--- sidebar_position: 3 title: Distribution --- import Admonition from "@theme/Admonition"; import TOCInline from "@theme/TOCInline"; Distributions are the flagship data type in Squiggle. The distribution type is a generic data type that contains one of three different formats of distributions. These subtypes are [point set](/docs/Api/DistPointSet), [sample set](/docs/Api/DistSampleSet), and symbolic. The first two of these have a few custom functions that only work on them. You can read more about the differences between these formats [here](/docs/Discussions/Three-Formats-Of-Distributions). Several functions below only can work on particular distribution formats. For example, scoring and pointwise math requires the point set format. When this happens, the types are automatically converted to the correct format. These conversions are lossy. ## Distribution Creation These are functions for creating primative distributions. Many of these could optionally take in distributions as inputs. In these cases, Monte Carlo Sampling will be used to generate the greater distribution. This can be used for simple hierarchical models. See a longer tutorial on creating distributions [here](/docs/Guides/DistributionCreation). ### normal ``` normal: (distribution|number, distribution|number) => distribution normal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution normal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution ``` **Examples** ```js normal(5, 1) normal({ p5: 4, p95: 10 }) normal({ mean: 5, stdev: 2 }) normal(5 to 10, normal(3, 2)) normal({ mean: uniform(5, 9), stdev: 3 }) ``` ### lognormal ``` lognormal: (distribution|number, distribution|number) => distribution lognormal: (dict<{p5: distribution|number, p95: distribution|number}>) => distribution lognormal: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution ``` **Examples** ```javascript lognormal(0.5, 0.8); lognormal({ p5: 4, p95: 10 }); lognormal({ mean: 5, stdev: 2 }); ``` ### uniform ``` uniform: (distribution|number, distribution|number) => distribution ``` **Examples** ```javascript uniform(10, 12); ``` ### beta ``` beta: (distribution|number, distribution|number) => distribution beta: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution ``` **Examples** ```javascript beta(20, 25); beta({ mean: 0.39, stdev: 0.1 }); ``` ### cauchy ``` cauchy: (distribution|number, distribution|number) => distribution ``` **Examples** ```javascript cauchy(5, 1); ``` ### gamma ```javascript gamma: (distribution|number, distribution|number) => distribution ``` **Examples** ```js gamma(5, 1); ``` ### logistic ``` logistic: (distribution|number, distribution|number) => distribution ``` **Examples** ```javascript gamma(5, 1); ``` ### exponential ``` exponential: (distribution|number) => distribution ``` **Examples** ```javascript exponential(2); ``` ### bernoulli ``` bernoulli: (distribution|number) => distribution ``` **Examples** ```javascript bernoulli(0.5); ``` ### triangular ```javascript triangular: (number, number, number) => distribution; ``` **Examples** ```javascript triangular(5, 10, 20); ``` ### to / credibleIntervalToDistribution 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. `To` is an alias for `credibleIntervalToDistribution`. However, because of its frequent use, it is recommended to use the shorter name. ``` to: (distribution|number, distribution|number) => distribution credibleIntervalToDistribution(distribution|number, distribution|number) => distribution ``` **Examples** ```javascript 5 to 10 to(5,10) -5 to 5 ``` ### mixture ``` mixture: (...distributionLike, weights?:list) => distribution mixture: (list, weights?:list) => distribution ``` **Examples** ```javascript mixture(normal(5, 1), normal(10, 1), 8); mx(normal(5, 1), normal(10, 1), [0.3, 0.7]); mx([normal(5, 1), normal(10, 1)], [0.3, 0.7]); ``` ## Functions ### sample One random sample from the distribution ``` sample: (distribution) => number ``` **Examples** ```javascript sample(normal(5, 2)); ``` ### sampleN N random samples from the distribution ``` sampleN: (distribution, number) => list ``` **Examples** ```javascript sampleN(normal(5, 2), 100); ``` ### mean The distribution mean ``` mean: (distribution) => number ``` **Examples** ```javascript mean(normal(5, 2)); ``` ### stdev Standard deviation. Only works now on sample set distributions (so converts other distributions into sample set in order to calculate.) ``` stdev: (distribution) => number ``` ### variance Variance. Similar to stdev, only works now on sample set distributions. ``` variance: (distribution) => number ``` ### mode ``` mode: (distribution) => number ``` ### cdf ``` cdf: (distribution, number) => number ``` **Examples** ```javascript cdf(normal(5, 2), 3); ``` ### pdf ``` pdf: (distribution, number) => number ``` **Examples** ```javascript pdf(normal(5, 2), 3); ``` ### quantile ``` quantile: (distribution, number) => number ``` **Examples** ```javascript quantile(normal(5, 2), 0.5); ``` ### toPointSet **TODO: Will soon be called "PointSet.make"** Converts a distribution to the pointSet format. ``` toPointSet: (distribution) => pointSetDistribution ``` **Examples** ```javascript toPointSet(normal(5, 2)); ``` ### toSampleSet **TODO: Will soon be called "SampleSet.make"** Converts a distribution to the sampleSet format, with n samples. ``` toSampleSet: (distribution, number) => sampleSetDistribution ``` **Examples** ```javascript toSampleSet(normal(5, 2), 1000); ``` ### truncateLeft Truncates the left side of a distribution. Returns either a pointSet distribution or a symbolic distribution. ``` truncateLeft: (distribution, l => number) => distribution ``` **Examples** ```javascript truncateLeft(normal(5, 2), 3); ``` ### truncateRight Truncates the right side of a distribution. Returns either a pointSet distribution or a symbolic distribution. ``` truncateRight: (distribution, r => number) => distribution ``` **Examples** ```javascript truncateLeft(normal(5, 2), 6); ``` ### klDivergence [Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between two distributions. ``` klDivergence: (distribution, distribution) => number ``` **Examples** ```javascript klDivergence(normal(5, 2), normal(5, 4)); // returns 0.57 ``` ## Display ### toString ``` toString: (distribution) => string ``` **Examples** ```javascript toString(normal(5, 2)); ``` ### sparkline Produce a sparkline of length n. For example, `▁▁▁▁▁▂▄▆▇██▇▆▄▂▁▁▁▁▁`. These can be useful for testing or quick text visualizations. ``` sparkline: (distribution, n = 20) => string ``` **Examples** ```javascript toSparkline(truncateLeft(normal(5, 2), 3), 20); // produces ▁▇█████▇▅▄▃▂▂▁▁▁▁▁▁▁ ``` ### inspect Prints the value of the distribution to the Javascript console, then returns the distribution. Useful for debugging. ``` inspect: (distribution) => distribution ``` **Examples** ```javascript inspect(normal(5, 2)); // logs "normal(5, 2)" to the javascript console and returns the distribution. ``` ## Normalization There are some situations where computation will return unnormalized distributions. This means that their cumulative sums are not equal to 1.0. Unnormalized distributions are not valid for many relevant functions; for example, klDivergence and scoring. The only functions that do not return normalized distributions are the pointwise arithmetic operations and the scalewise arithmetic operations. If you use these functions, it is recommended that you consider normalizing the resulting distributions. ### normalize Normalize a distribution. This means scaling it appropriately so that it's cumulative sum is equal to 1. ``` normalize: (distribution) => distribution ``` **Examples** ```javascript normalize(normal(5, 2)); ``` ### isNormalized Check of a distribution is normalized. Most distributions are typically normalized, but there are some commands that could produce non-normalized distributions. ``` isNormalized: (distribution) => bool ``` **Examples** ```javascript isNormalized(normal(5, 2)); // returns true ``` ### integralSum **Note: If you have suggestions for better names for this, please let us know.** Get the sum of the integral of a distribution. If the distribution is normalized, this will be 1.0. This is useful for understanding unnormalized distributions. ``` integralSum: (distribution) => number ``` **Examples** ```javascript integralSum(normal(5, 2)); ``` ## Regular Arithmetic Operations Regular arithmetic operations cover the basic mathematical operations on distributions. They work much like their equivalent operations on numbers. The infixes `+`,`-`, `*`, `/`, `^` are supported for addition, subtraction, multiplication, division, power, and unaryMinus. ```javascript pointMass(5 + 10) == pointMass(5) + pointMass(10); ``` ### add ``` add: (distributionLike, distributionLike) => distribution ``` **Examples** ```javascript normal(0, 1) + normal(1, 3); // returns normal(1, 3.16...) add(normal(0, 1), normal(1, 3)); // returns normal(1, 3.16...) ``` ### sum **Todo: Not yet implemented** ``` sum: (list) => distribution ``` **Examples** ```javascript sum([normal(0, 1), normal(1, 3), uniform(10, 1)]); ``` ### multiply ``` multiply: (distributionLike, distributionLike) => distribution ``` ### product ``` product: (list) => distribution ``` ### subtract ``` subtract: (distributionLike, distributionLike) => distribution ``` ### divide ``` divide: (distributionLike, distributionLike) => distribution ``` ### pow ``` pow: (distributionLike, distributionLike) => distribution ``` ### exp ``` exp: (distributionLike, distributionLike) => distribution ``` ### log ``` log: (distributionLike, distributionLike) => distribution ``` ### log10 ``` log10: (distributionLike, distributionLike) => distribution ``` ### unaryMinus ``` unaryMinus: (distribution) => distribution ``` **Examples** ```javascript -normal(5, 2); // same as normal(-5, 2) unaryMinus(normal(5, 2)); // same as normal(-5, 2) ``` ## Pointwise Arithmetic Operations

Pointwise arithmetic operations typically return unnormalized or completely invalid distributions. For example, the operation{" "} normal(5,2) .- uniform(10,12) results in a distribution-like object with negative probability mass.

Pointwise arithmetic operations cover the standard arithmetic operations, but work in a different way than the regular operations. These operate on the y-values of the distributions instead of the x-values. A pointwise addition would add the y-values of two distributions. The infixes `.+`,`.-`, `.*`, `./`, `.^` are supported for their respective operations. The `mixture` methods works with pointwise addition. ### dotAdd ``` dotAdd: (distributionLike, distributionLike) => distribution ``` ### dotMultiply ``` dotMultiply: (distributionLike, distributionLike) => distribution ``` ### dotSubtract ``` dotSubtract: (distributionLike, distributionLike) => distribution ``` ### dotDivide ``` dotDivide: (distributionLike, distributionLike) => distribution ``` ### dotPow ``` dotPow: (distributionLike, distributionLike) => distribution ``` ### dotExp ``` dotExp: (distributionLike, distributionLike) => distribution ``` ## Scale Arithmetic Operations

We're planning on removing scale operations in favor of more general functions soon.

Scale operations are similar to pointwise operations, but operate on a constant y-value instead of y-values coming from a distribution. You can think about this as scaling a distribution vertically by a constant. The following items would be equivalent. ```js scalePow(normal(5,2), 2) mapY(normal(5,2), {|y| y ^ 2}) // Not yet available ``` ### scaleMultiply ``` scaleMultiply: (distributionLike, number) => distribution ``` ### scalePow ``` scalePow: (distributionLike, number) => distribution ``` ### scaleExp ``` scaleExp: (distributionLike, number) => distribution ``` ### scaleLog ``` scaleLog: (distributionLike, number) => distribution ``` ### scaleLog10 ``` scaleLog10: (distributionLike, number) => distribution ```