Distribution
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.
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.
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
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
lognormal(0.5, 0.8);
lognormal({ p5: 4, p95: 10 });
lognormal({ mean: 5, stdev: 2 });
uniform
uniform: (distribution|number, distribution|number) => distribution
Examples
uniform(10, 12);
beta
beta: (distribution|number, distribution|number) => distribution
beta: (dict<{mean: distribution|number, stdev: distribution|number}>) => distribution
Examples
beta(20, 25);
beta({ mean: 0.39, stdev: 0.1 });
cauchy
cauchy: (distribution|number, distribution|number) => distribution
Examples
cauchy(5, 1);
gamma
gamma: (distribution|number, distribution|number) => distribution
Examples
gamma(5, 1);
logistic
logistic: (distribution|number, distribution|number) => distribution
Examples
gamma(5, 1);
exponential
exponential: (distribution|number) => distribution
Examples
exponential(2);
bernoulli
bernoulli: (distribution|number) => distribution
Examples
bernoulli(0.5);
triangular
triangular: (number, number, number) => distribution;
Examples
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
5 to 10
to(5,10)
-5 to 5
mixture
mixture: (...distributionLike, weights?:list<float>) => distribution
mixture: (list<distributionLike>, weights?:list<float>) => distribution
Examples
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
sample(normal(5, 2));
sampleN
N random samples from the distribution
sampleN: (distribution, number) => list<number>
Examples
sampleN(normal(5, 2), 100);
mean
The distribution mean
mean: (distribution) => number
Examples
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
cdf(normal(5, 2), 3);
pdf
pdf: (distribution, number) => number
Examples
pdf(normal(5, 2), 3);
quantile
quantile: (distribution, number) => number
Examples
quantile(normal(5, 2), 0.5);
truncate
Truncates both the left side and the right side of a distribution.
truncate: (distribution, left: number, right: number) => 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.
truncateLeft
Truncates the left side of a distribution.
truncateLeft: (distribution, left: number) => distribution
Examples
truncateLeft(normal(5, 2), 3);
truncateRight
Truncates the right side of a distribution.
truncateRight: (distribution, right: number) => distribution
Examples
truncateRight(normal(5, 2), 6);
klDivergence
Kullback–Leibler divergence between two distributions.
klDivergence: (distribution, distribution) => number
Examples
klDivergence(normal(5, 2), normal(5, 4)); // returns 0.57
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.
Dist.logScore: ({estimate: distribution, ?prior: distribution, answer: distribution|number}) => number
Examples
Dist.logScore({
estimate: normal(5, 2),
answer: normal(4.5, 1.2),
prior: normal(6, 4),
}); // returns -0.597.57
Display
toString
toString: (distribution) => string
Examples
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
toSparkline(truncateLeft(normal(5, 2), 3), 20); // produces ▁▇█████▇▅▄▃▂▂▁▁▁▁▁▁▁
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.
normalize: (distribution) => distribution
Examples
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
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
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.
pointMass(5 + 10) == pointMass(5) + pointMass(10);
add
add: (distributionLike, distributionLike) => distribution
Examples
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<distributionLike>) => distribution
Examples
sum([normal(0, 1), normal(1, 3), uniform(10, 1)]);
multiply
multiply: (distributionLike, distributionLike) => distribution
product
product: (list<distributionLike>) => 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
-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