On Thu, 29 Jul 2021, Benjamin Redelings wrote:

Hi Olaf,

I think you need to look at two things:

1. The Giry monad, and how it deals with continuous spaces.

I believe I understand the Giry monad well, and it is my measuring stick for functional probabilistic programming. Even more well-suited for programming ought to be the valuation monad, because that is a monad on domains, the [*] semantic spaces of the lambda calculus. Unfortunately, to my knowledge until now attempts were unsuccessful to find a cartesian closed category of domains which is also closed under this probabilistic powerdomain construction. [*] There are of course other semantics, domains being one option.

2. The paper "Practical Probabilistic Programming with Monads" - https://doi.org/10.1145/2804302.2804317

Wasn't that what you linked to in your original post? As said above, measure spaces is the wrong target category, in my opinion. There is too much non constructive stuff in there. See the work of Culbertson and Sturtz, which is categorically nice but not very constructive.

Also, observing 2.0 from a continuous distribution is not nonsensical.

-BenRI

Perhaps I am being too much of a point-free topologist here. Call me pedantic. Or I don't understand sampling at all. To me, a point is an idealised object, only open sets really exist and are observable. If the space is discrete, points are open sets. But on the real line you can not measure with infinite precision, so any observation must contain an interval. That aligns very nicely with functional programming, where only finite parts of infinite lazy structures are ever observable, and these finite parts are the open sets in the domain semantics. So please explain how observing 2.0 from a continuous distribution is not nonsensical. Olaf

On 7/21/21 11:15 PM, Olaf Klinke wrote:

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However, a lazy interpreter causes problems when trying to introduce *observation* statements (aka conditioning statements) into the monad [3].  For example,

run_lazy $ do    x <- normal 0 1    y <- normal x 1    z <- normal y 1    2.0 `observe_from` normal z 1    return y

In the above code fragment, y will be forced because it is returned, and y will force x.  However, the "observe_from" statement will never be forced, because it does not produce a result that is demanded.

I'm very confused. If the observe_from statement is never demanded, then what semantics should it have? What is the type of observe_from? It seems it is a -> m a -> m () for whatever monad m you are using. But conditioning usually is a function Observation a -> Dist a -> Dist a so you must use the result of the conditioning somehow. And isn't the principle of Monte Carlo to approximate the posterior by sampling from it? I tend to agree with your suggestion that observations and sampling can not be mixed (in the same do-notation) but the latter have to be collected in a prior, then conditioned by an observation.

What is the semantic connection between your sample and obsersvation monad? What is the connection between both and the semantic probability distributions? I claim that once you have typed everything, it becomes clear where the problem is.

Olaf

P.S. It has always bugged me that probabilists use elements and events interchangingly, while this can only be done on discrete spaces. So above I would rather like to write (2.0==) `observe_from` (normal 0 1) which still is a non-sensical statement if (normal 0 1) is a continuous distribution where each point set has probability zero.