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

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---
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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
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`(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 `5 to 10` 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)">
`5 to 10` does the same thing as `to(5,10)`.
<SquiggleEditor initialSquiggleString="to(5,10)" />
</TabItem>
<TabItem value="ex2" label="-5 to 5">
When `-5 to 5` 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
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- `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
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`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
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- `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.
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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
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`normal(mean:number, standardDeviation:number)`
Creates a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution) with the given mean and standard deviation.
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<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
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- `mean`: Number
- `standard deviation`: Number greater than zero
[Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution)
## Log-normal
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`lognormal(mu: number, sigma: number)`
Creates a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution) with the given mu and sigma.
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`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
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- `mu`: Number
- `sigma`: Number greater than zero
[Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)
<details>
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<summary>
❓ Understanding <bold>mu</bold> and <bold>sigma</bold>
</summary>
<p>
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The log of `lognormal(mu, sigma)` is a normal distribution with mean `mu`
and standard deviation `sigma`. For example, these two distributions are
identical:
</p>
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<SquiggleEditor
initialSquiggleString={`normalMean = 10
normalStdDev = 2
logOfLognormal = log(lognormal(normalMean, normalStdDev))
[logOfLognormal, normal(normalMean, normalStdDev)]`}
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/>
</details>
## Uniform
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`uniform(low:number, high:number)`
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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
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- `low`: Number
- `high`: Number greater than `low`
<Admonition type="caution" title="Caution">
<p>
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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>
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<p>
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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>
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## Delta
`delta(value:number)`
Creates a discrete distribution with all of its probability mass at point `value`.
Numbers are often cast into delta distributions automatically. 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, `mixture(1,2,normal(5,2))` is the same as `mixture(delta(1), delta(2),normal(5,2))`
<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="normal(5,2) .* 6">
<SquiggleEditor initialSquiggleString="normal(5,2) .* 6" />
</TabItem>
</Tabs>
### Arguments
- `value`: Number
## Beta
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`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>
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<TabItem value="ex2" label="beta(1000, 1000)">
<SquiggleEditor initialSquiggleString="beta(1000, 2000)" />
</TabItem>
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<TabItem value="ex3" label="beta(1, 10)">
<SquiggleEditor initialSquiggleString="beta(1, 10)" />
</TabItem>
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<TabItem value="ex4" label="beta(10, 1)">
<SquiggleEditor initialSquiggleString="beta(10, 1)" />
</TabItem>
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<TabItem value="ex5" label="beta(0.8, 0.8)">
<SquiggleEditor initialSquiggleString="beta(0.8, 0.8)" />
</TabItem>
</Tabs>
### Arguments
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- `alpha`: Number greater than zero
- `beta`: Number greater than zero
<Admonition type="caution" title="Caution with small numbers">
<p>
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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>
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<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
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`exponential(rate:number)`
Creates an [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) with the given rate.
<SquiggleEditor initialSquiggleString="exponential(4)" />
### Arguments
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- `rate`: Number greater than zero
## Triangular distribution
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`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.
#### Validity
### Arguments
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- `low`: Number
- `mode`: Number greater than `low`
- `high`: Number greater than `mode`
<SquiggleEditor initialSquiggleString="triangular(1, 2, 4)" />
## FromSamples
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`fromSamples(samples:number[])`
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Creates a sample set distribution using an array of samples.
<SquiggleEditor initialSquiggleString="fromSamples([1,2,3,4,6,5,5,5])" />
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### Arguments
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- `samples`: An array of at least 5 numbers.