233 lines
6.3 KiB
Plaintext
233 lines
6.3 KiB
Plaintext
---
|
|
title: "Distributions: Key Functions"
|
|
sidebar_position: 3
|
|
---
|
|
|
|
import { SquiggleEditor } from "../../src/components/SquiggleEditor";
|
|
|
|
## Operating on distributions
|
|
|
|
Here are the ways we combine distributions.
|
|
|
|
### Addition
|
|
|
|
A horizontal right shift. The addition operation represents the distribution of the sum of
|
|
the value of one random sample chosen from the first distribution and the value one random sample
|
|
chosen from the second distribution.
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist1 = 1 to 10
|
|
dist2 = triangular(1,2,3)
|
|
dist1 + dist2`}
|
|
/>
|
|
|
|
### Subtraction
|
|
|
|
A horizontal left shift. A horizontal right shift. The substraction operation represents
|
|
the distribution of the value of one random sample chosen from the first distribution minus
|
|
the value of one random sample chosen from the second distribution.
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist1 = 1 to 10
|
|
dist2 = triangular(1,2,3)
|
|
dist1 - dist2`}
|
|
/>
|
|
|
|
### Multiplication
|
|
|
|
A proportional scaling. The addition operation represents the distribution of the multiplication of
|
|
the value of one random sample chosen from the first distribution times the value one random sample
|
|
chosen from the second distribution.
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist1 = 1 to 10
|
|
dist2 = triangular(1,2,3)
|
|
dist1 * dist2`}
|
|
/>
|
|
|
|
We also provide concatenation of two distributions as a syntax sugar for `*`
|
|
|
|
<SquiggleEditor defaultCode="(0.1 to 1) * triangular(1,2,3)" />
|
|
|
|
### Division
|
|
|
|
A proportional scaling (normally a shrinking if the second distribution has values higher than 1).
|
|
The addition operation represents the distribution of the division of
|
|
the value of one random sample chosen from the first distribution over the value one random sample
|
|
chosen from the second distribution. If the second distribution has some values near zero, it
|
|
tends to be particularly unstable.
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist1 = 1 to 10
|
|
dist2 = triangular(1,2,3)
|
|
dist1 / dist2`}
|
|
/>
|
|
|
|
### Exponentiation
|
|
|
|
A projection over a contracted x-axis. The exponentiation operation represents the distribution of
|
|
the exponentiation of the value of one random sample chosen from the first distribution to the power of
|
|
the value one random sample chosen from the second distribution.
|
|
|
|
<SquiggleEditor defaultCode={`(0.1 to 1) ^ beta(2, 3)`} />
|
|
|
|
### Taking the base `e` exponential
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist = triangular(1,2,3)
|
|
exp(dist)`}
|
|
/>
|
|
|
|
### Taking logarithms
|
|
|
|
A projection over a stretched x-axis.
|
|
|
|
<SquiggleEditor
|
|
defaultCode={`dist = triangular(1,2,3)
|
|
log(dist)`}
|
|
/>
|
|
|
|
<SquiggleEditor defaultCode={`log10(5 to 10)`} />
|
|
|
|
Base `x`
|
|
|
|
<SquiggleEditor defaultCode={`log(5 to 10, 2)`} />
|
|
|
|
#### Validity
|
|
|
|
- `x` must be a scalar
|
|
- See [the current discourse](https://github.com/quantified-uncertainty/squiggle/issues/304)
|
|
|
|
### Pointwise addition
|
|
|
|
For every point on the x-axis, operate the corresponding points in the y axis of the pdf.
|
|
|
|
**Pointwise operations are done with `PointSetDist` internals rather than `SampleSetDist` internals**.
|
|
|
|
TODO: this isn't in the new interpreter/parser yet.
|
|
|
|
<SquiggleEditor defaultCode={`(1 to 10) .+ triangular(1,2,3)`} />
|
|
|
|
### Pointwise subtraction
|
|
|
|
TODO: this isn't in the new interpreter/parser yet.
|
|
|
|
<SquiggleEditor defaultCode={`(1 to 10) .- triangular(1,2,3)`} />
|
|
|
|
### Pointwise multiplication
|
|
|
|
<SquiggleEditor defaultCode={`(1 to 10) .* triangular(1,2,3)`} />
|
|
|
|
### Pointwise division
|
|
|
|
<SquiggleEditor defaultCode={`(uniform(0,10)) ./ normal(10,4)`} />
|
|
|
|
### Pointwise exponentiation
|
|
|
|
<SquiggleEditor defaultCode={`(1 to 10) .^ triangular(1,2,3)`} />
|
|
|
|
## Standard functions on distributions
|
|
|
|
### Probability density function
|
|
|
|
The `pdf(dist, x)` function returns the density of a distribution at the
|
|
given point x.
|
|
|
|
<SquiggleEditor defaultCode="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 `quantile`.
|
|
|
|
<SquiggleEditor defaultCode="cdf(normal(0,1),0)" />
|
|
|
|
#### Validity
|
|
|
|
- `x` must be a scalar
|
|
- `dist` must be a distribution
|
|
|
|
### Quantile
|
|
|
|
The `quantile(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`. In the literature, it
|
|
is also known as the quantiles function.
|
|
|
|
<SquiggleEditor defaultCode="quantile(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 defaultCode="mean(normal(5, 10))" />
|
|
|
|
### Sampling a distribution
|
|
|
|
The `sample(distribution)` samples a given distribution.
|
|
|
|
<SquiggleEditor defaultCode="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 defaultCode="SampleSet.fromDist(normal(5, 10))" />
|
|
|
|
Or `PointSet` format
|
|
|
|
<SquiggleEditor defaultCode="PointSet.fromDist(normal(5, 10))" />
|
|
|
|
#### Validity
|
|
|
|
- Second argument to `SampleSet.fromDist` must be a number.
|
|
|
|
## Normalization
|
|
|
|
Some distribution operations (like horizontal shift) return an unnormalized distriibution.
|
|
|
|
We provide a `normalize` function
|
|
|
|
<SquiggleEditor defaultCode="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 defaultCode="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 defaultCode="inspect(SampleSet.fromDist(0.1 to 1))" />
|
|
|
|
Save for a logging side effect, `inspect` does nothing to input and returns it.
|
|
|
|
## Truncate
|
|
|
|
You can cut off from the left
|
|
|
|
<SquiggleEditor defaultCode="truncateLeft(0.1 to 1, 0.5)" />
|
|
|
|
You can cut off from the right
|
|
|
|
<SquiggleEditor defaultCode="truncateRight(0.1 to 1, 0.5)" />
|
|
|
|
You can cut off from both sides
|
|
|
|
<SquiggleEditor defaultCode="truncate(0.1 to 1, 0.5, 1.5)" />
|