Fundamentals

[zenna]

What are the fundamental objects and corresponding semantics in Omega?

This discussion covers well-treaded ground, but I'm still uncertain.

I have two possible views:

Splitting

In this perspective:

• There is a finite (and small) set of globally unique random variables - StdUnif, StdNorm, etc
• There's a splitting operation that produces independent copies of these (and any) random variables

split :: RV -> (RV, RV)

Normal(μ, σ) = (StdNormal() .* σ) .+ μ 
X = Normal(0, 1)
function Y(i, ω)
  x_ = x(ω)
  ϵ = split(Normal(0, 1))
  # Problems here are that we don't know how many to split into, is infinite splitting allowed?
  x_ + ...
end

Alternative - infinite sequences

• There is a finite (and small) set of globally unique random variables - StdUnif, StdNorm, etc
• Can construct iid sequence from X
• Ability to compose indices together
• Operator to make random array / vector from a class

~ :: RV -> (ID ->  RV) # Cosntruct sequence, indexed by ID
i ~ X = (~ X)(i)

Normal(μ, σ) = (StdNormal() .* σ) .+ μ 

function Y(i, ω)
  x_ = x(ω)
  ϵ = @~ i Normal(0, 1)(ω)
  x_ + ϵ
end

This differs from current setup:

• StdUniform() etc are randvars as opposed to a class
• Primitives are randvars not classes
• For an explicit class, e.g. Y above we wouldn't use i |> Y,

Question:

• What's the relation between: 1 ~ Normal(3, 4) and 2 ~ Normal(5, 5)
• Is ~ the right symbol for making an iid sequence? Conventionally X ~ Y means X is distributed according to Y

Current setup

• Primitives are infinite classes

function Y(i, ω)
  x_ = x(ω)
  ϵ = @~ i Normal(0, 1)(ω)
  x_ + ϵ
end

Concrete implications are:

• Remove append from the ID API, think of IDs as keys that must satisfy compose. Define semantics of compose. Could be a list of tuples but not necessarily.