Distributions¶

Distribution objects represent probability distributions, they have two principle uses:

Samples can be generated from a distribution by passing a distribution object to the sample operator.
The logarithm of the probability (or density) that a distribution assigns to a value can be computed using dist.score(val). For example:
```
Bernoulli({p: .1}).score(true); // returns Math.log(.1)
```

Several primitive distributions are built into the language. Further distributions are created by performing marginal inference.

Primitives¶

Bernoulli({p: ...})¶

p: success probability (real [0, 1])

Distribution over {true, false}

Wikipedia entry

Beta({a: ..., b: ...})¶

a: shape (real (0, Infinity))
b: shape (real (0, Infinity))

Distribution over [0, 1]

Wikipedia entry

Binomial({p: ..., n: ...})¶

p: success probability (real [0, 1])
n: number of trials (int (>=1))

Distribution over the number of successes for n independent Bernoulli({p: p}) trials.

Wikipedia entry

Categorical({ps: ..., vs: ...})¶

ps: probabilities (can be unnormalized) (vector or real array [0, Infinity))
vs: support (any array)

Distribution over elements of vs with P(vs[i]) proportional to ps[i]. ps may be omitted, in which case a uniform distribution over vs is returned.

Wikipedia entry

Cauchy({location: ..., scale: ...})¶

location: (real)
scale: (real (0, Infinity))

Distribution over [-Infinity, Infinity]

Wikipedia entry

Delta({v: ...})¶

v: support element (any)

Discrete distribution that assigns probability one to the single element in its support. This is only useful in special circumstances as sampling from Delta({v: val}) can be replaced with val itself. Furthermore, a Delta distribution parameterized by a random choice should not be used with MCMC based inference, as doing so produces incorrect results.

DiagCovGaussian({mu: ..., sigma: ...})¶

mu: mean (tensor)
sigma: standard deviations (tensor (0, Infinity))

A distribution over tensors in which each element is independent and Gaussian distributed, with its own mean and standard deviation. i.e. A multivariate Gaussian distribution with diagonal covariance matrix. The distribution is over tensors that have the same shape as the parameters mu and sigma, which in turn must have the same shape as each other.

Dirichlet({alpha: ...})¶

alpha: concentration (vector (0, Infinity))

Distribution over probability vectors. If alpha has length d then the distribution is over probability vectors of length d.

Wikipedia entry

Discrete({ps: ...})¶

ps: probabilities (can be unnormalized) (vector or real array [0, Infinity))

Distribution over {0,1,...,ps.length-1} with P(i) proportional to ps[i]

Wikipedia entry

Exponential({a: ...})¶

a: rate (real (0, Infinity))

Distribution over [0, Infinity]

Wikipedia entry

Gamma({shape: ..., scale: ...})¶

shape: (real (0, Infinity))
scale: (real (0, Infinity))

Distribution over positive reals.

Wikipedia entry

Gaussian({mu: ..., sigma: ...})¶

mu: mean (real)
sigma: standard deviation (real (0, Infinity))

Distribution over reals.

Wikipedia entry

KDE({data: ..., width: ...})¶

data: data array
width: kernel width

A distribution based on a kernel density estimate of data. A Gaussian kernel is used, and both real and vector valued data are supported. When the data are vector valued, width should be a vector specifying the kernel width for each dimension of the data. When width is omitted, Silverman’s rule of thumb is used to select a kernel width. This rule assumes the data are approximately Gaussian distributed. When this assumption does not hold, a width should be specified in order to obtain sensible results.

Wikipedia entry

Laplace({location: ..., scale: ...})¶

location: (real)
scale: (real (0, Infinity))

Distribution over [-Infinity, Infinity]

Wikipedia entry

LogisticNormal({mu: ..., sigma: ...})¶

mu: mean (vector)
sigma: standard deviations (vector (0, Infinity))

A distribution over probability vectors obtained by transforming a random variable drawn from DiagCovGaussian({mu: mu, sigma: sigma}). If mu and sigma have length d then the distribution is over probability vectors of length d+1.

Wikipedia entry

LogitNormal({mu: ..., sigma: ..., a: ..., b: ...})¶

mu: location (real)
sigma: scale (real (0, Infinity))
a: lower bound (real)
b: upper bound (>a) (real)

A distribution over (a,b) obtained by scaling and shifting a standard logit-normal.

Wikipedia entry

Mixture({dists: ..., ps: ...})¶

dists: array of component distributions
ps: component probabilities (can be unnormalized) (vector or real array [0, Infinity))

A finite mixture of distributions. The component distributions should be either all discrete or all continuous. All continuous distributions should share a common support.

Multinomial({ps: ..., n: ...})¶

ps: probabilities (real array with elements that sum to one)
n: number of trials (int (>=1))

Distribution over counts for n independent Discrete({ps: ps}) trials.

Wikipedia entry

MultivariateBernoulli({ps: ...})¶

ps: probabilities (vector [0, 1])

Distribution over a vector of independent Bernoulli variables. Each element of the vector takes on a value in {0, 1}. Note that this differs from Bernoulli which has support {true, false}.

MultivariateGaussian({mu: ..., cov: ...})¶

mu: mean (vector)
cov: covariance (positive definite matrix)

Multivariate Gaussian distribution with full covariance matrix. If mu has length d and cov is a d-by-d matrix, then the distribution is over vectors of length d.

Wikipedia entry

Poisson({mu: ...})¶