New feature request: add support to dsl.py representing free and bound variables in a probabilistic expression

[rjc955]

Currently in Y0 we can associate values with variables and represent the probability of a set of variables in terms of other variables, but Y0 lacks a mechanism for representing $P(\mathbf{X} = \mathbf{x})$, where $\mathbf{X}$ denotes one or more variables and $\mathbf{x}$ denotes their corresponding values. As published, Algorithm 2 in Counterfactual Transportability: A Formal Approach requires this feature.

[cthoyt]

I see that, but I'm not convinced that the same can't be accomplished by having a pair of 1) the expression and 2) a dictionary mapping from the variable to its assignment

For example, you can have:

from y0.dsl import P, X, Variable

value = Variable('x')

expression = P(X)
assignments = {X: value}

If you had some more complicated expression that had multiple assignments, then maybe this is a different story, but I am not what that would look like besides $P(\mathbf{X}=\mathbf{x})*P(\mathbf{X}=\mathbf{x'})$, where no matter what the assignments are, this will be zero since X can't be two things

[djinnome]

The issue is that counterfactual transportability requires knowledge of whether a variable is bound or free.

Examples of bound variables:

$P(X=x, Y=y', Y_{x}=y \mid Z=z, W_{x}=w)$

P(X == -x, Y == +y, Y @ -x == -y | Z == -z, W @ -x == -w)

Examples of free variables:

$P(X, Y, Y_{x} \mid Z, W_{x})$

P(X, Y, Y @ -x | Z, W @ -x)

Here is a probabilistic expression with both free and bound variables:

$P(X, Y=y, Y_{x} \mid Z, W_{x}=w)$

P(X,  Y== +y, Y @ -x | Z, W @ -x  == -w)

The reason why this mixture can happen is that a variable that is free in an expression may be bound by an outer context such as a Sum or Product:

$\sum_X P(X, Y=y)$

Sum[X](P(X, Y == -y)) 

It turns out that the counterfactual transportability algorithm can also have free or bound variables in an intervention, too!

So we can have:

$\sum_X P(Y_X)$

Sum[X](P(Y @ X))

which is the same as:

P(Y @ +x) + P(Y @ -x)

Currently, interventions are always assumed to be bound variables, such that P(Y @ X) is automatically converted to P(Y @ -X). However, the counterfactual transportability algorithm does generate expressions where an intervention can be a free variable. Furthermore, for any counterfactual variable, some of the interventions could be free, and others could be bound:

$P(Y_{X=x, Z})$

P(Y @ (X == -x, Z))

[cthoyt]

I don't quite understand what the meaning of the minus or plus sign is anymore with this proposal. This is syntactically valid, but I don't know what it should mean:

P(-Y @ x)
P(+Y @ x)
P(-Y @ -x)
P(+Y @ -x)
P(-Y @ +x)
P(+Y @ +x)

[djinnome]

The current meaning of P(-Y @ x) is that the value of the CounterfactualVariable Y @ x is -y and the value of the Intervention x is -x
Note that P(-Y @ x) has the same meaning as P(-Y @ -x) because we have a restriction that if a Variable is on the right-hand-side of @ then we coerce it into an Intervention, which is not allowed to be a free variable (star must be True or False).
Note also that a Variable on the left-hand-side of @ can be a free variable (represented as having star=None).

We currently cannot represent counterfactual variables that have Interventions as free variables, and while we can imagine scenarios where this will become a problem in the future, it is not a blocker for implementing any of the algorithms so far.

[djinnome]

I think we can get 90% of what we want if we simply display dsl objects like P(-Y @ +x) as $P(Y_{do(X=x^\ast)}=y)$