# 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