266 lines
6.9 KiB
Plaintext
266 lines
6.9 KiB
Plaintext
---
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title: "Distribution Functions"
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sidebar_position: 3
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---
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import { SquiggleEditor } from "../../src/components/SquiggleEditor";
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## Operating on distributions
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Here are the ways we combine distributions.
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### Addition
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A horizontal right shift. The addition operation represents the distribution of the sum of
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the value of one random sample chosen from the first distribution and the value one random sample
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chosen from the second distribution.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 + dist2`}
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/>
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### Subtraction
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A horizontal left shift. A horizontal right shift. The substraction operation represents
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the distribution of the value of one random sample chosen from the first distribution minus
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the value of one random sample chosen from the second distribution.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 - dist2`}
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/>
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### Multiplication
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A proportional scaling. The addition operation represents the distribution of the multiplication of
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the value of one random sample chosen from the first distribution times the value one random sample
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chosen from the second distribution.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 * dist2`}
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/>
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We also provide concatenation of two distributions as a syntax sugar for `*`
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<SquiggleEditor squiggleString="(0.1 to 1) triangular(1,2,3)" />
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### Division
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A proportional scaling (normally a shrinking if the second distribution has values higher than 1).
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The addition operation represents the distribution of the division of
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the value of one random sample chosen from the first distribution over the value one random sample
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chosen from the second distribution. If the second distribution has some values near zero, it
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tends to be particularly unstable.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 / dist2`}
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/>
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### Exponentiation
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A projection over a contracted x-axis. The exponentiation operation represents the distribution of
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the exponentiation of the value of one random sample chosen from the first distribution to the power of
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the value one random sample chosen from the second distribution.
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<SquiggleEditor squiggleString={`(0.1 to 1) ^ beta(2, 3)`} />
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### Taking the base `e` exponential
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<SquiggleEditor
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squiggleString={`dist = triangular(1,2,3)
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exp(dist)`}
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/>
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### Taking logarithms
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A projection over a stretched x-axis.
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<SquiggleEditor
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squiggleString={`dist = triangular(1,2,3)
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log(dist)`}
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/>
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<SquiggleEditor
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squiggleString={`dist = beta(1,2)
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log10(dist)`}
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/>
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Base `x`
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<SquiggleEditor
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squiggleString={`x = 2
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dist = beta(2,3)
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log(dist, x)`}
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/>
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#### Validity
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- `x` must be a scalar
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- See [the current discourse](https://github.com/quantified-uncertainty/squiggle/issues/304)
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### Pointwise addition
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For every point on the x-axis, operate the corresponding points in the y axis of the pdf.
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**Pointwise operations are done with `PointSetDist` internals rather than `SampleSetDist` internals**.
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TODO: this isn't in the new interpreter/parser yet.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .+ dist2`}
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/>
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### Pointwise subtraction
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TODO: this isn't in the new interpreter/parser yet.
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .- dist2`}
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/>
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### Pointwise multiplication
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .* dist2`}
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/>
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### Pointwise division
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<SquiggleEditor
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squiggleString={`dist1 = uniform(0,20)
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dist2 = normal(10,8)
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dist1 ./ dist2`}
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/>
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### Pointwise exponentiation
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<SquiggleEditor
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squiggleString={`dist1 = 1 to 10
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dist2 = triangular(1,2,3)
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dist1 .^ dist2`}
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/>
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## Standard functions on distributions
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### Probability density function
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The `pdf(dist, x)` function returns the density of a distribution at the
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given point x.
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<SquiggleEditor squiggleString="pdf(normal(0,1),0)" />
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#### Validity
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- `x` must be a scalar
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- `dist` must be a distribution
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### Cumulative density function
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The `cdf(dist, x)` gives the cumulative probability of the distribution
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or all values lower than x. It is the inverse of `quantile`.
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<SquiggleEditor squiggleString="cdf(normal(0,1),0)" />
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#### Validity
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- `x` must be a scalar
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- `dist` must be a distribution
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### Quantile
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The `quantile(dist, prob)` gives the value x or which the probability for all values
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lower than x is equal to prob. It is the inverse of `cdf`. In the literature, it
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is also known as the quantiles function.
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<SquiggleEditor squiggleString="quantile(normal(0,1),0.5)" />
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#### Validity
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- `prob` must be a scalar (please only put it in `(0,1)`)
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- `dist` must be a distribution
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### Mean
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The `mean(distribution)` function gives the mean (expected value) of a distribution.
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<SquiggleEditor squiggleString="mean(normal(5, 10))" />
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### Sampling a distribution
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The `sample(distribution)` samples a given distribution.
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<SquiggleEditor squiggleString="sample(normal(0, 10))" />
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## Converting between distribution formats
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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
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<SquiggleEditor squiggleString="toSampleSet(normal(5, 10))" />
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Or `PointSet` format
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<SquiggleEditor squiggleString="toPointSet(normal(5, 10))" />
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### `toSampleSet` has two signatures
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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)
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<SquiggleEditor squiggleString="[toSampleSet(0.1 to 1, 100.1), toSampleSet(0.1 to 1, 5000), toSampleSet(0.1 to 1, 20000)]" />
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#### Validity
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- Second argument to `toSampleSet` must be a number.
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## Normalization
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Some distribution operations (like horizontal shift) return an unnormalized distriibution.
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We provide a `normalize` function
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<SquiggleEditor squiggleString="normalize((0.1 to 1) + triangular(0.1, 1, 10))" />
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#### Validity - Input to `normalize` must be a dist
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We provide a predicate `isNormalized`, for when we have simple control flow
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<SquiggleEditor squiggleString="isNormalized((0.1 to 1) * triangular(0.1, 1, 10))" />
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#### Validity
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- Input to `isNormalized` must be a dist
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## `inspect`
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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.
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<SquiggleEditor squiggleString="inspect(toSampleSet(0.1 to 1, 100))" />
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Save for a logging side effect, `inspect` does nothing to input and returns it.
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## Truncate
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You can cut off from the left
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<SquiggleEditor squiggleString="truncateLeft(0.1 to 1, 0.5)" />
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You can cut off from the right
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<SquiggleEditor squiggleString="truncateRight(0.1 to 1, 0.5)" />
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You can cut off from both sides
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<SquiggleEditor squiggleString="truncate(0.1 to 1, 0.5, 1.5)" />
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