Dist

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, sample set, 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.

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.

Distributions are created as sample sets by default. To create a symbolic distribution, use Sym. namespace: Sym.normal, Sym.beta and so on.

Distributions

These are functions for creating primitive 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.

make

Signatures

Dist.make(Dist) => Dist

Dist.make(Number) => SymbolicDist

Examples

Dist.make(5)

Dist.make(normal({p5: 4, p95: 10}))

mixture

The mixture function takes a list of distributions and a list of weights, and returns a new distribution that is a mixture of the distributions in the list. The weights should be positive numbers that sum to 1. If no weights are provided, the function will assume that all distributions have equal weight.

Note: If you want to pass in over 5 distributions, you must use the list syntax.

Namespace optional

Signatures

Dist.mixture(List(Dist|Number), List(Number)?) => Dist

Dist.mixture(Dist|Number) => Dist

Dist.mixture(Dist|Number, Dist|Number, [Number, Number]?) => Dist

Dist.mixture(Dist|Number, Dist|Number, Dist|Number, [Number, Number, Number]?) => Dist

Dist.mixture(Dist|Number, Dist|Number, Dist|Number, Dist|Number, [Number, Number, Number, Number]?) => Dist

Examples

mixture(1,normal(5,2))

mixture(normal(5,2), normal(10,2), normal(15,2), [0.3, 0.5, 0.2])

mixture([normal(5,2), normal(10,2), normal(15,2), normal(20,1)], [0.3, 0.5, 0.1, 0.1])

mx

Alias for mixture()

Namespace optional

Signatures

Dist.mx(List(Dist|Number), List(Number)?) => Dist

Dist.mx(Dist|Number) => Dist

Dist.mx(Dist|Number, Dist|Number, [Number, Number]?) => Dist

Dist.mx(Dist|Number, Dist|Number, Dist|Number, [Number, Number, Number]?) => Dist

Dist.mx(Dist|Number, Dist|Number, Dist|Number, Dist|Number, [Number, Number, Number, Number]?) => Dist

Examples

mx(1,normal(5,2))

normal

Namespace optional

Signatures

Dist.normal(Dist|Number, Dist|Number) => SampleSetDist

Dist.normal({p5: Number, p95: Number}) => SampleSetDist

Dist.normal({p10: Number, p90: Number}) => SampleSetDist

Dist.normal({p25: Number, p75: Number}) => SampleSetDist

Dist.normal({mean: Number, stdev: Number}) => SampleSetDist

Examples

normal(5,1)

normal({p5: 4, p95: 10})

normal({p10: 4, p90: 10})

normal({p25: 4, p75: 10})

normal({mean: 5, stdev: 2})

lognormal

Namespace optional

Signatures

Dist.lognormal(Dist|Number, Dist|Number) => SampleSetDist

Dist.lognormal({p5: Number, p95: Number}) => SampleSetDist

Dist.lognormal({p10: Number, p90: Number}) => SampleSetDist

Dist.lognormal({p25: Number, p75: Number}) => SampleSetDist

Dist.lognormal({mean: Number, stdev: Number}) => SampleSetDist

Examples

lognormal(0.5, 0.8)

lognormal({p5: 4, p95: 10})

lognormal({p10: 4, p90: 10})

lognormal({p25: 4, p75: 10})

lognormal({mean: 5, stdev: 2})

uniform

Namespace optional

Signatures

Dist.uniform(Dist|Number, Dist|Number) => SampleSetDist

Examples

uniform(10, 12)

beta

Namespace optional

Signatures

Dist.beta(Dist|Number, Dist|Number) => SampleSetDist

Dist.beta({mean: Number, stdev: Number}) => SampleSetDist

Examples

beta(20, 25)

beta({mean: 0.39, stdev: 0.1})

cauchy

Namespace optional

Signatures

Dist.cauchy(Dist|Number, Dist|Number) => SampleSetDist

Examples

cauchy(5, 1)

gamma

Namespace optional

Signatures

Dist.gamma(Dist|Number, Dist|Number) => SampleSetDist

Examples

gamma(5, 1)

logistic

Namespace optional

Signatures

Dist.logistic(Dist|Number, Dist|Number) => SampleSetDist

Examples

logistic(5, 1)

to

The "to" function is a shorthand for lognormal({p5:min, p95:max}). It does not accept values of 0 or less, as those are not valid for lognormal distributions.

infix: to

Namespace optional

Signatures

Dist.to(Dist|Number, Dist|Number) => SampleSetDist

Examples

5 to 10

to(5,10)

exponential

Namespace optional

Signatures

Dist.exponential(Dist|Number) => SampleSetDist

Examples

exponential(2)

bernoulli

Namespace optional

Signatures

Dist.bernoulli(Dist|Number) => SampleSetDist

Examples

bernoulli(0.5)

triangular

Namespace optional

Signatures

Dist.triangular(Number, Number, Number) => SampleSetDist

Examples

triangular(3, 5, 10)

Basic Functions

mean

Namespace optional

Signatures

Dist.mean(Dist) => Number

median

Namespace optional

Signatures

Dist.median(Dist) => Number

stdev

Namespace optional

Signatures

Dist.stdev(Dist) => Number

variance

Namespace optional

Signatures

Dist.variance(Dist) => Number

min

Namespace optional

Signatures

Dist.min(Dist) => Number

max

Namespace optional

Signatures

Dist.max(Dist) => Number

mode

Namespace optional

Signatures

Dist.mode(Dist) => Number

sample

Namespace optional

Signatures

Dist.sample(Dist) => Number

sampleN

Namespace optional

Signatures

Dist.sampleN(Dist, Number) => List(Number)

exp

Namespace optional

Signatures

Dist.exp(Dist) => Dist

cdf

Namespace optional

Signatures

Dist.cdf(Dist, Number) => Number

pdf

Namespace optional

Signatures

Dist.pdf(Dist, Number) => Number

inv

Namespace optional

Signatures

Dist.inv(Dist, Number) => Number

quantile

Namespace optional

Signatures

Dist.quantile(Dist, Number) => Number

truncate

Truncates both the left side and the right side of a distribution.

Sample set distributions are truncated by filtering samples, but point set distributions are truncated using direct geometric manipulation. Uniform distributions are truncated symbolically. Symbolic but non-uniform distributions get converted to Point Set distributions.

Namespace optional

Signatures

Dist.truncate(Dist, Number, Number) => Dist

truncateLeft

Namespace optional

Signatures

Dist.truncateLeft(Dist, Number) => Dist

truncateRight

Namespace optional

Signatures

Dist.truncateRight(Dist, Number) => Dist

Algebra (Dist)

add

infix: +

Namespace optional

Signatures

Dist.add(Dist, Number) => Dist

Dist.add(Number, Dist) => Dist

Dist.add(Dist, Dist) => Dist

multiply

infix: *

Namespace optional

Signatures

Dist.multiply(Dist, Number) => Dist

Dist.multiply(Number, Dist) => Dist

Dist.multiply(Dist, Dist) => Dist

subtract

infix: -

Namespace optional

Signatures

Dist.subtract(Dist, Number) => Dist

Dist.subtract(Number, Dist) => Dist

Dist.subtract(Dist, Dist) => Dist

divide

infix: /

Namespace optional

Signatures

Dist.divide(Dist, Number) => Dist

Dist.divide(Number, Dist) => Dist

Dist.divide(Dist, Dist) => Dist

pow

infix: ^

Namespace optional

Signatures

Dist.pow(Dist, Number) => Dist

Dist.pow(Number, Dist) => Dist

Dist.pow(Dist, Dist) => Dist

log

Namespace optional

Signatures

Dist.log(Dist, Number) => Dist

Dist.log(Number, Dist) => Dist

Dist.log(Dist, Dist) => Dist

log

Namespace optional

Signatures

Dist.log(Dist, Number) => Dist

Dist.log(Number, Dist) => Dist

Dist.log(Dist, Dist) => Dist

log10

Namespace optional

Signatures

Dist.log10(Dist) => Dist

unaryMinus

unary: -

Namespace optional

Signatures

Dist.unaryMinus(Dist) => Dist

Algebra (List)

sum

Namespace optional

Signatures

Dist.sum(List(Dist|Number)) => Dist

product

Namespace optional

Signatures

Dist.product(List(Dist|Number)) => Dist

cumsum

Namespace optional

Signatures

Dist.cumsum(List(Dist|Number)) => List(Dist)

cumprod

Namespace optional

Signatures

Dist.cumprod(List(Dist|Number)) => List(Dist)

diff

Namespace optional

Signatures

Dist.diff(List(Dist|Number)) => List(Dist)

Pointwise Algebra

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. Mixture works using pointwise addition.

Pointwise operations work on Point Set distributions, so will convert other distributions to Point Set ones first. 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.

dotAdd

Signatures

Dist.dotAdd(Dist, Number) => Dist

Dist.dotAdd(Number, Dist) => Dist

Dist.dotAdd(Dist, Dist) => Dist

dotMultiply

Signatures

Dist.dotMultiply(Dist, Number) => Dist

Dist.dotMultiply(Number, Dist) => Dist

Dist.dotMultiply(Dist, Dist) => Dist

dotSubtract

Signatures

Dist.dotSubtract(Dist, Number) => Dist

Dist.dotSubtract(Number, Dist) => Dist

Dist.dotSubtract(Dist, Dist) => Dist

dotDivide

Signatures

Dist.dotDivide(Dist, Number) => Dist

Dist.dotDivide(Number, Dist) => Dist

Dist.dotDivide(Dist, Dist) => Dist

dotPow

Signatures

Dist.dotPow(Dist, Number) => Dist

Dist.dotPow(Number, Dist) => Dist

Dist.dotPow(Dist, Dist) => Dist

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. This only impacts Point Set distributions, because those are the only ones that can be non-normlized.

Namespace optional

Signatures

Dist.normalize(Dist) => Dist

isNormalized

Check if a distribution is normalized. This only impacts Point Set distributions, because those are the only ones that can be non-normlized. Most distributions are typically normalized, but there are some commands that could produce non-normalized distributions.

Namespace optional

Signatures

Dist.isNormalized(Dist) => Bool

integralSum

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.

Namespace optional

Signatures

Dist.integralSum(Dist) => Number

Utility

sparkline

Produce a sparkline of length n. For example, ▁▁▁▁▁▂▄▆▇██▇▆▄▂▁▁▁▁▁. These can be useful for testing or quick visualizations that can be copied and pasted into text.

Namespace optional

Signatures

Dist.sparkline(Dist, Number?) => String

Scoring

klDivergence

Kullback–Leibler divergence between two distributions.

Note that this can be very brittle. If the second distribution has probability mass at areas where the first doesn't, then the result will be infinite. Due to numeric approximations, some probability mass in point set distributions is rounded to zero, leading to infinite results with klDivergence.

Signatures

Dist.klDivergence(Dist, Dist) => Number

Examples

Dist.klDivergence(Sym.normal(5,2), Sym.normal(5,1.5))

logScore

A log loss score. Often that often acts as a scoring rule. Useful when evaluating the accuracy of a forecast.

Note that it is fairly slow.

Signatures

Dist.logScore({estimate: Dist, answer: Dist|Number, prior?: Dist}) => Number

Examples

Dist.logScore({estimate: Sym.normal(5,2), answer: Sym.normal(5.2,1), prior: Sym.normal(5.5,3)})

Dist.logScore({estimate: Sym.normal(5,2), answer: Sym.normal(5.2,1)})

Dist.logScore({estimate: Sym.normal(5,2), answer: 4.5})

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