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Starting to pull out distributions functionality

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Ozzie Gooen 2022-04-30 21:47:54 -04:00
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---
title: "Creating Distributions"
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: float) to (95thPercentile: float)`
`to(5thPercentile: float, 95thPercentile: float)`
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)" default>
`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
- `5thPercentile`: Float
- `95thPercentile`: Float
<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?: float[])`
`mx(...distributions: Distribution[], weights?: float[])`
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" default>
<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" default>
<SquiggleEditor initialSquiggleString="mixture(1 to 5, 8 to 10, 1, 3, 20)" />
</TabItem>
</Tabs>
### Arguments
- `distributions`: A set of distributions or floats, each passed as a paramater. Floats will be converted into Delta distributions.
- `weights`: An optional array of floats, 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 assignment. 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>
<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 assignment. I think I have an 80% chance of doing it.
</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:float, standardDeviation:float)`
<Tabs>
<TabItem value="ex1" label="normal(5,1)" default>
<SquiggleEditor initialSquiggleString="normal(5, 1)" />
</TabItem>
<TabItem value="ex2" label="normal(10m, 10m)" default>
<SquiggleEditor initialSquiggleString="normal(100000000000, 100000000000)" />
</TabItem>
</Tabs>
### Arguments
- `mean`: Float
- `standard deviation`: Float greater than zero
[Wikipedia entry](https://en.wikipedia.org/wiki/Normal_distribution)
## Log-normal
The log of `lognormal(mu, sigma)` is a normal distribution with mean `mu` and standard deviation `sigma`.
`lognormal(mu: float, sigma: float)`
<SquiggleEditor initialSquiggleString="lognormal(0, 0.7)" />
### Arguments
- `mu`: Float
- `sigma`: Float greater than zero
[Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)
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`
## Uniform
`normal(low:float, high:float)`
<Tabs>
<TabItem value="ex1" label="uniform(3,7)" default>
<SquiggleEditor initialSquiggleString="uniform(3,7)" />
</TabItem>
<TabItem value="ex2" label="invalid: uniform(7,5)" default>
<SquiggleEditor initialSquiggleString="uniform(7,5)" />
</TabItem>
</Tabs>
### Arguments
- `low`: Float
- `high`: Float greater than `low`
## Beta
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
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.

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@ -113,31 +113,6 @@ For `fromSamples(xs)`,
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