In my last post I went over the categorical and measure-theoretic foundations of the Giry monad, the ‘canonical’ probability monad that operates on the level of probability measures.

In this post I’ll pick up from where I left off and talk about a neat and faithful (if impractical) implementation of the Giry monad that one can put together in Haskell.

Measure, Integral, and Continuation

So. For a quick review, we’ve established the Giry monad as a triple $(\mathcal{P}, \mu, \eta)$, where $\mathcal{P}$ is an endofunctor on the category of measurable spaces $\textbf{Meas}$, $\mu$ is a marginalizing integration operation defined by:

and $\eta$ is a monoidal identity, defined by the Dirac measure at a point:

How do we actually implement this beast? If we’re looking to be suitably general then it is unlikely that we’re going to be able to easily represent something like a $\sigma$-algebra over some space of measures on a computer, so that route is sort of a non-starter.

But it can be done. The key to implementing a general-purpose Giry monad is to notice that the fundamental operation involved in it is integration, and that we can avoid working with $\sigma$-algebras and measurable spaces directly if we focus on dealing with measurable functions instead of measurable sets.

Consider the integration map on measurable functions $\tau_f$ that we’ve been using this whole time. For some measurable function $f$, $\tau_f$ takes a measure on some measurable space $M = (X, \mathcal{X})$ and uses it to integrate $f$ over $X$. In other words:

A measure in $\mathcal{P}(M)$ has type $X \to \mathbb{R}$, so $\tau_f$ has corresponding type $(X \to \mathbb{R}) \to \mathbb{R}$.

This might look familiar to you; it’s very similar to the type signature for a continuation:

newtype Cont a r = Cont ((a -> r) -> r)


Indeed, if we restrict the carrier type of ‘Cont’ to the reals, we can be really faithful to the type:

newtype Integral a = Integral ((a -> Double) -> Double)


Now, let’s overload notation and call the integration map $\tau_f$ itself a measure. That is, $\tau_f$ is a mapping $\nu \mapsto \int_{X}fd\nu$, so we’ll just interpret the notation $\nu(f)$ to mean the same thing - $\int_{X}fd\nu$. This is convenient because we can dispense with $\tau$ and just pretend measures can be applied directly to measurable functions. There’s no way we can get confused here; measures operate on sets, not functions, so notation like $\nu(f)$ is not currently in use. We just set $\nu(f) = \tau_f(\nu)$ and that’s that. Let’s rename the ‘Integral’ type to match:

newtype Measure a = Measure ((a -> Double) -> Double)


We can extract a very nice shallowly-embedded language for integration here, the core of which is a single term:

integrate :: (a -> Double) -> Measure a -> Double
integrate f (Measure nu) = nu f


Note that this is the same way we’d express integration mathematically; we specify that we want to integrate a measurable function $f$ with respect to some measure $\nu$:

The only subtle difference here is that we don’t specify the space we’re integrating over in the integral expression - instead, we’ll bake that into the definition of the measures we create themselves. Details in a bit.

What’s interesting here is that the Giry monad is the continuation monad with the carrier type restricted to the reals. This isn’t surprising when you think about what’s going on here - we’re representing measures as integration procedures, that is, programs that take a measurable function as input and then compute its integral in some particular way. A measure, as we’ve implemented it here, is just a ‘program with a missing piece’. And this is exactly the essence of the continuation monad in Haskell.

Typeclass Instances

We can fill out the functor, applicative, and monad instances mechanically by reference to the a standard continuation monad implementation, and each instance gives us some familiar conceptual structure or operation on probability measures. Let’s take a look.

The functor instance lets us transform the support of a measurable space while keeping its density structure invariant. If we have:

then mapping a measurable function over the measure corresponds to:

The functor structure allows us to precisely express a pushforward measure or distribution of $\nu$ under $g$. It lets us ‘adapt’ a measure to other measurable spaces, just like a good functor should.

In Haskell, the functor instance corresponds exactly to the math:

instance Functor Measure where
fmap g nu = Measure $\f -> integrate (f . g) nu  The monad instance is exactly the Giry monad structure that we developed previously, and it allows us to sequence probability measures together by marginalizing one into another. We’ll write it in terms of bind, of course, which went like: The Haskell translation is verbatim: instance Monad Measure where return x = Measure (\f -> f x) rho >>= g = Measure$ \f ->
integrate (\m -> integrate f (g m)) rho


Finally there’s the Applicative instance, which as I mentioned in the last post is sort of conceptually weird here. So in the spirit of that comment, I’m going to dodge any formal justification for now and just use the following instance which works in practice:

instance Applicative Measure where
pure x = Measure (\f -> f x)
Measure g <*> Measure h = Measure $\f -> g (\k -> h (f . k))  Conceptual Example It’s worth taking a look at an example of how things should conceivably work here. Consider the following probabilistic model: It’s a standard hierarchical presentation. A ‘compound’ measure can be obtained here by marginalizing over the beta measure $\pi$, and that’s called the beta-binomial measure. Let’s find it. The beta distribution has support on the $[0, 1]$ subset of the reals, and the binomial distribution with argument $n$ has support on the $\{0, \ldots, n\}$ subset of the integers, so we know that things should proceed like so: Eliding some theory of integration, I can tell you that $\pi$ is absolutely continuous with respect to Lebesgue measure and that $\mu(p)$ is absolutely continuous w/respect to counting measure for appropriate $p$. So, $\pi$ admits a density $d\pi/dx = g_\pi$ and $\mu(p)$ admits a density $d\mu(p)/d\# = g_{\mu(p)}$, defined as: and respectively, for $B$ the beta function and $\binom{n}{x}$ a binomial coefficient. Again, we can reduce the integral as follows, transforming the outermost integral into a standard Riemann integral and the innermost integral into a simple sum of products: where $dx$ denotes Lebesgue measure. I could expand this further or simplify things a little more (the beta and binomial are conjugates) but you get the point, which is that we have a way to evaluate the integral. What is really required here then is to be able to encode into the definitions of measures like $\pi$ and $\mu(p)$ the method of integration to use when evaluating them. For measures absolutely continuous w/respect to Lebesgue measure, we can use the Riemann integral over the reals. For measures absolutely continuous w/respect to counting measure, we can use a sum of products. In both cases, we’ll also need to supply the density or mass function by which the integral should be evaluated. Creating Measures Recall that we are representing measures as integration procedures. So to create one is to define a program by which we’ll perform integration. Let’s start with the conceptually simpler case of a probability measure that’s absolutely continuous with respect to counting measure. We need to provide a support (the region for which probability is greater than 0) and a probability mass function (so that we can weight every point appropriately). Then we just want to integrate a function by evaluating it at every point in the support, multiplying the result by that point’s probability mass, and summing everything up. In code, this translates to: fromMassFunction :: (a -> Double) -> [a] -> Measure a fromMassFunction f support = Measure$ \g ->
foldl' (\acc x -> acc + f x * g x) 0 support


So if we want to construct a binomial measure, we can do that like so (where choose comes from Numeric.SpecFunctions):

binomial :: Int -> Double -> Measure Int
binomial n p = fromMassFunction (pmf n p) [0..n] where
pmf n p x
| x < 0 || n < x = 0
| otherwise = choose n x * p ^^ x * (1 - p) ^^ (n - x)


The second example involves measures over the real line that are absolutely continuous with respect to Lebesgue measure. In this case we want to evaluate a Riemann integral over the entire real line, which is going to necessitate approximation on our part. There are a bunch of methods out there for approximating integrals, but a simple one for one-dimensional problems like this is quadrature, an implementation for which Ed Kmett has handily packaged up in his integration package:

fromDensityFunction :: (Double -> Double) -> Measure Double
fromDensityFunction d = Measure $\f -> quadratureTanhSinh (\x -> f x * d x) where quadratureTanhSinh = result . last . everywhere trap  Here we’re using quadrature to approximate the integral, but otherwise it has a similar form as ‘fromMassFunction’. The difference here is that we’re integrating over the entire real line, and so don’t have to supply a support explicitly. We can use this to create a beta measure (where logBeta again comes from Numeric.SpecFunctions): beta :: Double -> Double -> Measure Double beta a b = fromDensityFunction (density a b) where density a b p | p < 0 || p > 1 = 0 | otherwise = 1 / exp (logBeta a b) * p ** (a - 1) * (1 - p) ** (b - 1)  Note that since we’re going to be integrating over the entire real line and the beta distribution has support only over $[0, 1]$, we need to implicitly define the support here by specifying which regions of the domain will lead to a density of 0. In any case, now that we’ve constructed those things we can just use a monadic bind to create the beta-binomial measure we described before. It masks a lot of under-the-hood complexity. betaBinomial :: Int -> Double -> Double -> Measure Int betaBinomial n a b = beta a b >>= binomial n  There are a couple of other useful ways to create measures, but the most notable is to use a sample in order to create an empirical measure. This is equivalent to passing in a specific support for which the mass function assigns equal probability to every element; I’ll use Gabriel Gonzalez’s foldl package here as it’s pretty elegant: fromSample :: Foldable f => f a -> Measure a fromSample = Measure . flip weightedAverage weightedAverage :: (Foldable f, Fractional r) => (a -> r) -> f a -> r weightedAverage f = Foldl.fold (weightedAverageFold f) where weightedAverageFold :: Fractional r => (a -> r) -> Fold a r weightedAverageFold f = Foldl.premap f averageFold averageFold :: Fractional a => Fold a a averageFold = (/) <$> Foldl.sum <*> Foldl.genericLength


Using ‘fromSample’ you can create an empirical measure using just about anything you’d like:

data Foo = Foo | Bar | Baz

foos :: [Foo]
foos = [Foo, Foo, Bar, Foo, Baz, Foo, Bar, Foo, Foo, Foo, Bar]

nu :: Measure Foo
nu = fromSample foos


Though I won’t demonstrate it here, you can use this approach to also create measures from sampling functions or random variables that use a source of randomness - just draw a sample from the function and pipe the result into ‘fromSample’.

Querying Measures

To query a measure is to simply get some result out of it, and we do that by integrating some measurable function against it. The easiest thing to do is to just take a straightforward expectation by integrating the identity function; for example, here’s the expected value of a beta(10, 10) measure:

> integrate id (beta 10 10)
0.49999999999501316


The expected value of a beta($\alpha$, $\beta$) distribution is $\alpha / (\alpha + \beta)$, so we can verify analytically that the result should be 0.5. We observe a bit of numerical imprecision here because, if you’ll recall, we’re just approximating the integral via quadrature. For measures created via ‘fromMassFunction’ we don’t need to use quadrature, so we won’t observe the same kind of approximation error. Here’s the expected value of a binomial(10, 0.5) measure, for example:

> integrate fromIntegral (binomial 10 0.5)
5.0


Note here that we’re integrating the ‘fromIntegral’ function against the binomial measure. This is because the binomial measure is defined over the integers, rather than the reals, and we always need to evaluate to a real when we integrate. That’s part of the definition of a measure!

Let’s calculate the expectation of the beta-binomial distribution with $n = 10$, $\alpha = 1$, and $\beta = 8$:

> integrate fromIntegral (betaBinomial 10 1 8)
1.108635884924813


Neato. And since we can integrate like this, we can really compute any of the moments of a measure. The first raw moment is what we’ve been doing here, and is called the expectation:

expectation :: Measure Double -> Double
expectation = integrate id


The second (central) moment is the variance. Here I mean variance in the moment-based sense, rather than as the possibly better-known sample variance:

variance :: Measure Double -> Double
variance nu = integrate (^ 2) nu - expectation nu ^ 2


The variance of a binomial($n$, $p$) distribution is known to be $np(1-p)$, so for $n = 10$ and $p = 0.5$ we should get 2.5:

> variance (binomial 10 0.5)
<interactive>:87:11: error:
• Couldn't match type ‘Int’ with ‘Double’
Expected type: Measure Double
Actual type: Measure Int
• In the first argument of ‘variance’, namely ‘(binomial 10 0.5)’
In the expression: variance (binomial 10 0.5)
In an equation for ‘it’: it = variance (binomial 10 0.5)


Ahhh, but remember: the binomial measure is defined over the integers, so we can’t integrate it directly. No matter - the functorial structure allows us to adapt it to any other measurable space via a measurable function:

> variance (fmap fromIntegral (binomial 10 0.5))
2.5


Expectation and variance (and other moments) are pretty well-known, but you can do more exotic things as well. You can calculate the moment generating function for a measure, for example:

momentGeneratingFunction :: Measure Double -> Double -> Double
momentGeneratingFunction nu t = integrate (\x -> exp (t * x)) nu


and the cumulant generating function follows naturally:

cumulantGeneratingFunction :: Measure Double -> Double -> Double
cumulantGeneratingFunction nu = log . momentGeneratingFunction nu


A particularly useful construct is the cumulative distribution function for a measure, which calculates the probability of a region less than or equal to some number:

cdf :: Measure Double -> Double -> Double
cdf nu x = integrate (negativeInfinity to x) nu

negativeInfinity :: Double
negativeInfinity = negate (1 / 0)

to :: (Num a, Ord a) => a -> a -> a -> a
to a b x
| x >= a && x <= b = 1
| otherwise        = 0


The beta(2, 2) distribution is symmetric around its mean 0.5, so the probability of the region $[0, 0.5]$ should itself be 0.5. This checks out as expected, modulo approximation error due to quadrature:

> cdf (beta 2 2) 0.5
0.4951814897381374


Similarly for measurable spaces without any notion of order, there’s a simple CDF analogue that calculates the probability of a region that contains the given points:

containing :: (Num a, Eq b) => [b] -> b -> a
containing xs x
| x elem xs = 1
| otherwise   = 0


And probably the least interesting query of all is the simple ‘volume’, which calculates the total measure of a space. For any probability measure this must obviously be one, so it can at least be used as a quick sanity check:

volume :: Measure Double -> Double
volume = integrate (const 1)


Convolution and Friends

I mentioned in the last post that applicativeness corresponds to independence in some sense, and that independent measures over the same measurable space can be convolved together, à la:

for measures $\nu$ and $\zeta$ on $M$. In Haskell-land it’s well-known that any applicative instance gives you a free ‘Num’ instance, and the story is no different here:

instance Num a => Num (Measure a) where
(+)         = liftA2 (+)
(-)         = liftA2 (-)
(*)         = liftA2 (*)
abs         = fmap abs
signum      = fmap signum
fromInteger = pure . fromInteger


There are a few neat ways to demonstrate this kind of thing. Let’s use a Gaussian measure here as a running example:

gaussian :: Double -> Double -> Measure Double
gaussian m s = fromDensityFunction (density m s) where
density m s x
| s <= 0    = 0
| otherwise =
1 / (s * sqrt (2 * pi)) * exp (negate ((x - m) ^^ 2) / (2 * (s ^^ 2)))


First, consider a chi-squared measure with $k$ degrees of freedom. We could create this directly using a density function, but instead we can represent it by summing up independent squared Gaussian measures:

chisq :: Int -> Measure Double
chisq k = sum (replicate k normal) where
normal = fmap (^ 2) (gaussian 0 1)


To sanity check the result, we can compute the mean and variance of a $\chi^2(2)$ measure, which should be $k$ and $2k$ respectively for $k = 2$:

> expectation (chisq 2)
2.0000000000000004
> variance (chisq 2)
4.0


As a second example, consider a product of independent Gaussian measures. This is a trickier distribution to deal with analytically (see here), but we can use some well-known identities for general independent measures in order to verify our results. For any independent measures $\mu$ and $\nu$, we have:

and

for the expectation and variance of their product. So for a product of independent Gaussians w/parameters (1, 2) and (2, 3) respectively, we expect to see 2 for its expectation and 61 for its variance:

> expectation (gaussian 1 2 * gaussian 2 3)
2.0000000000000001
> variance (gaussian 1 2 * gaussian 2 3)
61.00000000000003


Woop!

Wrapping Up

And there you have it, a continuation-based implementation of the Giry monad. You can find a bunch of code with similar functionality to this packaged up in my old measurable library on GitHub if you’d like to play around with the concepts.

That library has accumulated a few stars since I first pushed it up in 2013. I think a lot of people are curious about these weird measure things, and this framework at least gives you the ability to play around with a representation for measures directly. I found it particularly useful for really grokking, say, that integrating some function $f$ against a probability measure $\nu$ is identical to integrating the identity function against the probability measure $\texttt{fmap} \, f \, \nu$. And there are a few similar concepts there that I find really pop out when you play with measures directly, rather than when one just works with them on paper.

But let me now tell you why the Giry monad sucks in practice.

Take a look at this integral expression, which is brought about due to a monadic bind:

For simplicitly, let’s assume that $M$ is discrete and has cardinality $|M|$. This means that the integral reduces to

for $d\mu(m)$ and $d\nu$ the appropriate Radon-Nikodym derivatives. You can see that the total number of operations involved in the integral is $O(|M|^2)$, and indeed, for $p$ monadic binds the computational complexity involved in evaluating all the integrals involved is exponential, on the order of $|M|^p$. It was no coincidence that I demonstrated a variance calculation for a $\chi^2(2)$ distribution instead of for a $\chi^2(10)$.

This isn’t really much of a surprise - the cottage industry of approximating integrals exists because integration is hard in practice, and integration is surely best avoided whenever one can get away with doing so. Vikash Mansinghka’s quote on this topic is fitting: “don’t calculate probabilities - sample good guesses.” I’ll also add: relegate the measures to measure theory, where they seem to belong.

The Giry monad is a lovely abstract construction for formalizing the monadic structure of probability, and as canonical probabilistic objects, measures and integrals are tremendously useful when working theoretically. But they’re a complete non-starter when it comes to getting anything nontrivial done in practice. For that, there are far more useful representations for probability distributions in Haskell - notably, the sampling function or random variable representation found in things like mwc-probability/mwc-random-monad and random-fu, or even better, the structural representation based on free or operational monads like I’ve written about before, or that you can find in something like monad-bayes.

The intuitions gleaned from playing with the Giry monad carry over precisely to other representations for the probability monad. In all cases, ‘return’ will correspond, semantically, to constructing a Dirac distribution at a point, while ‘bind’ will correspond to a marginalizing operator. The same is true for the underlying (applicative) functor structure: ‘fmap’ always corresponds to a density-preserving transformation of the support, while applicativeness corresponds to independence (yielding convolution, etc.). And you have to admit, the connection to continuations is pretty cool.

There is clearly some connection to the codensity monad as well, but I think I’ll let someone else figure out the specifics of that one. Something something right-Kan extension..

The Giry monad is the canonical probability monad that operates on the level of measures, which are the abstract constructs that canonically represent probability distributions. It’s sort of the baseline by which all other probability monads can be judged.

In this article I’m going to go through the categorical and measure-theoretic foundations of the Giry monad. In another article, I’ll describe how you can implement it in a very faithful sense in Haskell.

I was putting some notes together for another project and wound up writing up things up in a somewhat blog-friendly style, but this isn’t intended to be a tutorial per se. Really this isn’t the kind of content I’d usually post here, but since I’ve jotted everything up, I figured I may as well. If you like extremely dry mathematics and computer science, you’re in the right place.

I won’t define everything under the sun here - for properties or coherence conditions or other things that I’ve elided details on, check out something like Mac Lane or Aliprantis & Border. I’ll include some references at the end.

This is the game plan we’re working with:

• Define monads and their supporting machinery in a categorical sense.
• Define probability measures and some required background around that.
• Construct the functor that maps a measurable space to the collection of all probability measures on that space.
• Demonstrate that it’s a monad.

Let’s get started.

Categorical Foundations

A category $C$ is a collection of objects and morphisms between them. So if $W$, $X$, $Y$, and $Z$ are objects in $C$, then $f : W \to X$, $g : X \to Y$, and $h : Y \to Z$ are examples of morphisms. These morphisms can be composed in the obvious associative way, i.e.

and there exist identity morphisms (or automorphisms) that simply map objects to themselves.

A functor is a mapping between categories (equivalently, it’s a morphism in the category of so-called ‘small’ categories). The functor $F : C \to D$ takes every object in $C$ to some object in $D$, and every morphism in $C$ to some morphism in $D$, such that the structure of morphism composition is preserved. An endofunctor is a functor from a category to itself, and a bifunctor is a functor from a pair of categories to another category, i.e. $F : A \times B \to C$.

A natural transformation is a mapping between functors. So for two functors $F, G : C \to D$, a natural transformation $\epsilon : F \to G$ associates to every object $c$ in $C$ a morphism $\epsilon_c : F(c) \to G(c)$ in $D$.

A monoidal category $C$ is a category with some additional monoidal structure, namely an identity object $I$ and a bifunctor $\otimes : C \times C \to C$ called the tensor product, plus several natural isomorphisms that provide the associativity of the tensor product and its right and left identity with the identity object $I$.

A monoid $(M, \mu, \eta)$ in a monoidal category $C$ is an object $M$ in $C$ together with two morphisms (obeying the standard associativity and identity properties) that make use of the category’s monoidal structure: the associative binary operator $\mu : M \otimes M \to M$, and the identity $\eta : I \to M$.

A monad is (infamously) a ‘monoid in the category of endofunctors’. So take the category of endofunctors $\mathcal{F}$ whose objects are endofunctors and whose morphisms are natural transformations between them. This is a monoidal category; there exists an identity endofunctor $1_\mathcal{F}(F) = F$ for all $F$ in $\mathcal{F}$, plus a tensor product $\otimes : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ defined by functor composition such that the required associativity and identity properties hold. $\mathcal{F}$ is thus a monoidal category, and any specific monoid $(F, \mu, \eta)$ we construct on it is a specific monad.

Probabilistic Foundations

A measurable space $(X, \mathcal{X})$ is a set $X$ equipped with a topology-like structure called a $\sigma$-algebra $\mathcal{X}$ that essentially contains every well-behaved subset of $X$ in some sense. A measure $\nu : \mathcal{X} \to \mathbb{R}$ is a particular kind of set function from the $\sigma$-algebra to the nonnegative real line. A measure just assigns a generalized notion of area or volume to well-behaved subsets of $X$. In particular, if the total possible area or volume of the underlying set is 1 then we’re dealing with a probability measure. A measurable space completed with a measure, e.g. $(X, \mathcal{X}, \nu)$ is called a measure space, and a measurable space completed with a probability measure is called a probability space.

There is a lot of overloaded lingo around the word ‘measurable’. A ‘measurable set’ is an element of a $\sigma$-algebra in a measurable space. A measurable mapping is a mapping between measurable spaces. Given a ‘source’ measurable space $(X, \mathcal{X})$ and ‘target’ measurable space $(Y, \mathcal{Y})$, a measurable mapping $(X, \mathcal{X}) \to (Y, \mathcal{Y})$ is a map $T : X \to Y$ with the property that, for any measurable set in the target, the inverse image is measurable in the source. Or, formally, for any $B$ in $\mathcal{Y}$, you have that $T^{-1}(B)$ is in $\mathcal{X}$.

The Space of Probability Measures on a Measurable Space

If you consider the collection of all measurable spaces and measurable mappings between them, you get a category. Define $\textbf{Meas}$ to be the category of measurable spaces. So, objects are measurable spaces and morphisms are the measurable mappings between them.

For any specific measurable space $M$ in $\textbf{Meas}$, we can consider the space of all possible probability measures that could be placed on it and denote that $\mathcal{P}(M)$. To be clear, $\mathcal{P}(M)$ is a space of measures - that is, a space in which the points themselves are probability measures.

What’s remarkable about $\mathcal{P}(M)$ is that it is itself a measurable space. Let me explain.

As a probability measure, any element of $\mathcal{P}(M)$ is a function from measurable subsets of $M$ to the interval $[0, 1]$ in $\mathbb{R}$. That is: if $M$ is the measurable space $(X, \mathcal{X})$, then a point $\nu$ in $\mathcal{P}(M)$ is a function $\mathcal{X} \to \mathbb{R}$. For any measurable $A$ in $M$, there just naturally exists a sort of ‘evaluation’ mapping I’ll call $\tau_A: \mathcal{P}(M) \to \mathbb{R}$ that takes a measure on $M$ and evaluates it on the set $A$. To be explicit: if $\nu$ is a measure in $\mathcal{P}(M)$, then $\tau_A$ simply evaluates $\nu(A)$. It ‘runs’ the measure in a sense; in Haskell, $\tau_A$ would be analogous to a function like \f -> f a for some a.

This evaluation map $\tau_A$ corresponds to an integral. If you have a measurable space $(X, \mathcal{X})$, then for any $A$ a subset in $\mathcal{X}$, $\tau_A(\nu) = \nu(A) = \int_{X}\chi_A d\nu$ for $\chi$ the characteristic or indicator function of $A$ (where $\chi(x)$ is $1$ if $x$ is in $A$, and is $0$ otherwise). And we can actually extend $\tau$ to operate over measurable mappings from $(X, \mathcal{X})$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ is a suitable $\sigma$-algebra on $\mathbb{R}$. Here we typically use what’s called the Borel $\sigma$-algebra, which takes a topology on the set and then generates a $\sigma$-algebra from the open sets in the topology (for $\mathbb{R}$ we can just use the ‘usual’ topology generated by the Euclidean metric). For $f : X \to \mathbb{R}$ a measurable function, we can define the evaluation mapping $\tau_f : \mathcal{P}(M) \to \mathbb{R}$ as $\tau_f(\nu) = \int_X f d\nu$.

We can abuse notation here a bit and just use $\tau$ to refer to ‘duck typed’ mappings that evaluate measures over measurable sets or measurable functions depending on context. If we treat $\tau_A(\nu)$ as a function $\tau(\nu)(A)$, then $\tau(\nu)$ has type $\mathcal{X} \to \mathbb{R}$. If we treat $\tau_f(\nu)$ as a function $\tau(\nu)(f)$, then $\tau(\nu)$ has type $(X \to \mathbb{R}) \to \mathbb{R}$. I’ll say $\tau_{\{A, f\}}$ to refer to the mappings that accept either measurable sets or functions.

In any case. For a measurable space $M$, there exists a topology on $\mathcal{P}(M)$ called the weak-* topology that makes all the evaluation mappings $\tau_{\{A, f\}}$ continuous for any measurable set $A$ or measurable function $f$. From there, we can generate the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{P}(M))$ that makes the evaluation functions $\tau_{\{A, f\}}$ measurable. The result is that $(\mathcal{P}(M), \mathcal{B}(\mathcal{P}(M)))$ is itself a measurable space, and thus an object in $\textbf{Meas}$.

The space $\mathcal{P}(M)$ actually has all sorts of insane properties that one wouldn’t expect - there are implications on convexity, completeness, compactness and such that carry over from $M$. But I digress.

$\mathcal{P}$ is a Functor

So: for any $M$ an object in $\textbf{Meas}$, we have that $\mathcal{P}(M)$ is also an object in $\textbf{Meas}$. And if you look at $\mathcal{P}$ like a functor, you notice that it takes objects of $\textbf{Meas}$ to objects of $\textbf{Meas}$. Indeed, you can define an analogous procedure on morphisms in $\textbf{Meas}$ as follows. Take $N$ to be another object (read: measurable space) in $\textbf{Meas}$ and $T : M \to N$ to be a morphism (read: measurable mapping) between them. Now, for any measure $\nu$ in $\mathcal{P}(M)$ we can define $\mathcal{P}(T)(\nu) = \nu \circ T^{-1}$ (this is called the image, distribution, or pushforward of $\nu$ under $T$). For some $T$ and $\nu$, $\mathcal{P}(T)(\nu)$ thus takes measurable sets in $N$ to a value in the interval $[0, 1]$ - that is, it is a measure on $\mathcal{P}(N)$. So we have that:

and so $\mathcal{P}$ is an endofunctor on $\textbf{Meas}$.

$\mathcal{P}$ is a Monad

See where we’re going here? If we can define natural transformations $\mu$ and $\eta$ such that $(\mathcal{P}, \mu, \eta)$ is a monoid in the category of endofunctors, we’ll have defined a monad. We thus need to come up with a suitable monoidal structure, et voilà.

First the identity. We want a natural transformation $\eta$ between the identity functor $1_{\mathcal{F}}$ and the functor $\mathcal{P}$ such that $\eta_M : 1_{\mathcal{F}}(M) \to \mathcal{P}(M)$ for any measurable space $M$ in $\textbf{Meas}$. Evaluating the identity functor simplifies things to $\eta_M : M \to \mathcal{P}(M)$.

We can define this concretely as follows. Grab a measurable space $M$ in $\textbf{Meas}$ and define $\eta(x)(A) = \chi_A(x)$ for any point $x \in M$ and any measurable set $A \subseteq M$. $\eta(x)$ is thus a probability measure on $M$ - we assign $1$ to measurable sets that contain $x$, and 0 to those that don’t. If we peel away another argument, we have that $\eta : M \to \mathcal{P}(M)$, as required.

So $\eta$ takes points in measurable spaces to probability measures on those spaces. In technical parlance, it takes a point $x$ to the Dirac measure at $x$ - the probability measure that places the entirety of its mass at $x$.

Now for the other part of the monoidal structure, $\mu$. I initially found this next part to be a bit of a mind fuck, but let me see what I can do about that.

Recall that the category of endofunctors, $\mathcal{F}$, is monoidal, so there exists a tensor product $\otimes : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ that we can deal with, which here just corresponds to functor composition. We’re looking for a natural transformation:

which is often written as:

Take $M = (X, \mathcal{X})$ a measurable space in $\textbf{Meas}$ and then consider the space of probability measures over it, $\mathcal{P}(M)$. Then take the space of probability measures over the space of probability measures on $M$, $\mathcal{P}(\mathcal{P}(M))$. Since $\mathcal{P}$ is an endofunctor, this is again a measurable space, and for any measurable subset $A$ of $M$ we again have a family of mappings $\tau_A$ that take a probability measure in $\mathcal{P}(\mathcal{P}(M))$ and evaluate it on $A$. We want $\mu$ to be the thing that turns a measure over measures $\rho$ into a plain old probability measure on $\mathcal{P}(M)$.

In the context of probability theory, this kind of semigroup action is a marginalizing operator. We’re taking the ‘uncertainty’ captured in $\mathcal{P}(\mathcal{P}(M))$ via the probability measure $\rho$ and smearing it into the probability measures in $\mathcal{P}(M)$.

Take $\rho$ in $\mathcal{P}(\mathcal{P}(M))$ and some $A$ a measurable subset of $M$. We can define $\mu$ as follows:

Using some lambda calculus notation to see the argument for $\tau_A$, we can expand the integrals to get the following gnarly expression:

Notice what’s happening here. For $M$ a measurable space, we’re integrating over $\mathcal{P}(M)$ the space of probability measures on $M$, with respect to the probability measure $\rho$, which itself is a point in the space of probability measures over probability measures on $M$, $\mathcal{P}(\mathcal{P}(M))$. Whew.

The spaces we’re integrating over here are unusual, but $\rho$ is still a probability measure, so when applied to a measurable set in $\mathcal{B}(\mathcal{P}(M))$ it results in a probability in $[0, 1]$. So, peeling back an argument, we have that $\mu(\rho)$ has type $\mathcal{X} \to \mathbb{R}$. In other words, it’s a probability measure on $M$, and thus is in $\mathcal{P}(M)$. And if we peel back another argument, we find that:

so, as required, that

It’s also worth noting that we can overload the notation for $\mu$ in the same way we did for $\tau$, i.e. to supply measurable functions in addition to measurable sets:

Combining the three components, we get $(\mathcal{P}, \mu, \eta)$, the canonical Giry monad.

In Haskell, when we’re dealing with monads we typically use the bind operator $\gg\!\!=$ instead of manually dealing with the functorial structure and $\mu$ (called ‘join’). Bind has the type:

and for illustration, we can define $\gg\!\!=$ for the Giry monad like so:

Here $\rho$ is in $\mathcal{P}(M)$, $g$ is in $M \to \mathcal{P}(N)$, and $f$ is in $N \to \mathbb{R}$, so note that we potentially simplify the outermost integral enormously. It now operates over a general measurable space, rather than a space of measures in particular, and this will come in handy when we get to implementation details in the next post.

Wrapping Up

That’s about it for now. It’s worth noting as a kind of footnote here that the existence of the Giry monad also obviously implies the existence of a Giry applicative functor. But the official situation for applicative functors seems kind of weird in this context, and I’m not yet up to the task of dealing with it formally.

Intuitively, one should be able to define the binary applicative operator characteristic of its lax monoidal structure as follows:

But this has some really weird measure-theoretic implications - namely, that it assumes the existence of a space of probability measures over the space of all measurable functions $M \to N$, which is not trivial to define and indeed may not even exist. It seems like some people are looking into this problem as I just happened to stumble on this paper on the arXiv while doing some googling. I notice that some people on e.g. nLab require categories with additional structure beyond $\textbf{Meas}$ for the development of the Giry monad as well, for example the category of Polish (separable, completely metrizable) spaces $\textbf{Pol}$, so maybe the extra structure there takes care of the quirks.

Anyway. Applicatives are neat here because applicative probability measures are independent probability measures. And the existence of applicativeness means you can do all the things with independent probability measures that you might be used to. Measure convolution and friends are good examples. Given a measurable space $M$ that supports some notion of addition and two probability measures $\nu$ and $\zeta$ in $\mathcal{P}(M)$, we can add measures together via:

where $x$ and $y$ are both points in $M$. Subtraction and multiplication translate trivially as well.

In another article I’ll detail how the Giry monad can be implemented in Haskell and point out some neat extensions. There are some cool connections to continuations and codensity monads, and seemingly de Finetti’s theorem and exchangeability. That kind of thing. It’d also be worth trying to justify independence of probability measures from a categorical perspective, which seems easier than resolving the nitty-gritty measurability qualms I mentioned above.

‘Til then! Thanks to Jason Forbes for sifting through this stuff and providing some great comments.

Rotating Squares

Here’s a short one.

I use Colin Percival’s Hacker News Daily to catch the top ten articles of the day on Hacker News. Today an article called Why Recursive Data Structures? popped up, which illustrates that recursive algorithms can become both intuitive and borderline trivial when a suitable data structure is used to implement them. This is exactly the motivation for using recursion schemes.

In the above article, Reginald rotates squares by representing them via a quadtree. If we have a square of bits, something like:

.x..
..x.
xxx.
....


then we want to be able to easily rotate it 90 degrees clockwise, for example. So let’s define a quadtree in Haskell:

{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE LambdaCase #-}

import Data.Functor.Foldable
import Data.List.Split

NodeF r r r r
| LeafF a
| EmptyF
deriving (Show, Functor)



The four fields of the ‘NodeF’ constructor correspond to the upper left, upper right, lower right, and lower left quadrants of the tree respectively.

Gimme some embedded language terms:

node :: QuadTree a -> QuadTree a -> QuadTree a -> QuadTree a -> QuadTree a
node ul ur lr ll = Fix (NodeF ul ur lr ll)

leaf :: a -> QuadTree a
leaf = Fix . LeafF

empty = Fix EmptyF


That lets us define quadtrees easily. Here’s the tree that the previous diagram corresponds to:

tree :: QuadTree Bool
tree = node ul ur lr ll where
ul = node (leaf False) (leaf True) (leaf False) (leaf False)
ur = node (leaf False) (leaf False) (leaf False) (leaf True)
lr = node (leaf True) (leaf False) (leaf False) (leaf False)
ll = node (leaf True) (leaf True) (leaf False) (leaf False)


Rotating is then really easy - we rotate each quadrant recursively. Just reach for a catamorphism:

rotate :: QuadTree a -> QuadTree a
rotate = cata $\case NodeF ul ur lr ll -> node ll ul ur lr LeafF a -> leaf a EmptyF -> empty  Notice that you just have to shift each field of ‘NodeF’ rightward, with wraparound. Then if you rotate and render the original tree you get: .x.. .x.x .xx. ....  Rotating things more times yields predictable results. If you want to rotate another structure - say, a flat list - you can go through a quadtree as an intermediate representation using the same pattern I described in Sorting with Style. Build yourself a coalgebra and algebra pair: builder :: [a] -> QuadTreeF a [a] builder = \case [] -> EmptyF [x] -> LeafF x xs -> NodeF a b c d where [a, b, c, d] = chunksOf (length xs div 4) xs consumer :: QuadTreeF a [a] -> [a] consumer = \case EmptyF -> [] LeafF a -> [a] NodeF ul ur lr ll -> concat [ll, ul, ur, lr]  and then glue them together with a hylomorphism: rotateList :: [a] -> [a] rotateList = hylo consumer builder  Neato. For a recent recursion scheme resource I’ve spotted on the Twitters, check out Pascal Hartig’s compendium in progress. Promorphisms, Pre and Post To the.. uh, ‘layperson’, pre- and postpromorphisms are probably well into the WTF category of recursion schemes. This is a mistake - they’re simple and useful, and I’m going to try and convince you of this in short order. Preliminaries: {-# LANGUAGE DeriveFunctor #-} {-# LANGUAGE LambdaCase #-} import Data.Functor.Foldable import Prelude hiding (sum)  For simplicity, let’s take a couple of standard interpreters on lists. We’ll define ‘sumAlg’ as an interpreter for adding up list contents and ‘lenAlg’ for just counting the number of elements present: sumAlg :: Num a => ListF a a -> a sumAlg = \case Cons h t -> h + t Nil -> 0 lenAlg :: ListF a Int -> Int lenAlg = \case Cons h t -> 1 + t Nil -> 0  Easy-peasy. We can use cata to make these things useful: sum :: Num a => [a] -> a sum = cata sumAlg len :: [a] -> Int len = cata lenAlg  Nothing new there; ‘sum [1..10]’ will give you 55 and ‘len [1..10]’ will give you 10. An interesting twist is to consider only small elements in some sense; say, we only want to add or count elements that are less than or equal to 10, and ignore any others. We could rewrite the previous interpreters, manually checking for the condition we’re interested in and handling it accordingly: smallSumAlg :: (Ord a, Num a) => ListF a a -> a smallSumAlg = \case Cons h t -> if h <= 10 then h + t else 0 Nil -> 0 smallLenAlg :: (Ord a, Num a) => ListF a Int -> Int smallLenAlg = \case Cons h t -> if h <= 10 then 1 + t else 0 Nil -> 0  And you get ‘smallSum’ and ‘smallLen’ by using ‘cata’ on them respectively. They work like you’d expect - ‘smallLen [1, 5, 20]’ ignores the 20 and just returns 2, for example. You can do better though. Enter the prepromorphism. Instead of writing additional special-case interpreters for the ‘small’ case, consider the following natural transformation on the list base functor. It maps the list base functor to itself, without needing to inspect the carrier type: small :: (Ord a, Num a) => ListF a b -> ListF a b small Nil = Nil small term@(Cons h t) | h <= 10 = term | otherwise = Nil  A prepromorphism is a ‘cata’-like recursion scheme that proceeds by first applying a natural transformation before interpreting via a supplied algebra. That’s.. surprisingly simple. Here are ‘smallSum’ and ‘smallLen’, defined without needing to clumsily create new special-case algebras: smallSum :: (Ord a, Num a) => [a] -> a smallSum = prepro small sumAlg smallLen :: (Ord a, Num a) => [a] -> Int smallLen = prepro small lenAlg  They work great: > smallSum [1..100] 55 > smallLen [1..100] 10  In pseudo category-theoretic notation you visualize how a prepromorphism works via the following commutative diagram: The only difference, when compared to a standard catamorphism, is the presence of the natural transformation applied via the looping arrow in the top left. The natural transformation ‘h’ has type ‘forall r. Base t r -> Base t r’, and ‘embed’ has type ‘Base t t -> t’, so their composition gets you exactly the type you need for an algebra, which is then the input to ‘cata’ there. Mapping the catamorphism over the type ‘Base t t’ brings it right back to ‘Base t t’. A postpromorphism is dual to a prepromorphism. It’s ‘ana’-like; proceed with your corecursive production, applying natural transformations as you go. Here’s a streaming coalgebra: streamCoalg :: Enum a => a -> ListF a a streamCoalg n = Cons n (succ n)  A normal anamorphism would just send this thing shooting off into infinity, but we can use the existing ‘small’ natural transformation to cap it at 10: smallStream :: (Ord a, Num a, Enum a) => a -> [a] smallStream = postpro small streamCoalg  You get what you might expect: > smallStream 3 [3,4,5,6,7,8,9,10]  And similarly, you can visualize a postpromorphism like so: In this case the natural transformation is applied after mapping the postpromorphism over the base functor (hence the ‘post’ namesake). Comonadic Markov Chain Monte Carlo Some time ago I came across a way to in-principle perform inference on certain probabilistic programs using comonadic structures and operations. I decided to dig it up and try to use it to extend the simple probabilistic programming language I talked about a few days ago with a stateful, experimental inference backend. In this post we’ll • Represent probabilistic programs as recursive types parameterized by a terminating instruction set. • Represent execution traces of probabilistic programs via a simple transformation of our program representation. • Implement the Metropolis-Hastings algorithm over this space of execution traces and thus do some inference. Let’s get started! Representing Programs That Terminate I like thinking of embedded languages in terms of instruction sets. That is: I want to be able to construct my embedded language by first defining a collection of abstract instructions and then using some appropriate recursive structure to represent programs over that set. In the case of probabilistic programs, our instructions are probability distributions. Last time we used the following simple instruction set to define our embedded language: data ModelF r = BernoulliF Double (Bool -> r) | BetaF Double Double (Double -> r) deriving Functor  We then created an embedded language by just wrapping it up in the higher-kinded Free type to denote programs of type Model. data Free f a = Pure a | Free (f (Free f a)) type Model = Free ModelF  Recall that Free represents programs that can terminate, either by some instruction in the underlying instruction set, or via the Pure constructor of the Free type itself. The language defined by Free ModelF is expressive enough to easily construct a ‘forward-sampling’ interpreter, as well as a simple rejection sampler for performing inference. Notice that we don’t have a terminating instruction in ModelF itself - if we’re using it, then we need to rely on the Pure constructor of Free to terminate programs. Otherwise they’d just have to recurse forever. This can be a bit limiting if we want to transform a program of type Free ModelF to something else that doesn’t have a notion of termination baked-in (Fix, for example). Let’s tweak the ModelF type to get the following: data ModelF a r = BernoulliF Double (Bool -> r) | BetaF Double Double (Double -> r) | NormalF Double Double (Double -> r) | DiracF a deriving Functor  Aside from adding another foundational distribution - NormalF - we’ve also added a new constructor, DiracF, which carries a parameter with type a. We need to incorporate this carrier type in the overall type of ModelF as well, so ModelF itself also gets a new type parameter to carry around. The DiracF instruction is a terminating instruction; it has no recursive point and just terminates with a value of type a when reached. It’s structurally equivalent to the Pure a branch of Free that we were relying on to terminate our programs previously - the only thing we’ve done is add it to our instruction set proper. Why DiracF? A Dirac distribution places the entirety of its probability mass on a single point, and this is the exact probabilistic interpretation of the applicative pure or monadic return that one encounters with an appropriate probability type. Intuitively, if I sample a value $x$ from a uniform distribution, then that is indistinguishable from sampling $x$ from said uniform distribution and then sampling from a Dirac distribution with parameter $x$. Make sense? If not, it might be helpful to note that there is no difference between any of the following (to which uniform and dirac are analogous): > action :: m a > action >>= return :: m a > action >>= return >>= return >>= return :: m a  Wrapping ModelF a up in Free, we get the following general type for our programs: type Program a = Free (ModelF a)  And we can construct a bunch of embedded language terms in the standard way: beta :: Double -> Double -> Program a Double beta a b = liftF (BetaF a b id) bernoulli :: Double -> Program a Bool bernoulli p = liftF (BernoulliF p id) normal :: Double -> Double -> Program a Double normal m s = liftF (NormalF m s id) dirac :: a -> Program a b dirac x = liftF (DiracF x)  Program is a general type, capturing both terminating and nonterminating programs via its type parameters. What do I mean by this? Note that in Program a b, the a type parameter can only be concretely instantiated via use of the terminating dirac term. On the other hand, the b type parameter is unaffected by the dirac term; it can only be instantiated by the other nonterminating terms: beta, bernoulli, normal, or compound expressions of these. We can thus distinguish between terminating and nonterminating programs at the type level, like so: type Terminating a = Program a Void type Model b = forall a. Program a b  Void is the uninhabited type, brought into scope via Data.Void or simply defined via data Void = Void Void. Any program that ends via a dirac instruction must be Terminating, and any program that doesn’t end with a dirac instruction can not be Terminating. We’ll just continue to call a nonterminating program a Model, as before. Good. So if it’s not clear: from a user’s perspective, nothing has changed. We still write probabilistic programs using simple monadic language terms. Here’s a Gaussian mixture model where the mixing parameter follows a beta distribution, for example: mixture :: Double -> Double -> Model Double mixture a b = do prob <- beta a b accept <- bernoulli prob if accept then normal (negate 2) 0.5 else normal 2 0.5  Meanwhile the syntax tree generated looks something like the following. It’s more or less a traditional probabilistic graphical model description of our program: It’s important to note that in this embedded framework, the only pieces of the syntax tree that we can observe are those related directly to our primitive instructions. For our purposes this is excellent - we can focus on programs entirely at the level of their probabilistic components, and ignore the deterministic parts that would otherwise be distractions. To collect samples from mixture, we can first interpret it into a sampling function, and then simulate from it. The toSampler function from last time doesn’t change much: toSampler :: Program a a -> Prob IO a toSampler = iterM$ \case
BernoulliF p f -> Prob.bernoulli p >>= f
BetaF a b f    -> Prob.beta a b >>= f
NormalF m s f  -> Prob.normal m s >>= f
DiracF x       -> return x


Sampling from mixture 2 3 a thousand times yields the following

> simulate (toSampler (mixture 2 3))


Note that the rightmost component gets more traffic due to the hyperparameter combination of 2 and 3 that we provided to mixture.

Also, a note - since we have general recursion in Haskell, so-called ‘terminating’ programs here can actually.. uh, fail to terminate. They must only terminate as far as we can express the sentiment at the embedded language level. Consider the following, for example:

foo :: Terminating a
foo = (loop 1) >>= dirac where
loop a = do
p <- beta a 1
loop p


foo here doesn’t actually terminate. But at least this kind of weird case can be picked up in the types:

> :t simulate (toSampler foo)
simulate (toSampler foo) :: IO Void


If you try to sample from a distribution over Void or forall a. a then I can’t be held responsible for what you get up to. But there are other cases, sadly, where we’re also out of luck:

trollGeometric :: Double -> Model Int
trollGeometric p = loop where
loop = do
accept <- return False
if   accept
then return 1
else fmap succ loop


A geometric distribution that actually used its argument $p$, for $% $, could be guaranteed to terminate with probability 1. This one doesn’t, so trollGeometric undefined >>= dirac won’t.

At the end of the day we’re stuck with what our host language offers us. So, take the termination guarantees for our embedded language with a grain of salt.

Stateful Inference

In the previous post we used a simple rejection sampler to sample from a conditional distribution. ‘Vanilla’ Monte Carlo algorithms like rejection and importance sampling are stateless. This makes them nice in some ways - they tend to be simple to implement and are embarrassingly parallel, for example. But the curse of dimensionality prevents them from scaling well to larger problems. I won’t go into detail on that here - for a deep dive on the topic, you probably won’t find anything better than this phenomenal couple of talks on MCMC that Iain Murray gave at a MLSS session in Cambridge in 2009. I think they’re unparalleled to this day.

The point is that in higher dimensions we tend to get a lot out of state. Essentially, if one finds an interesting region of high-dimensional parameter space, then it’s better to remember where that is, rather than forgetting it exists as soon as one stumbles onto it. The manifold hypothesis conjectures that interesting regions of space tend to be near other interesting regions of space, so exploring neighbourhoods of interesting places tends to pay off. Stateful Monte Carlo methods - namely, the family of Markov chain Monte Carlo algorithms - handle exactly this, by using a Markov chain to wander over parameter space. I’ve written on MCMC in the past - you can check out some of those articles if you’re interested.

In the stateless rejection sampler we just performed conditional inference via the following algorithm:

• Sample from a parameter model.
• Sample from a data model, using the sample from the parameter model as input.
• If the sample from the data model matches the provided observations, return the sample from the parameter model.

By repeating this many times we get a sample of arbitrary size from the appropriate conditional, inverse, or posterior distribution (whatever you want to call it).

In a stateful inference routine - here, the good old Metropolis-Hastings algorithm - we’re instead going to do the following repeatedly:

• Sample from a parameter model, recording the way the program executed in order to return the sample that it did.
• Compute the cost, in some sense, of generating the provided observations, using the sample from the parameter model as input.
• Propose a new sample from the parameter model by perturbing the way the program executed and recording the new sample the program outputs.
• Compute the cost of generating the provided observations using this new sample from the parameter model as input.
• Compare the costs of generating the provided observations under the respective samples from the parameter models.
• With probability depending on the ratio of the costs, flip a coin. If you see a head, then move to the new, proposed execution trace of the program. Otherwise, stay at the old execution trace.

This procedure generates a Markov chain over the space of possible execution traces of the program - essentially, plausible ways that the program could have executed in order to generate the supplied observations.

Implementations of Church use variations of this method to do inference, the most famous of which is a low-overhead transformational compilation procedure described in a great and influential 2011 paper by David Wingate et al.

Representing Running Programs

To perform inference on probabilistic programs according to the aforementioned Metropolis-Hastings algorithm, we need to represent executing programs somehow, in a form that enables us to examine and modify their internal state.

How can we do that? We’ll pluck another useful recursive structure from our repertoire and consider the humble Cofree:

data Cofree f a = a :< f (Cofree f a)


Recall that Cofree allows one to annotate programs with arbitrary information at each internal node. This is a great feature; if we can annotate each internal node with important information about its state - its current value, the current state of its generator, the ‘cost’ associated with it - then we can walk through the program and examine it as required. So, it can capture a ‘running’ program in exactly the way we need.

Let’s describe running programs as values having the following Execution type:

type Execution a = Cofree (ModelF a) Node


The Node type is what we’ll use to describe the internal state of each node on the program. I’ll define it like so:

data Node = Node {
nodeCost    :: Double
, nodeValue   :: Dynamic
, nodeSeed    :: MWC.Seed
, nodeHistory :: [Dynamic]
} deriving Show


I’ll elaborate on this type below, but you can see that it captures a bunch of information about the state of each node.

One can mechanically transform any Free-encoded program into a Cofree-encoded program, so long as the original Free-encoded program can terminate of its own accord, i.e. on the level of its own instructions. Hence the need for our Terminating type and all that.

In our case, setting everything up just right takes a bit of code, mainly around handling pseudo-random number generators in a pure fashion. So I won’t talk about every little detail of it right here. The general idea is to write a function that takes instructions to the appropriate state captured by a Node value, like so:

initialize :: Typeable a => MWC.Seed -> ModelF a b -> Node
initialize seed = \case
BernoulliF p _ -> runST $do (nodeValue, nodeSeed) <- samplePurely (Prob.bernoulli p) seed let nodeCost = logDensityBernoulli p (unsafeFromDyn nodeValue) nodeHistory = mempty return Node {..} BetaF a b _ -> runST$ do
(nodeValue, nodeSeed) <- samplePurely (Prob.beta a b) seed
let nodeCost    = logDensityBeta a b (unsafeFromDyn nodeValue)
nodeHistory = mempty
return Node {..}

...


You can see that for each node, I sample from it, calculate its cost, and then initialize its ‘history’ as an empty list.

Here it’s worth going into a brief aside.

There are two mildly annoying things we have to deal with in this situation. First, individual nodes in the program typically sample values at different types, and second, we can’t easily use effects when annotating. This means that we have to pack heterogeneously-typed things into a homogeneously-typed container, and also use pure random number generation facilities to sample them.

A quick-and-dirty answer for the first case is to just use dynamic typing when storing the values. It works and is easy, but of course is subject to the standard caveats. I use a function called unsafeFromDyn to convert dynamically-typed values back to a typed form, so you can gauge the safety of all this for yourself.

For the second case, I just use the ST monad, along with manual state snapshotting, to execute and iterate a random number generator. Pretty simple.

Also: in terms of efficiency, keeping a node’s history on-site at each execution falls into the ‘completely insane’ category, but let’s not worry much about efficiency right now. Prototypes gonna prototype and all that.

Anyway.

Given this initialize function, we can transform a terminating program into a running program by simple recursion. Again, we can only transform programs with type Terminating a because we need to rule out the case of ever visiting the Pure constructor of Free. We handle that by the absurd function provided by Data.Void:

execute :: Typeable a => Terminating a -> Execution a
execute = annotate defaultSeed where
defaultSeed         = (42, 108512)
annotate seeds term = case term of
Pure r -> absurd r
Free instruction ->
let (nextSeeds, generator) = xorshift seeds
seed  = MWC.toSeed (V.singleton generator)
node  = initialize seed instruction
in  node :< fmap (annotate nextSeeds) instruction


And there you have it - execute takes a terminating program as input and returns a running program - an execution trace - as output. The syntax tree we had previously gets turned into something like this:

Perturbing Running Programs

Given an execution trace, we’re able to step through it sequentially and investigate the program’s internal state. But to do inference we also need to modify it as well. What’s the answer here?

Just as Free has a monadic structure that allows us to write embedded programs using built-in monadic combinators and do-notation, Cofree has a comonadic structure that is amenable to use with the various comonadic combinators found in Control.Comonad. The most important for our purposes is the comonadic ‘extend’ operation that’s dual to monad’s ‘bind’:

extend :: Comonad w => (w a -> b) -> w a -> w b
extend f = fmap f . duplicate


To perturb a running program, we can thus write a function that perturbs any given annotated node, and then extend it over the entire execution trace.

The perturbNode function can be similar to the initialize function from earlier; it describes how to perturb every node based on the instruction found there:

perturbNode :: Execution a -> Node
perturbNode (node@Node {..} :< cons) = case cons of
BernoulliF p _ -> runST $do (nvalue, nseed) <- samplePurely (Prob.bernoulli p) nodeSeed let nscore = logDensityBernoulli p (unsafeFromDyn nvalue) return$! Node nscore nvalue nseed nodeHistory

BetaF a b _ -> runST $do (nvalue, nseed) <- samplePurely (Prob.beta a b) nodeSeed let nscore = logDensityBeta a b (unsafeFromDyn nvalue) return$! Node nscore nvalue nseed nodeHistory

...


Note that this is a very crude way to perturb nodes - we’re just sampling from whatever distribution we find at each one. A more refined procedure would sample from each node on a more local basis, sampling from its respective domain in a neighbourhood of its current location. For example, to perturb a BetaF node we might sample from a tiny Gaussian bubble around its current location, repeating the process if we happen to ‘fall off’ the support. I’ll leave matters like that for another post.

Perturbing an entire trace is then as easy as I claimed it to be:

perturb :: Execution a -> Execution a
perturb = extend perturbNode


For some comonadic intuition: when we ‘extend’ a function over an execution, the trace itself gets ‘duplicated’ in a comonadic context. Each node in the program becomes annotated with a view of the rest of the execution trace from that point forward. It can be difficult to visualize at first, but I reckon the following image is pretty faithful:

Each annotation then has perturbNode applied to it, which reduces the trace back to the standard annotated version we saw before.

Iterating the Markov Chain

So: to move around in parameter space, we’ll propose state changes by perturbing the current state, and then accept or reject proposals according to local economic conditions.

If you already have no idea what I’m talking about, then the phrase ‘local economic conditions’ probably didn’t help you much. But it’s a useful analogy to have in one’s head. Each state in parameter space has a cost associated with it - the cost of generating the observations that we’re conditioning on while doing inference. If certain parameter values yield a data model that is unlikely to generate the provided observations, then those observations will be expensive to generate when measured in terms of log-likelihood. Parameter values that yield data models more likely to generate the supplied observations will be comparatively cheaper.

If a proposed execution trace is significantly cheaper than the trace we’re currently at, then we usually want to move to it. We allow some randomness in our decision to keep everything nice and measure-preserving.

We can thus construct the conditional distribution over execution traces using the following invert function, using the same nomenclature as the rejection sampler we used previously. To focus on the main points, I’ll elide some of its body:

invert
:: (Eq a, Typeable a, Typeable b)
=> Int -> [a] -> Model b -> (b -> a -> Double)
-> Model (Execution b)
invert epochs obs prior ll = loop epochs (execute terminated) where
terminated = prior >>= dirac
loop n current
| n == 0    = return current
| otherwise = do
let proposal = perturb current

-- calculate costs and movement probability here

accept <- bernoulli prob
let next = if accept then proposal else stepGenerators current
loop (pred n) (snapshot next)


There are a few things to comment on here.

First, notice how the return type of invert is Model (Execution b)? Using the semantics of our embedded language, it’s literally a standard model over execution traces. The above function returns a first-class value that is completely uninterpreted and abstract. Cool.

We’re also dealing with things a little differently from the rejection sampler that we built previously. Here, the data model is expressed by a cost function; that is, a function that takes a parameter value and observation as input, and returns the cost of generating the observation (conditional on the supplied parameter value) as output. This is the approach used in the excellent Practical Probabilistic Programming with Monads paper by Adam Scibior et al and also mentioned by Dan Roy in his recent talk at the Simons Institute. Ideally we’d just reify the cost function here from the description of a model directly (to keep the interface similar to the one used in the rejection sampler implementation), but I haven’t yet found a way to do this in a type-safe fashion.

Regardless of whether or not we accept a proposed move, we need to snapshot the current value of each node and add it to that node’s history. This can be done using another comonadic extend:

snapshotValue :: Cofree f Node -> Node
snapshotValue (Node {..} :< cons) = Node { nodeHistory = history, .. } where
history = nodeValue : nodeHistory

snapshot :: Functor f => Cofree f Node -> Cofree f Node
snapshot = extend snapshotValue


The other point of note is minor, but an extremely easy detail to overlook. Since we’re handling random value generation at each node purely, using on-site PRNGs, we need to iterate the generators forward a step in the event that we don’t accept a proposal. Otherwise we’d propose a new execution based on the same generator states that we’d used previously! For now I’ll just iterate the generators by forcing a sample of a uniform variate at each node, and then throwing away the result. To do this we can use the now-standard comonadic pattern:

stepGenerator :: Cofree f Node -> Node
stepGenerator (Node {..} :< cons) = runST \$ do
(nval, nseed) <- samplePurely (Prob.beta 1 1) nodeSeed
return Node {nodeSeed = nseed, ..}

stepGenerators :: Functor f => Cofree f Node -> Cofree f Node
stepGenerators = extend stepGenerator


Inspecting Execution Traces

Alright so let’s see how this all works. Let’s write a model, condition it on some observations, and do inference.

We’ll choose our simple Gaussian mixture model from earlier, where the mixing probability follows a beta distribution, and cluster assignment itself follows a Bernoulli distribution. We thus choose the ‘leftmost’ component of the mixture with the appropriate mixture probability.

We can break the mixture model up as follows:

prior :: Double -> Double -> Model Bool
prior a b = do
p <- beta a b
bernoulli p

likelihood :: Bool -> Model Double
likelihood left
| left      = normal (negate 2) 0.5
| otherwise = normal 2 0.5


Let’s take a look at some samples from the marginal distribution. This time I’ll flip things and assign hyperparameters of 3 and 2 for the prior:

> simulate (toSampler (prior 3 2 >>= likelihood))


It looks like we’re slightly more likely to sample from the left mixture component than the right one. Again, this makes sense - the mean of a beta(3, 2) distribution is 0.6.

Now, what about inference? I’ll define the conditional model as follows:

posterior :: Model (Execution Bool)
posterior = invert 1000 obs prior ll where
obs = [ -1.7, -1.8, -2.01, -2.4
, 1.9, 1.8
]

ll left
| left      = logDensityNormal (negate 2) 0.5
| otherwise = logDensityNormal 2 0.5


Here we have four observations that presumably arise from the leftmost component, and only two that match up with the rightmost. Note also that I’ve replaced the likelihood model with its appropriate cost function due to reasons I mentioned in the last section. (It would be easy to reify this model as its cost function, but doing it for general models is trickier)

Anyway, let’s sample from the conditional distribution:

> simulate (toSampler posterior)


Sampling returns a running program, of course, and we can step through it to examine its structure. We can use the supplied values recorded at each node to ‘automatically’ step through execution, or we can supply our own values to investigate arbitrary branches.

The conditional distribution we’ve found over the mixing probability is as follows:

Looks like we’re in the right ballpark.

We can examine the traces of other elements of the program as well. Here’s the recorded distribution over component assignments, for example - note that the rightmost bar here corresponds to the leftmost component in the mixture:

You can see that whenever we wandered into the rightmost component, we’d swiftly wind up jumping back out of it:

This is a fun take on probabilistic programming. In particular I find a few aspects of the whole setup to be pretty attractive:

We use a primitive, limited instruction set to parameterize both programs - via Free - and running programs - via Cofree. These off-the-shelf recursive types are used to wrap things up and provide most of our required control flow automatically. It’s easy to transparently add structure to embedded programs built in this way; for example, we can statically encode independence by replacing our ModelF a type with something like:

data InstructionF a = Coproduct (ModelF a) (Ap (ModelF a))


This can be hidden from the user so that we’re left with the same simple monadic syntax we presently enjoy, but we also get to take independence into account when performing inference, or any other structural interpretation for that matter.

When it comes to inference, the program representation is completely separate from whatever inference backend we choose to augment it with. We can deal with traces as first-class values that can be directly stored, inspected, manipulated, and so on. And everything is done in a typed and purely-functional framework. I’ve used dynamic typing functionality from Data.Dynamic to store values in execution traces here, but we could similarly just define a concrete Value type with the appropriate constructors for integers, doubles, bools, etc., and use that to store everything.

At the same time, this is a pretty early concept - doing inference efficiently in this setting is another matter, and there are a couple of computational and statistical issues here that need to be ironed out to make further progress.

The current way I’ve organized Markov chain generation and iteration is just woefully inefficient. Storing the history of each node on-site is needlessly costly and I’m sure results in a ton of unnecessary allocation. On a semantic level, it also ‘complects’ state and identity: why, after all, should a single execution trace know anything about traces that preceded it? Clearly this should be accumulated in another data structure. There is a lot of other low-hanging fruit around strictness and PRNG management as well.

From a more statistical angle, the present implementation does a poor job when it comes to perturbing execution traces. Some changes - such as improving the proposal mechanism for a given instruction - are easy to implement, and representing distributions as instructions indeed makes it easy to tailor local proposal distributions in a context-independent way. But another problem is that, by using a ‘blunt’ comonadic extend, we perturb an execution by perturbing every node in it. In general it’s better to make small perturbations rather than large ones to ensure a reasonable acceptance ratio, but to do that we’d need to perturb single nodes (or at least subsets of nodes) at a time.

There may be some inroads here via comonad transformers like StoreT or lenses that would allow us to zoom in on a particular node and perturb it, rather than perturbing everything at once. But my comonad-fu is not yet quite at the required level to evaluate this, so I’ll come back to that idea some other time.

I’m interested in playing with this concept some more in the future, though I’m not yet sure how much I expect it to be a tenable way to do inference at scale. If you’re interested in playing with it, I’ve dumped the code from this post into this gist.

Thanks to Niffe Hermansson and Fredrik Olsen for reviewing a draft of this post and providing helpful comments.