# particles.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:

## Univariate distributions¶

The module defines the following classes of univariate continuous distributions:

class (with signature)

Beta(a=1., b=1.)

Dirac(loc=0.)

Dirac mass at point loc

FlatNormal(loc=0.)

Normalp 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 intervalp [a,b]

Uniform(a=0., b=1.)

uniform over intervalp [a,b]

and the following classes of univariate discrete distributions:

class (with signature)

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 allp 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 labelp “missing”, and another component:

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

This particular distribution is usefulp 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 fullp 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 allp 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 callp 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 modelp 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 modelp 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 modelp 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:

Distribution

Corresponding model

Normalp

N(theta, sigma^2),

sigma fixed (passed as extra argument)

TruncNormalp

same

Gamma

N(0, 1/theta)

InvGamma

N(0, theta)

MvNormalp

N(theta, Sigma)

Sigma fixed (passed as extra argument)

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 willp 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 (Sequentialp quasi Monte Carlo).
 `IID`(law, k) Joint distribution of k iid (independent and identically distributed) variables.
 `Beta`([a, b]) Beta(a,b) distribution. `Binomial`([n, p]) Binomial(n,p) distribution. `Categorical`([p]) Categorical distribution. `Cond`(law[, dim, dtype]) Conditionalp distributions. `Dirac`([loc]) Dirac mass. Base class for discrete probability distributions. `DiscreteUniform`([lo, hi]) Discrete uniform distribution. `FlatNormal`([loc]) Normalp with infinite variance. `Gamma`([a, b]) Gamma(a,b) distribution, scale=1/b. `Geometric`([p]) Geometric(p) distribution. `IndepProd`(*dists) Product of independent univariate distributions. `InvGamma`([a, b]) Inverse Gamma(a,b) distribution. `Laplace`([loc, scale]) Laplace(loc,scale) distribution. `LinearD`(base_dist[, a, b]) Distribution of Y = a*X + b. `LocScaleDist`([loc, scale]) Base class for location-scale distributions. `LogD`(base_dist) Distribution of Y = log(X). `LogNormal`([mu, sigma]) Distribution of Y=e^X, with X ~ N(mu, sigma^2). `Logistic`([loc, scale]) Logistic(loc, scale) distribution. `LogitD`(base_dist[, a, b]) Distributions of Y=logit((X-a)/(b-a)). `MixMissing`([pmiss, base_dist]) Mixture between a given distribution and 'missing'. `Mixture`(pk, *components) Mixture distributions. `MvNormal`([loc, scale, cov]) Multivariate Normalp distribution. `NegativeBinomial`([n, p]) Negative Binomialp distribution. `Normal`([loc, scale]) N(loc, scale^2) distribution. `Poisson`([rate]) Poisson(rate) distribution. Base class for probability distributions. `StructDist`(laws) A distribution such that inputs/outputs are structured arrays. `Student`([df, loc, scale]) Student distribution. `TransformedDist`(base_dist) Base class for transformed distributions. `TruncNormal`([mu, sigma, a, b]) Normal(mu, sigma^2) truncated to [a, b] interval. `Uniform`([a, b]) Uniform([a,b]) distribution. `VaryingCovNormal`([loc, cov]) Multivariate Normalp (varying covariance matrix).