Distribution Creation

Normal

normal(mean: number, standardDeviation: number)
normal({mean: number, standardDeviation: number})
normal({p5: number, p95: number})
normal({p10: number, p90: number})
normal({p25: number, p75: number})

Creates a normal distribution (opens in a new tab) with the given mean and standard deviation.

Wikipedia (opens in a new tab)

Lognormal

lognormal(mu: number, sigma: number)
lognormal({mean: number, standardDeviation: number})
lognormal({p5: number, p95: number})
lognormal({p10: number, p90: number})
lognormal({p25: number, p75: number})

Creates a lognormal distribution (opens in a new tab) with the given mu and sigma.

Mu and sigma represent the mean and standard deviation of the normal which results when you take the log of our lognormal distribution. They can be difficult to directly reason about. However, there are several alternative ways to specify a lognormal distribution which are often easier to reason about.

Wikipedia (opens in a new tab)

❓ Understanding mu and sigma

The log of lognormal(mu, sigma) is a normal distribution with mean mu and standard deviation sigma. For example, these two distributions are identical:

To

(5thPercentile: number) to (95thPercentile: number)
to(5thPercentile: number, 95thPercentile: number)

The to function is an easy way to generate lognormal distributions using predicted 5th and 95th percentiles. It's the same as lognormal({p5, p95}), but easier to write and read.

Arguments

5thPercentile: number
95thPercentile: number, greater than 5thPercentile

💡

Tip
"To" is a great way to generate probability distributions very quickly from your intuitions. It's easy to write and easy to read. It's often a good place to begin an estimate.

⚠️

Caution
If you haven't tried calibration training (opens in a new tab), you're likely to be overconfident. We recommend doing calibration training to get a feel for what a 90 percent confident interval feels like.

Uniform

uniform(low:number, high:number)

Creates a uniform distribution (opens in a new tab) with the given low and high values.

Arguments

low: Number
high: Number greater than low

⚠️

Caution
While uniform distributions are very simple to understand, we find it rare to find uncertainties that actually look like this. Before using a uniform distribution, think hard about if you are really 100% confident that the paramater will not wind up being just outside the stated boundaries.
One good example of a uniform distribution uncertainty would be clear physical limitations. You might have complete complete uncertainty on what time of day an event will occur, but can say with 100% confidence it will happen between the hours of 0:00 and 24:00.

Point Mass

pointMass(value:number)

Creates a discrete distribution with all of its probability mass at point value.

Few Squiggle users call the function pointMass() directly. Numbers are often (but not always) converted into point mass distributions automatically, when it is appropriate.

For example, in the function mixture(1,2,normal(5,2)), the first two arguments will get converted into point mass distributions with values at 1 and 2. Therefore, this is the same as mixture(pointMass(1),pointMass(2),pointMass(5,2)).

pointMass() distributions are currently the only discrete distributions accessible in Squiggle.

Arguments

value: Number

Beta

beta(alpha:number, beta:number)
beta({mean: number, stdev: number})

Creates a beta distribution (opens in a new tab) with the given alpha and beta values. For a good summary of the beta distribution, see this explanation (opens in a new tab) on Stack Overflow.

Arguments

alpha: Number greater than zero
beta: Number greater than zero

⚠️

Caution with small numbers
Squiggle struggles to show beta distributions when either alpha or beta are below 1.0. This is because the tails at ~0.0 and ~1.0 are very high. Using a log scale for the y-axis helps here.

Examples

Mixture

mixture(...distributions: Distribution[], weights?: number[])
mixture(distributions: Distribution[], weights?: number[])
mx(...distributions: Distribution[], weights?: number[])
mx(distributions: Distribution[], weights?: number[])

The mixture mixes combines multiple distributions to create a mixture. You can optionally pass in a list of proportional weights.

Arguments

distributions: A set of distributions or numbers, each passed as a paramater. Numbers will be converted into point mass distributions.
weights: An optional array of numbers, each representing the weight of its corresponding distribution. The weights will be re-scaled to add to 1.0. If a weights array is provided, it must be the same length as the distribution paramaters.

Aliases

mx

Special Use Cases of Mixtures

🕐 Zero or Continuous

One common reason to have mixtures of continous and discrete distributions is to handle the special case of 0. Say I want to model the time I will spend on some upcoming project. I think I have an 80% chance of doing it.

In this case, I have a 20% chance of spending 0 time with it. I might estimate my hours with,

🔒 Model Uncertainty Safeguarding

One technique several Foretold.io users used is to combine their main guess, with a "just-in-case distribution". This latter distribution would have very low weight, but would be very wide, just in case they were dramatically off for some weird reason.

SampleSet.fromList

SampleSet.fromList(samples:number[])

Creates a sample set distribution using an array of samples.

Samples are converted into PDFs automatically using kernel density estimation (opens in a new tab) and an approximated bandwidth. This is an approximation and can be error-prone.

Arguments

samples: An array of at least 5 numbers.

PointSet.makeContinuous

PointSet.makeContinuous(points:{x: number, y: number})

Creates a continuous point set distribution using a list of points.

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Caution!
Distributions made with makeContinuous are not automatically normalized. We suggest normalizing them manually using the normalize function.

Arguments

points: An array of at least 3 coordinates.

PointSet.makeDiscrete

PointSet.makeDiscrete(points:{x: number, y: number})

Creates a discrete point set distribution using a list of points.

Arguments

points: An array of at least 1 coordinate.

Introduction Distribution Functions