibration training
    to get a feel for what a 90 percent confident interval feels like.
  </p>
</Admonition>

## Mixture

`mixture(...distributions: Distribution[], weights?: number[])`  
`mx(...distributions: Distribution[], weights?: number[])`

The `mixture` mixes combines multiple distributions to create a mixture. You can optionally pass in a list of proportional weights.

<Tabs>
  <TabItem value="ex1" label="Simple" default>
    <SquiggleEditor initialSquiggleString="mixture(1 to 2, 5 to 8, 9 to 10)" />
  </TabItem>
  <TabItem value="ex2" label="With Weights">
    <SquiggleEditor initialSquiggleString="mixture(1 to 2, 5 to 8, 9 to 10, [0.1, 0.1, 0.8])" />
  </TabItem>
  <TabItem value="ex3" label="With Continuous and Discrete Inputs">
    <SquiggleEditor initialSquiggleString="mixture(1 to 5, 8 to 10, 1, 3, 20)" />
  </TabItem>
</Tabs>

### Arguments

- `distributions`: A set of distributions or numbers, each passed as a paramater. Numbers will be converted into Delta distributions.
- `weights`: An optional array of numbers, each representing the weight of its corresponding distribution. The weights will be re-scaled to add to `1.0`. If a weights array is provided, it must be the same length as the distribution paramaters.

### Aliases

- `mx`

### Special Use Cases of Mixtures

<details>
  <summary>🕐 Zero or Continuous</summary>
  <p>
    One common reason to have mixtures of continous and discrete distributions is to handle the special case of 0.
    Say I want to model the time I will spend on some upcoming project. I think I have an 80% chance of doing it.
  </p>

  <p>
    In this case, I have a 20% chance of spending 0 time with it. I might estimate my hours with,
  </p>
  <SquiggleEditor
    initialSquiggleString={`hours_the_project_will_take = 5 to 20
chance_of_doing_anything = 0.8
mx(hours_the_project_will_take, 0, [chance_of_doing_anything, 1 - chance_of_doing_anything])`}
  />
</details>

<details>
  <summary>🔒 Model Uncertainty Safeguarding</summary>
  <p>
  One technique several <a href="https://www.foretold.io/">Foretold.io</a> users used is to combine their main guess, with a
  "just-in-case distribution". This latter distribution would have very low weight, but would be
  very wide, just in case they were dramatically off for some weird reason.
  </p>
<SquiggleEditor
  initialSquiggleString={`forecast = 3 to 30
chance_completely_wrong = 0.05
forecast_if_completely_wrong = -100 to 200
mx(forecast, forecast_if_completely_wrong, [1-chance_completely_wrong, chance_completely_wrong])`}
/>

</details>

## Normal

`normal(mean:number, standardDeviation:number)`

Creates a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution) with the given mean and standard deviation.

<Tabs>
  <TabItem value="ex1" label="normal(5,1)" default>
    <SquiggleEditor initialSquiggleString="normal(5, 1)" />
  </TabItem>
  <TabItem value="ex2" label="normal(100000000000, 100000000000)">
    <SquiggleEditor initialSquiggleString="normal(100000000000, 100000000000)" />
  </TabItem>
</Tabs>

### Arguments

- `mean`: Number
- `standard deviation`: Number greater than zero

[Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution)

## Log-normal

`lognormal(mu: number, sigma: number)`

Creates a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution) with the given mu and sigma.

`Mu` and `sigma` can be difficult to directly reason about. Because of this complexity, we recommend typically using the <a href="#to">to</a> syntax instead of estimating `mu` and `sigma` directly.

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

### Arguments

- `mu`: Number
- `sigma`: Number greater than zero

[Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)

<details>
  <summary>
    ❓ Understanding <bold>mu</bold> and <bold>sigma</bold>
  </summary>
  <p>
    The log of <code>lognormal(mu, sigma)</code> is a normal distribution with
    mean <code>mu</code>
    and standard deviation <code>sigma</code>. For example, these two distributions
    are identical:
  </p>
  <SquiggleEditor
    initialSquiggleString={`normalMean = 10
normalStdDev = 2
logOfLognormal = log(lognormal(normalMean, normalStdDev))
[logOfLognormal, normal(normalMean, normalStdDev)]`}
  />
</details>

## Uniform

`uniform(low:number, high:number)`

Creates a [uniform distribution](<https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)>) with the given low and high values.

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

### Arguments

- `low`: Number
- `high`: Number greater than `low`

<Admonition type="caution" title="Caution">
  <p>
    While uniform distributions are very simple to understand, we find it rare
    to find uncertainties that actually look like this. Before using a uniform
    distribution, think hard about if you are really 100% confident that the
    paramater will not wind up being just outside the stated boundaries.
  </p>

  <p>
    One good example of a uniform distribution uncertainty would be clear
    physical limitations. You might have complete complete uncertainty on what
    time of day an event will occur, but can say with 100% confidence it will
    happen between the hours of 0:00 and 24:00.
  </p>
</Admonition>

## Delta

`delta(value:number)`

Creates a discrete distribution with all of its probability mass at point `value`.

Few Squiggle users call the function `delta()` directly. Numbers are converted into delta distributions automatically, when it is appropriate.

For example, in the function `mixture(1,2,normal(5,2))`, the first two arguments will get converted into delta distributions
with values at 1 and 2. Therefore, this is the same as `mixture(delta(1),delta(2),normal(5,2))`.

`Delta()` distributions are currently the only discrete distributions accessible in Squiggle.

<Tabs>
  <TabItem value="ex1" label="delta(3)" default>
    <SquiggleEditor initialSquiggleString="delta(3)" />
  </TabItem>
  <TabItem value="ex3" label="mixture(1,3,5)">
    <SquiggleEditor initialSquiggleString="mixture(1,3,5)" />
  </TabItem>
  <TabItem value="ex2" label="normal(5,2) * 6">
    <SquiggleEditor initialSquiggleString="normal(5,2) * 6" />
  </TabItem>
  <TabItem value="ex4" label="dotAdd(normal(5,2), 6)">
    <SquiggleEditor initialSquiggleString="dotAdd(normal(5,2), 6)" />
  </TabItem>
  <TabItem value="ex5" label="dotMultiply(normal(5,2), 6)">
    <SquiggleEditor initialSquiggleString="dotMultiply(normal(5,2), 6)" />
  </TabItem>
</Tabs>

### Arguments

- `value`: Number

## Beta

`beta(alpha:number, beta:number)`

Creates a [beta distribution](https://en.wikipedia.org/wiki/Beta_distribution) with the given `alpha` and `beta` values. For a good summary of the beta distribution, see [this explanation](https://stats.stackexchange.com/a/47782) on Stack Overflow.

<Tabs>
  <TabItem value="ex1" label="beta(10, 20)" default>
    <SquiggleEditor initialSquiggleString="beta(10,20)" />
  </TabItem>
  <TabItem value="ex2" label="beta(1000, 1000)">
    <SquiggleEditor initialSquiggleString="beta(1000, 2000)" />
  </TabItem>
  <TabItem value="ex3" label="beta(1, 10)">
    <SquiggleEditor initialSquiggleString="beta(1, 10)" />
  </TabItem>
  <TabItem value="ex4" label="beta(10, 1)">
    <SquiggleEditor initialSquiggleString="beta(10, 1)" />
  </TabItem>
  <TabItem value="ex5" label="beta(0.8, 0.8)">
    <SquiggleEditor initialSquiggleString="beta(0.8, 0.8)" />
  </TabItem>
</Tabs>

### Arguments

- `alpha`: Number greater than zero
- `beta`: Number greater than zero

<Admonition type="caution" title="Caution with small numbers">
  <p>
    Squiggle struggles to show beta distributions when either alpha or beta are
    below 1.0. This is because the tails at ~0.0 and ~1.0 are very high. Using a
    log scale for the y-axis helps here.
  </p>
  <details>
    <summary>Examples</summary>
    <Tabs>
      <TabItem value="ex1" label="beta(0.3, 0.3)" default>
        <SquiggleEditor initialSquiggleString="beta(0.3, 0.3)" />
      </TabItem>
      <TabItem value="ex2" label="beta(0.5, 0.5)">
        <SquiggleEditor initialSquiggleString="beta(0.5, 0.5)" />
      </TabItem>
      <TabItem value="ex3" label="beta(0.8, 0.8)">
        <SquiggleEditor initialSquiggleString="beta(.8,.8)" />
      </TabItem>
      <TabItem value="ex4" label="beta(0.9, 0.9)">
        <SquiggleEditor initialSquiggleString="beta(.9,.9)" />
      </TabItem>
    </Tabs>
  </details>
</Admonition>

## Exponential

`exponential(rate:number)`

Creates an [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) with the given rate.

<SquiggleEditor initialSquiggleString="exponential(4)" />

### Arguments

- `rate`: Number greater than zero

## Triangular distribution

`triangular(low:number, mode:number, high:number)`

Creates a [triangular distribution](https://en.wikipedia.org/wiki/Triangular_distribution) with the given low, mode, and high values.

### Arguments

- `low`: Number
- `mode`: Number greater than `low`
- `high`: Number greater than `mode`

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

## FromSamples

`fromSamples(samples:number[])`

Creates a sample set distribution using an array of samples.

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

### Arguments

- `samples`: An array of at least 5 numbers.

<Admonition type="caution" title="Caution!">
  <p>
    Samples are converted into{" "}
    <a href="https://en.wikipedia.org/wiki/Probability_density_function">PDF</a>{" "}
    shapes automatically using{" "}
    <a href="https://en.wikipedia.org/wiki/Kernel_density_estimation">
      kernel density estimation
    </a>{" "}
    and an approximated bandwidth. Eventually Squiggle will allow for more
    specificity.
  </p>
</Admonition>