# distributions¶

Probability distributions as Python objects.

## Overview¶

This module lets users define probability distributions as Python objects.

The probability distributions defined in this module may be used:

• to define state-space models (see module state_space_models);
• to define a prior distribution, in order to perform parameter estimation (see modules smc_samplers and mcmc).

## Univariate distributions¶

The module defines the following classes of univariate continuous distributions:

Beta(a=1., b=1.)
Dirac(loc=0.) Dirac mass at point loc
FlatNormal(loc=0.) Normal with inf variance (missing data)
Gamma(a=1., b=1.) scale = 1/b
InvGamma(a=1., b=1.) Distribution of 1/X for X~Gamma(a,b)
Laplace(loc=0., scale=1.)
Logistic(loc=0., scale=1.)
LogNormal(mu=0., sigma=1.) Dist of Y=e^X, X ~ N(μ, σ^2)
Normal(loc=0., scale=1.) N(loc,scale^2) distribution
Student(loc=0., scale=1., df=3)
TruncNormal(mu=0, sigma=1., a=0., b=1.) N(mu, sigma^2) truncated to interval [a,b]
Uniform(a=0., b=1.) uniform over interval [a,b]

and the following classes of univariate discrete distributions:

Binomial(n=1, p=0.5)
Categorical(p=None) returns i with prob p[i]
DiscreteUniform(lo=0, hi=2) uniform over a, …, b-1
Geometric(p=0.5)
Poisson(rate=1.) Poisson with expectation rate

Note that all the parameters of these distributions have default values, e.g.:

some_norm = Normal(loc=2.4)  # N(2.4, 1)
some_gam = Gamma()  # Gamma(1, 1)


## Mixture distributions (new in version 0.4)¶

A (univariate) mixture distribution may be specified as follows:

mix = Mixture([0.5, 0.5], Normal(loc=-1), Normal(loc=1.))


The first argument is the vector of probabilities, the next arguments are the k component distributions.

See also MixMissing for defining a mixture distributions, between one component that generates the label “missing”, and another component:

mixmiss = MixMissing(pmiss=0.1, base_dist=Normal(loc=2.))


This particular distribution is useful to specify a state-space model where the observation may be missing with a certain probability.

## Transformed distributions¶

To further enrich the list of available univariate distributions, the module lets you define transformed distributions, that is, the distribution of Y=f(X), for a certain function f, and a certain base distribution for X.

class name (and signature) description

LinearD(base_dist, a=1., b=0.) Y = a * X + b
LogD(base_dist) Y = log(X)
LogitD(base_dist, a=0., b=1.) Y = logit( (X-a)/(b-a) )

A quick example:

from particles import distributions as dists
d = dists.LogD(dists.Gamma(a=2., b=2.))  # law of Y=log(X), X~Gamma(2, 2)


Note

These transforms are often used to obtain random variables defined over the full real line. This is convenient in particular when implementing random walk Metropolis steps.

## Multivariate distributions¶

The module implements one multivariate distribution class, for Gaussian distributions; see MvNormal.

Furthermore, the module provides two ways to construct multivariate distributions from a collection of univariate distributions:

## Under the hood¶

Probability distributions are represented as objects of classes that inherit from base class ProbDist, and implement the following methods:

• logpdf(self, x): computes the log-pdf (probability density function) at point x;
• rvs(self, size=None): simulates size random variates; (if set to None, number of samples is either one if all parameters are scalar, or the same number as the common size of the parameters, see below);
• ppf(self, u): computes the quantile function (or Rosenblatt transform for a multivariate distribution) at point u.

A quick example:

some_dist = dists.Normal(loc=2., scale=3.)
x = some_dist.rvs(size=30)  # a (30,) ndarray containing IID N(2, 3^2) variates
z = some_dist.logpdf(x)  # a (30,) ndarray containing the log-pdf at x


By default, the inputs and outputs of these methods are either scalars or Numpy arrays (with appropriate type and shape). In particular, passing a Numpy array to a distribution parameter makes it possible to define “array distributions”. For instance:

some_dist = dists.Normal(loc=np.arange(1., 11.))
x = some_dist.rvs(size=10)


generates 10 Gaussian-distributed variates, with respective means 1., …, 10. This is how we manage to define “Markov kernels” in state-space models; e.g. when defining the distribution of X_t given X_{t-1} in a state-space model:

class StochVol(ssm.StateSpaceModel):
def PX(self, t, xp, x):
return stats.norm(loc=xp)
### ... see module state_space_models for more details


Then, in practice, in e.g. the bootstrap filter, when we generate particles X_t^n, we call method PX and pass as an argument a numpy array of shape (N,) containing the N ancestors.

Note

ProbDist objects are roughly similar to the frozen distributions of package scipy.stats. However, they are not equivalent. Using such a frozen distribution when e.g. defining a state-space model will return an error.

## Posterior distributions¶

A few classes also implement a posterior method, which returns the posterior distribution that corresponds to a prior set to self, a model which is conjugate for the considered class, and some data. Here is a quick example:

from particles import distributions as dists
prior = dists.InvGamma(a=.3, b=.3)
data = random.randn(20)  # 20 points generated from N(0,1)
post = prior.posterior(data)
# prior is conjugate wrt model X_1, ..., X_n ~ N(0, theta)
print("posterior is Gamma(%f, %f)" % (post.a, post.b))


Here is a list of distributions implementing posteriors:

If you would like to create your own univariate probability distribution, the easiest way to do so is to sub-class ProbDist, for a continuous distribution, or DiscreteDist, for a discrete distribution. This will properly set class attributes dim (the dimension, set to one, for a univariate distribution), and dtype, so that they play nicely with StructDist and so on. You will also have to properly define methods rvs, logpdf and ppf. You may omit ppf if you do not plan to use SQMC (Sequential quasi Monte Carlo).
 DiscreteDist Base class for discrete probability distributions. Beta Beta(a,b) distribution. Dirac Dirac mass. FlatNormal Normal with infinite variance. Gamma Gamma(a,b) distribution, scale=1/b. InvGamma Inverse Gamma(a,b) distribution. Laplace Laplace(loc,scale) distribution. Logistic Logistic(loc, scale) distribution. LogNormal Distribution of Y=e^X, with X ~ N(mu, sigma^2). Normal N(loc, scale^2) distribution. Student Student distribution. TruncNormal Normal(mu, sigma^2) truncated to [a, b] interval. Uniform Uniform([a,b]) distribution. Binomial Binomial(n,p) distribution. Geometric Geometric(p) distribution. Poisson Poisson(rate) distribution. LinearD Distribution of Y = a*X + b. LogD Distribution of Y = log(X). LogitD Distributions of Y=logit((X-a)/(b-a)). MvNormal Multivariate Normal distribution. IndepProd Product of independent univariate distributions. ProbDist Base class for probability distributions. StructDist A distribution such that inputs/outputs are structured arrays.