Conditional Independence

A class in Omega is a function of the form f(id, ω::Ω). It represents a sequence of random variables. To get the nth member of a class use the function nth.

There are primitive random variable classes in Omega.

A1 = nth(StdNormal, 1)
A2 = nth(StdNormal, 2)
A3 = nth(StdNormal, 3)

Or equivalently, use ~:

A1 = 1 ~ StdNormal, 1
A2 = 2 ~ StdNormal, 2
A3 = 3 ~ StdNormal, 3

Of course, you can specify your own classes simply by constructing a function. In

using Omega, Distributions
μ = 1 ~ StdNormal{Float64}()
function Xs(id, ω)
  id ~ Normal(ω, μ(ω), 1) 
end
x1 = 1 ~ Xs
x2 = 2 ~ Xs

To construct a random variable over collections from a class, use Mv

A very important property of classes is that the members of a class are conditionally independent, given the shared parents. In the above exmaple, x1 and x2 are conditionally independent given μ.

Independence

Sometimes we need to construct random variables that are independent. The function iid constructs a class of random variables that are independent.