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Random Variables

What I cannot create, I do not understand - Richard Feynman

Summary

Since version 1.4 a new package called probability has been added to the core api with an aim to aid in the modeling of random variables and measurable functions.

Random variables and probability distributions form the bedrock of modern statistical based approaches to inference. Furthermore, analytically tractable inference is only possible for a small number of models while a wealth of interesting model structures don't yield themselves to analytical inference and approximate sampling based approaches are often employed.

Random Variable API

Although both random variable with tractable and intractable distributions can be constructed, the emphasis is on the sampling capabilities of random variable objects.

The probability package class hierarchy consists of classes and traits which represent continuous and discrete random variables along with ability to endow them with distributions.

DynaML Random Variable

The RandomVariable[Domain] forms the top of the class hierarchy in the probability package. It is a light weight trait which takes a form like so.

abstract class RandomVariable[Domain] {

  val sample: DataPipe[Unit, Domain]

  def :*[Domain1](other: RandomVariable[Domain1]): RandomVariable[(Domain, Domain1)] = {
    val sam = this.sample
    RandomVariable(BifurcationPipe(sam,other.sample))
  }
}

A RandomVariable instance is defined by its type parameter Domain, in Mathematics this is the underlying space (referred to as the support) over which the random variable is defined ($\mathbb{R}^p$ for continuos variables, $\mathbb{N}$ for discrete variables).

The two main functionalities are as follows.

• sample which is a data pipe having no input and outputs a sample from the random variables distribution whenever invoked.

• :* the 'composition' operator between random variables, evaluating an expression like randomVar1 :* randomVar2 creates a new random variable whose domain is a cartesian product of the domains of randomVar1 and randomVar2.

Continuous and discrete distribution random variables are implemented through the ContinuousDistrRV[Domain] and DiscreteDistrRV[Domain] respectively.

Creating Random Variables

Creating random variables can be created by a number of ways.

import breeze.stats.distributions._
import spire.implicits._


//Create a sampling function
val sampF: () => Double = ...
val rv = RandomVariable(sampF)

//Also works with a pipe
val sampF: DataPipe[Unit, Double] = ...
val rv = RandomVariable(sampF)

Sampling is the core functionality of the classes extending RandomVariable but in some cases representing random variables having an underlying (tractable and known) distribution is a requirement, for that purpose there exists the RandomVarWithDistr[Domain, Dist] trait which is a bare bones extension of RandomVariable; it contains only one other member, underlyingDist which is of abstract type Dist.

The type Dist can be any breeze distribution, which is either contained in the package breeze.stats.distributions or a user written extension of a breeze probability distribution.

Creating random variables from breeze distributions

Creating a random variable backed by a breeze distribution is easy, simply pass the breeze distribution to the RandomVariable companion object.

val p = RandomVariable(new Beta(7.5, 7.5))

The RandomVariable object recognizes the breeze distribution passed to it and creates a continuous or discrete random variable accordingly.

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