Cheat Sheet

# Cheat Sheet

## Core Functions

The major functions that you will use in Omega are:

• ~x : that is equal in distribution to `x` but conditionally independent given parents
• cond(x, y) : condition random variable `x` on condition `y`
• cond(ω, ab) : condition random variable that contains statement by force `ab` to be `true`]
• rand(x, n; alg = Alg) : `n` samples from (possibly conditioned) random variable `x` using algorithm `ALG`
• replace(x, θold => θnew) : causal intervention in random variable
• rid(x, θ) : random interventional distribution of `x` given `θ`
• rcd(x, θ) or x ∥ θ : random conditional distribution of `x` given `θ`

## Terminology

• Causal Inference:
• Conditioning: Restricts a RandVar to be consistent with a predicate. Conceptually, conditioning is the mechanism to add knowledge (observations, declarative facts, etcs) to a model.
• Intervention: A change to a model. Interventions support counterfactual "what if" questions.
• Lift: To lift a function means to construct a new function that transforms random variables.
• Model: A collection of Random Variables.
• Prior: Unconditioned distribution. In Bayesian inference terms, prior to having observed data
• Posterior: Technically identical to conditional distribution. The term posterior is used commonly in the context of Bayesian inference where the conditional distribution is having observed more data.
• Random Variable: a random variable is one kind of representation of a probability distribution.
• Realization (or outcome) space: Space (or type) of values that a random variable can take. Since Random Variable are functions, technically this is its domain. In Omega: `elemtype(x)` is its realization space
• Realization of a random variable: a value in the realization space, typically understood to be drawn according to its distribution. In Omega, the result of `rand(x)` is a realizataion of `x`
• Probability Space: A tuple \$(Ω, Σ, μ)\$ where \$Ω\$ is a sample space, Σ is a sigma algebra (roughly, set of all subsets of Ω, and μ is a probability measure). In Omega:

## Built-in Distributions

bernoulli(w) boolbernoulli(w) betarv categorical constant exponential gammarv invgamma kumaraswamy logistic poisson normal uniform rademacher

## Built-in Inference Algorithms

RejectionSample MI SSMH SSMHDrift HMC HMCFAST