Do we need both ciid and iid?

[zenna]

Even though I've thought about this alot, I'm still not certain.

In the aircraft example I modelled the initial state as

function init_state(ω)
  x = 2 ~ Uniform(xlb, xub)
  y = 3 ~ Uniform(ylb, yub)
  vx = 4 ~ Normal(0, 1)
  vy = 5 ~ Normal(0, 1)
  Aircraft((x(ω), y(ω)), (vx(ω), vy(ω)))
end

C = iid(1, init_state)
D = iid(2, init_state)
E = iid(3, init_state)

Then I made several copies using iid. The issue was that suppose I wanted to introduce some correlations between these components with a shared parents, I'd then need to switch to ciid

a = 1 ~ Normal(0, 1) 
function init_state_plate(id, ω)
  x = (id, 1) ~ Uniform(xlb, xub) + a
  y = (id, 2) ~ Uniform(ylb, yub) + a
  vx = (id, 3) ~ Normal(0, 1)
  vy = (id, 4) ~ Normal(0, 1)
  Aircraft((x(ω), y(ω)), (vx(ω), vy(ω)))
end

C = ciid(1, init_state)
D = ciid(2, init_state)
E = ciid(3, init_state)

Why not just start with the plate in the first place? Then you get to this same conclusion which is that why not just make everything a plate. Well, if you want to share structure, you need to actually make random variables

some_other_func(w) = a(w) + 10 + normal...

a cannot be a plate here, because thats the point of a plate. I could make a a plate and then init_state_plate and some_other_func use the same id for the same effect but that would be no good.

The issue remains. If I decide there should be some shared structure then the type has to change from a randvar to a plate, which means everything downstream has to be updated. That's bad for compositionality.

Now I want to actually make 3 iid copies, but the most likely behaviour is that I don't. Rather, if you add parent structure then I want to share it.

You could imagine an operator xiid(x), which is maybe what tilde should represent, which is iid(x) if x is a randvar and ciid(x) if x is a plate.

The problem is that I don't know that can implement this in Julia as we can't tell from the type, other than with a generated function.

Could making a plate take a pair of an omega and an id be the answer?

X = Normal(0, 1)
function f((id, w))
  x_ = X(w)
  a = Normal(0, 1)
end

The alternative, is to

Q: What if I want to iid a plate?

• Of course I could just take the fiid ciid element then iid that, but it seems a little hacky.

[zenna]

I think there are a few competing designs. Let me try to enumerate them:

Dual functions

In this pattern (the current choice):

• There are random variables, which are functions of omega f(w)
• There are claasses, which is an function f(id, w)
• I'm somewhat agnostic about the type of id, except that primitives are defined on a particular type
• ciid(i,x) = w -> x(i,w) -- ciid is just partial application of a class
• iid changes the subspace

Old Omega

• There are only random variables f(w)
• the primitives are hence random variables (not classes), eg unif(w) = w[(unif, w)]
• ciid(i, x) = w->x(lproj(w, i)) does a "local" projects onto i. The difference between a local projection and that used above by iid is that a local projection replaces the subspace instead of adding to it. This means that the parent structure is preserved because the parents will themselves project onto their own subspace
• iid is not possible (without some tricks), since parents will always steal omega
• some primitives dont project, allowing iid copies

unif(ω) = ω[(unif, 1)]

id ~ x = ω -> x(lproj(id, ω))

function f(ω)
  a = (1 ~ unif)(ω)
  b = (2 ~ unif)(ω)
  c = unif(ω)
  a + b + c
end

a(ω) = (f(ω), f(ω), (4 ~ f)(ω), (3 ~ unif)(ω))

• Parent structure speciied within f
• Bad because its hard to specify something as being within the plate (c in this case) because I need some guarantees that it has no shared parents

Explicit within f spec

• Specify which are within/outside

unif(ω) = ω[(unif, 1)]

id ~ x = ω -> x(setid(id, ω))
id ≗ x = ω -> x(proj((id, getid(ω)), ω))

function f(ω)
  a = (1 ≗ unif)(ω)
  b = (2 ≗ unif)(ω)
  c = unif(ω)
  a + b + c
end

seq = 1 ~ f, 2 ~ f, 3 ~ f # ciid given c
a(ω) = (f(ω), f(ω), (4 ~ f)(ω), (3 ~ unif)(ω))

-- What is 1 ~ 2 ~ f?

Questions

• Specify plate structure within or outside of f
• Type distinction between plate and random variable?

[zenna]

What if we changed iid so that it created a sequence

function init_state(ω)
  x = 2 ~ Uniform(xlb, xub)
  y = 3 ~ Uniform(ylb, yub)
  vx = 4 ~ Normal(0, 1)
  vy = 5 ~ Normal(0, 1)
  Aircraft((x(ω), y(ω)), (vx(ω), vy(ω)))
end

C = 1 ~ iid(init_state)
D = 2 ~ iid(init_state)
E = 3 ~ iid(init_state)

a = ..
function init_state(id, ω)
  x =  id, 2 ~ Uniform(xlb, xub) + a(ω)
  y =  id, 3 ~ Uniform(ylb, yub)
  vx = id, 4 ~ Normal(0, 1)
  vy = id, 5 ~ Normal(0, 1)
  Aircraft((x(ω), y(ω)), (vx(ω), vy(ω)))
end

C = 1 ~ init_state
D = 2 ~ init_state
E = 3 ~ init_state

I think it should be the following

id ~ xs -- idth member of sequence `xs` which is function `f(id, w)`
iid(x) -- sequence of independent copies of `x`
ciid(x, p) -- sequence of copies of `x` conditionally independent given (variable in) p

function init_state(ω)
  x = 2 ~ Uniform(xlb, xub)
  y = 3 ~ Uniform(ylb, yub)
  vx = 4 ~ Normal(0, 1) + a(w)
  vy = 5 ~ Normal(0, 1)
  Aircraft((x(ω), y(ω)), (vx(ω), vy(ω)))
end
xs = ciid(init_state, a)