<div dir="ltr">The `random-fu` package has a monad called `Random` to encapsulate/construct random variables (i.e., for simulation).<br><div><br></div><div>Whether monads are a "perfect" solution to your problem depends on what your random variables represent. In most "finite" cases of jointly distributed random variables, you can get away with just applicative functors. But this wouldn't do if you were modeling something like a normal random walk.</div><div><br></div><div>The sequencing property isn't something you "need" to care about when you're not using it (unless it causes performance issues). After all, even hand-written math is written down "in order", even mathematical expressions have parse trees with implicit partial orders, etc.</div></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Jan 11, 2018 at 8:27 AM, Benjamin Redelings <span dir="ltr"><<a href="mailto:benjamin.redelings@gmail.com" target="_blank">benjamin.redelings@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Hi,<br>
<br>
0. Does anyone know of any simple extensions of the HM type system to non-deterministic functions? The reason that I'm asking is that for probabilistic programming in the lambda calculus, there are two ways of writing expressions:<br>
<br>
(a) stochastic: let x = sample $ normal 0 1 in x*x<br>
<br>
or simply (sample $ normal 0 1)^2<br>
<br>
(b) "mochastic": do {x <- normal 0 1; return (x*x)}<br>
<br>
The "mo" in the second one refers to the use of monads. That is the approach taken in the paper "Practical Probabilistic Programming with monads" (<a href="http://mlg.eng.cam.ac.uk/pub/pdf/SciGhaGor15.pdf" rel="noreferrer" target="_blank">http://mlg.eng.cam.ac.uk/pub/<wbr>pdf/SciGhaGor15.pdf</a>) which I really enjoyed.<br>
<br>
However, I am interested in the stochastic form here. There are a number of reasons, such as the fact that the monadic representation forces serialization on things that need not be serial. In fact, though, I'm not trying to prove which one is best, I am just interested in exploring the non-monadic approach as well.<br>
<br>
1. So, is it possible to do a simple extension to the type system to express non-determinism? I found this paper (Implicit Self-Adjusting Computation for Purely Functional Programs) that uses "level" tags on types to express either (i) security or (ii) changeability. The first idea (for example) is that each type is tagged with one of two "levels", say Public and Secure, so that we actually have Int[Public] or Int[Secure]. Any function that consumes a Secure value must (i) must return a Secure type and (ii) has the arrow in its type labelled with [Secure]. I won't explain the "changeable" idea because its kind of complicated, but I am very interested in it.<br>
<br>
2. This is kind of tangential to the point of my question, but to explain the examples below, it might be important to distinguish sampling from a distribution from the distribution itself. So, normal 0 1 won't generate a random sample. Instead, normal 0 1 () will generate a random sample. This allows us to pass (normal 0 1) to another function which applies it multiple times to generate multiple samples from the same distribution.<br>
<br>
-- sample from a distribution dist<br>
sample dist = dist ()<br>
<br>
--- take n samples from a distribution dist<br>
iid n dist = take n (map sample $ repeat dist)<br>
<br>
Here we see some of the value of using the stochastic approach, versus the "mochastic" approach: we can use normal Haskell syntax to handle lists of random values!<br>
<br>
3. So, I'm wondering if its possible to extend the HM type system to handle non-determinism in a similar fashion by either (i) having some function types be non-deterministic and/or (ii) having term types be non-deterministic. Taking the second approach, I suggest tagging each type with level [D] (for deterministic) or [N] (for non-deterministic). Notation-wise, if a determinism level is unspecified, then this means (I think) quantifying over determinism levels. A function that samples from the normal distribution we would get a type like:<br>
<br>
normal:: double -> double -> () -> double[N]<br>
<br>
Our goal would be that an expression that consumes a non-deterministic expression must itself be non-deterministic, and any function that takes a non-deterministic input must have a non-deterministic output. We could implement that using rules something like this, where {a,b} are type variables and {l1,l2} are level variables.<br>
<br>
x:a[l1] :: a[l1]<br>
\x:a[l1] -> E:b[l2] :: a[l1] -> b[max l1 l2]<br>
E[a[l1]->b[l2]] E[a[l]] :: b[l2]<br>
<br>
The idea is that max l1 l2 would yield N (non-deterministic) if either l1=N or l2=N, because N > D.<br>
<br>
4. Putting non-determinism into the type system would affect GHC in a few ways:<br>
<br>
(a) we shouldn't pull non-deterministic expressions out of lambdas:<br>
<br>
We should NOT change<br>
\x -> let y=sample $ normal 0 1 in y+x<br>
into<br>
let y = sample $ normal 0 1 in \x -> y+x<br>
<br>
(b) we should merge variables with identical values if the types are non-deterministic.<br>
<br>
For example it is OK to change<br>
let {x=normal 0 1; y = normal 0 1 in (sample x * sample y)}<br>
into<br>
let {x=normal 0 1} in sample x<br>
<br>
However it is NOT OK to change<br>
let {x=sample $ normal 0 1; y = sample $ normal 0 1} in x*y<br>
into<br>
let {x=sample $ normal 0 1} in x*x<br>
<br>
Perhaps this would be useful in other contexts?<br>
<br>
5. If what I've written makes sense, then the types of the functions 'sample' and 'iid' would be:<br>
<br>
sample:: (()->a[N]) -> a[N]<br>
<br>
iid:: Int -> (() -> a[N]) -> [a[N]]<br>
<br>
6. This is quite a long e-mail, so to summarize, I am interested in whether or not there are any simple systems for putting non-determinism into HM. Is the use of tagged types known NOT to work? Is there are work on this that I should be aware of?<br>
<br>
Any help much appreciated! :-)<br>
<br>
take care,<br>
<br>
-BenRI<br>
<br>
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