Overview of proposed ciid changes

[zenna]

This issue proposes a big change to the structure of Omega and the semantics of ciid. #120

• It removes the distinction between random variables and functions of omega (with some caveats, discussed below)
• It removes the notion of a sample space projection, and is more random variable centric
• It supports constructing arbitrary conditional independence classes
• it adds an i.i.d. operator as a special case of a more general operator
• The code is currently written in Cassette. This is conceptually clean but can become inefficient because the contextual environments can become nested. I believe it could be implemented efficiently using the same mechanisms we use currently for attaching metadata to ω.

Overall setup

• Assume there exists an infinite sequence of primitive uniformly distributed random variables, unif1, unif2, ... that are i.i.d.
• New random variables can be constructed by either
  • Applying a functional transformation. e.g., y(ω) = f(unif(ω))
  • There is a generic operator ~ that constructs "copies" of random variables. These copies are identically distributed and either independent or conditionally independent.
• ~ has the following general form i ~ x <| (x, y, z, ....). This constructs a random variable that is the ith element in a sequence of random variables that have the same functional form as x, share x, y and z as parents. Consequently:
  • Random variables in this sequence are are identically distributed with x and conditionally independent given x, y, z, ....
  • The sequence 1 ~ x, 2 ~ x, ... are all i.i.d.

Independence

Operationally, it works as follows. First, consider the simplest case, the i.i.d. case, and the following example:

a = 1 ~ unif
b = 2 ~ a

• Recall the existence of sequence of primitive standard uniforms: unif1, unif2, ...
• Note due to the existence of a bijection between the integers and tuples of integers, we can index this set with tuples. Let unif_gen be a mapping from tuples of integers to random variables in this sequence.
• Assume the environment contains a context, which is a tuple of integers
• The context has one purpose, to reinterpret the evaluate of unif(ω). In a context s, unif(ω) is interpreted as unif_gen(s)(ω)
• Hence conceptually, in our example a has the form a(ω) = context_apply(1, unif, ω).
• context_apply(i, x, ω) means to evaluates x(ω) in a modified environment where i is appended onto the context
• Consequently, evaluating a(ω) will ultimately evaluate unif(ω) within the context (1,).
• contexts nest: when context_apply is applied within another context, the contexts are merged.
• This, in fact makes ~ a generic operator to construct i.i.d random variables

Conditional independence

For example, in the following snippet. m1 and m2 are conditionally independent given x, and m3 is completely independent from them both,

x = 1 ~ uniform(0, 19)
measure(ω) = x(ω) + uniform(0, 1)(ω)
m1 = 3 ~ measure <| (x,)
m2 = 4 ~ measure <| (x,)
m3 = 5 ~ measure

Operationally, conditional independence is possible with a single modification. To illustrate, in order for m1 to be ciid given x, the the evaluation of m1(ω) we should evaluate x(ω) in the context that x uses ordinarily, and not whatever context m1 has introduced. Hence, we simply to evaluate x(ω) with out any context, a new context will be reintroduced by x.

Prototype

module MiniOmega

using Cassette
import Base:~
export sample, unif, pointwise, <|, rt, Ω

const ID = NTuple{N, Int} where N

"Ω is a hypercube"
struct Ω
  data::Dict{ID, Real}
end

"Sample a random ω ∈ Ω"
sample(::Type{Ω}) = Ω(Dict{ID, Real}())
Base.getindex(ω::Ω, id) = get!(ω.data, id, rand())

sample(f) = f(sample(Ω))

unif(i::ID) = ω -> ω[@show(i)]

"Single primitive random variable"
unif(ω::Ω) = ω[(1,)]

# (Conditional) independence 

# Use cassette to augment enivonrment with extra state
Cassette.@context CIIDCtx 

"""Conditionally independent and identically distributed given `shared` parents

Returns the `id`th element of an (exchangeable) sequence of random variables
that are identically distributed with `f` but conditionally independent given
random variables in `shared`.

This is essentially constructs a plate where all variables in shared are shared,
and all other parents are not.
"""
function ciid(f, id::Integer, shared)
  let ctx = CIIDCtx(metadata = (shared = shared, id = (id,)))
    ω -> withctx(ctx, f, ω)
  end
end

withctx(ctx, f, ω) = Cassette.overdub(ctx, f, ω)
function Cassette.overdub(ctx::CIIDCtx, ::typeof(withctx), ctxinner, f, ω)
  # Merge the context
  id = (ctxinner.metadata.id..., ctx.metadata.id...)
  shared = (ctxinner.metadata.shared..., ctx.metadata.shared...)
  Cassette.overdub(CIIDCtx(metadata = (shared = shared, id = id)), f, ω)
end

"I.I.D.: `ciid` with nothing shar ed"
ciid(f, id::Integer) = ciid(f, id, ())

Cassette.overdub(ctx::CIIDCtx, ::typeof(unif), ω::Ω) =
  @show unif(ctx.metadata.id)(ω)

function Cassette.overdub(ctx::CIIDCtx, x, ω::Ω)
  if x in ctx.metadata.shared
    x(ω)
  else
    Cassette.recurse(ctx, x, ω)
  end
end

# Syntactic Sugar (to make model-building nicer)

"Random tuple"
rt(fs...) = ω -> map(f -> f(ω), fs)

"""Supports notation `i ~ x <| (y,z,)`
which is the ith element of an (exchangeable) sequence of random variables that are
identically distributed with x but conditionally independent given y and z.
"""
struct Plate{F, S}
  f::F
  shared::S
end

@inline <|(f, shared::Tuple) = Plate(f, shared)
~(id::Integer, f) = ciid(f, id)
~(id::Integer, plate::Plate) = ciid(plate.f, id, plate.shared)

# Pointwise
Cassette.@context PWCtx
Lifted = Union{map(typeof, (+, -, /, *))...}
Cassette.overdub(::PWCtx, op::Lifted, x::Function) = ω -> op(x(ω))
Cassette.overdub(::PWCtx, op::Lifted, x::Function, y::Function) = ω -> op(x(ω), y(ω))
Cassette.overdub(::PWCtx, op::Lifted, x::Function, y) = ω -> op(x(ω), y)
Cassette.overdub(::PWCtx, op::Lifted, x, y::Function) = ω -> op(x, y(ω))
pointwise(f) = Cassette.overdub(PWCtx(), f)

# AutoId
Cassette.@context AutoIdCtx

end

using .MiniOmega

function simpletest()
  a = 1 ~ unif
  b = 2 ~ a
  sample(b)
end

function test()
  uniform(a, b) = ω -> unif(ω) * (b - a) + a

  x = 1 ~ uniform(0, 1)
  y = 2 ~ uniform(0, 1)
  d = 7 ~ y
  z = ω -> (x(ω), x(ω), y(ω), d(ω))
  @show sample(z)
  function a(ω)
    x_ = x(ω)
    d = 3 ~ uniform(0, 4)
    e = 4 ~ uniform(0, 4)
    x_ + d(ω) + e(ω)
  end

  @show sample(rt(x, y, z, a))

  # Conditional Independence
  x = 1 ~ uniform(0, 19)
  measure(ω) = x(ω) + uniform(0, 1)(ω)
  m1 = 3 ~ measure <| (x,)
  m2 = 4 ~ measure <| (x,)
  m3 = 5 ~ measure

  @show sample(rt(m1, m2, m3))

  # Linaer regression 
  α = 1 ~ uniform(0, 1) 
  β = 2 ~ uniform(0, 1)
  f(x, i) = i ~ (ω -> x*α(ω) + β(ω) + uniform(0.0, 1.0)(ω)) <| (α, β)
  y1 = f(0.3, 3)
  y2 = f(0.3, 4)
  @show sample(rt(y1, y2))

  y1b, y2b = pointwise() do
    f2(x, i) = i ~ (x * α + β + uniform(0.0, 1.0)) <| (α, β)
    y1b = f2(0.3, 3)
    y2b = f2(0.3, 4)
    y1b, y2b
  end

  @show sample(rt(y1, y2, y1b, y2b))

  # A bunch of random variables with a shared parents
  pointwise() do
    x = 1 ~ unifQD
    y = x * 100
    function a(ω)
      y(ω) + 5
    end
    function b(ω)
      y(ω) + 10
    end
    c = 2 ~ b <| (y,)
    d = 3 ~ b
    @show sample(rt(y, a, b, c, d))
  end
end

Port to Omega?

There may be some small bugs (please check) in this approach but I'm fairly confident it is both mostly correct, more general, and simpler than what we are currently doing. So, I should probably port the change to real Omega.

A few concerns?

• What need is there for RandVars
  • RandVars have explicit ids. These ids are used to check random variable equivalence for (i) memoization, and causal interventions. The above code uses Julia's function equivalence checking. It's not clear exactly what that is doing, but I imagine it's doing some kind of memory check. It's not clear that integer ID's are much better because: we can define the same random variable (functionally) and give them different IDs.
  • Random variables support pointwise operation. It would be bad type-piracy to extend pointwise notation to all functions. A middle ground is what I have done here which is define particular pointwise contexts
  • Some RandVars have type information that can be useful, e.g. for analytic computation of expectations.

Overall, I'm inclined to say we support any object that "purely" implements (::Type)(ω::Ω). We can still use typed and/or id'd objects for the above reasons, but not if we don't need that.

• Would this break existing code? Currently a lot of omega code is of the form

x = normal(0, 1)
f_(ω) = uniform(ω, 0, 1) + 10 * 2 + x(ω)
f = ciid(f)   # unexpected! x in f will be independent normal

This code would break, or more precisely, do something different to what it currently does. The reason is that currently this code would share the value of x, but ciid (aka ~) without any shared parameters would create an i.i.d. variable. Fortunately though, because we no longer have a distinction between random variables and functions, this does not mean we always need to explicitly stat what we want the parents to be, we in fact do not need the ciid at all in this example

x = normal(0, 1)
f(ω) = 1 ~ uniform(ω, 0, 1) + 10 * 2 + x(ω) # just use f! x will be a parent in f and in g below
g(ω) = 2 ~ normal(ω, 0, 1) + 2 + x(ω)

[zenna]

A quick primer on Cassette. Mostly for @jburroni

Cassette allows you to implement some (actually many) kinds of non-standard interpretations.

The general approach is to first create a context, and then add new definitions to that context.

using Cassette
Cassette.@context MyCtx   # Creates a new context type called MyCtx

In this new context I will replace sin with cos

# adding methods to Cassette.overdub is how to define new semantics
function Cassette.overdub(::MyCtx, ::typeof(sin), x)
  println("I will now do cos instead of sin")
  cos(x)
end

Try it out

myprogram() = sin(3)

# execute myprogram in MyCtx
Cassette.overdub(MyCtx(), myprogram)

One thing it quite easily allows you to do something equivalent to adding extra state to the environment, in the operational semantics sense. This is done by adding data ('metadata') to your context. In this example I'll add metadata which just counts the number of function calls

mutable struct Counter
  count::Int
end

# This will intercept __every__ function call (except one's in Base that Cassette has defaults to not recurse into)
function Cassette.overdub(ctx::MyCtx, f, args...)
  ctx.metadata.count += 1
  # Now Continue running 
  Cassette.recurse(ctx, f, args...)
end

# Let's try it out
ctx = MyCtx(metadata = Counter(1))
Cassette.overdub(ctx, myprogram)
println("Number of function calls is $(ctx.metadata)")

[zenna]

Better in discussions