You can think of something like x↦f(x,Y) as a python dictionary mapping x-values to the corresponding f(x,Y) values. That whole dictionary would be a function of Y. In the case of something like (y,r↦P∗[R=r|X,Y=y]), it’s a partial policy mapping each second-input-value y and regulator output value r to the probability that the regulator chooses that output value on that input value, and we’re thinking of that whole partial policy as a function of the first input value X. So, it’s a function which is itself a random variable constructed from X.
The reason I need something like this is because sometimes I want to say e.g. “two policies (x,r↦P[R=r|X=x]) are identical” (i.e. P[R=r|X=x] is the the same for all r, x), sometimes I want to say “two distributions (r↦P[R=r|X]) are identical” (i.e. two X-values yield the same output distribution), etc, and writing it all out in terms of quantifiers makes it hard to see what’s going on conceptually.
I’ve been trying to figure out a good notation for this, and I haven’t settled on one, so I’d be interested in peoples’ thoughts on it. Thankyou to TurnTrout for some good advice already; I’ve updated the post based on that. The notation remains somewhat cumbersome and likely confusing for people not accustomed to dense math notation; I’m interested in suggestions to improve both of those problems.
Note on notation...
You can think of something like x↦f(x,Y) as a python dictionary mapping x-values to the corresponding f(x,Y) values. That whole dictionary would be a function of Y. In the case of something like (y,r↦P∗[R=r|X,Y=y]), it’s a partial policy mapping each second-input-value y and regulator output value r to the probability that the regulator chooses that output value on that input value, and we’re thinking of that whole partial policy as a function of the first input value X. So, it’s a function which is itself a random variable constructed from X.
The reason I need something like this is because sometimes I want to say e.g. “two policies (x,r↦P[R=r|X=x]) are identical” (i.e. P[R=r|X=x] is the the same for all r, x), sometimes I want to say “two distributions (r↦P[R=r|X]) are identical” (i.e. two X-values yield the same output distribution), etc, and writing it all out in terms of quantifiers makes it hard to see what’s going on conceptually.
I’ve been trying to figure out a good notation for this, and I haven’t settled on one, so I’d be interested in peoples’ thoughts on it. Thankyou to TurnTrout for some good advice already; I’ve updated the post based on that. The notation remains somewhat cumbersome and likely confusing for people not accustomed to dense math notation; I’m interested in suggestions to improve both of those problems.