The n-Category Café

David wrote:

For example, ask me about a coin you’re about to toss, and, so long as I take all exchangeable sequences (permutations) to be equally likely, I can represent my belief in a distribution over p, the binomial parameter. Ask me for my probability that heads will be tossed and I’ll calculate the expectation of that distribution.

Indeed, if you do not begin with the apriori assumption that the coin is unbiased but only that your state of ignorance (since you have yet to flip any coins) implies that the conditional probability is 1/2, but Laplace’s rule of succession implies that if your first flip turns up heads, then you are less likely to receive another heads in the next flip, but you still can and usually do, but if you do, then you are even LESS likely to receive another, but it is still possible, in fact there is a non-zero chance that you could flip a coin and get heads each time but Laplace’s rule ensures that that usually never happens and is a law in some sense. Frequentist probabability is totally bogus based on such a revelation because it ignores the “time-dynamics of probability”. Good gamblers know this well. Now, extending these concepts to continuous-time is quite a challenge because the problem immediately becomes infinite dimensional, as in, how do you separate ‘instants’ and project expectations for the future? It can be done but it is very unintuitive. Naely, analytical number theory and analytic continuation comes in handy by using the fact that the gamma function is the real/complex generalization of factorials(combinatorics)! Now, build a sigma-algebra or delta-ring out of all this..

Yes you’re right that the first equation is Bayes’ theorem. (Paper 15 here.) However, Bayes’ theorem is not intrinsically Bayesian. It’s just a truth about probabilities.

On the other hand, the Theorem is used often by Bayesians as a means of updating their probability assignments (or ‘degrees of belief’) in situations where it makes no sense to a frequentist. For example, one uses $P(h|e) = P(e|h).P(h)/P(e)$ to find the support given to hypothesis $h$ by evidence $e$. For many hypotheses $P(h)$ will make no sense for the frequentist. Further, but there are provisos, if $e$ is all you learn, the new $P(h)$ will be the old $P(h|e)$.

Doberkat doesn’t discuss what kinds of element are to be found in his Polish spaces so may be a frequentist for all I know. My point about the monad picture fitting nicely with Bayesianism is that the map $\mu_{X}: P P(X) \to P(X)$ is something the Bayesian can understand directly, while for the frequentist I can’t see that it could be more than a tool for composing conditional probabilities.

Does anyone know more about Voevodsky’s approach? It was announced here:

…a categorical study of probability theory where “categorical” is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces.

One thing that’s interesting about conditionalizing is that it is an example of projection from the initial distribution onto a subspace of distributions satisfying a constraint. I discuss a paper by Harremoes on this in this post. Projection like this happens very frequently in inference such as when matching a moment of an empirical sample. Inference takes the form of passing down a geodesic in the space of distributions.

I wish I had a better sense of how category theory and the information geometry picture could be made to work together. An arrow between two objects $A$ and $B$ in the Kleisi category of the Giry monad corresponds to a conditional probability $P(b_i|a_j)$. Could one see the category as enriched over the category of some differential geometric spaces? Perhaps Cencov’s book might help.

Anyhow, conditionalizing on $A = a$, is a little like you’ve taken the Kleisli category of Giry’s monad, and then turned $A$ into a single element set, such that an old distribution on $B$, $f: 1 \to B$, becomes $f': 1 \to B$, $P(b_i|a)$.

Hi David

Thanks for your answer. I had a look a the paper. Generally I am interested in how the Bayesian language seems to be a naturally higher-order probability language, and in understanding all these confusing statements about “outliers”, “confidence intervals”, “moderately strong evidence for a significant level of something” …

Following on your earlier post I thought the Giry monad could be a nice way to clarify the grammar of beliefs and learning. And then, for instance, to connect this with the various notions of approximations of Markov chains Panangaden and others (including myself) have developed over the years.

Anyway, here is my rephrasing of Bayes learning.

Say we have the following ingredients:
- X a finite measurable space with discrete sigma-algebra (say)
- p ∈ GX a hidden probability on X
- P = ⊕ f_i p_i ∈ GGX a belief represented as a probability on GX
- s an observation on multisets over X

We can equally write P(p_i)=f_i (or more rigorously P({p_i})=f_i).

By multiplication, we have μP(A) = \sum_i f_i p_i(A) - a sort of majority vote where f_i is the weight accorded to p_i in the prediction. The idea for learning is that we sample repeatedly from the hidden p, which gives us the observation s above, and we modify the weights in the majority vote of P in order to get closer to the real p.

Specifically, one modifies f_i (note that the support of P ′ remains the same as that of P) as follows:

f_i′/f_i = p_i(s)/μP(s) (1)

I am sligthly abusing notation here since p and μP are not really defined on multisets, but we can extend them using the “independence map” GX → G(multiset(X)); defined as p(s)=\prod x\in X p(x)^(s(x)) where s(x) is the number of occurrences of x in s.

P is called the prior, P’ the posterior. If we rewrite the above as a sort of Bayes formula

P(p_i | s) · μP(s) = P(s | p_i) · P(pi | ∅) (1’)

we see that up to the normalising factor μP(s), we are saying that P(p_i) has to be “nudged” by how likely s -which is what we get- is, assuming p_i.

Write s · P for P’. I like that notation since it reminds me that the observable acts on GGX, ie gives me a deterministic update instruction.

This defines a Markov chain (MC) on GGX defined as

K(P, s · P ) = p(s) (2)

that is to say we are ‘walking’ stochastically on GGX, so the kernel K ∈ [GGX;GGGX] might have a steady state in GGGX - but in fact the interesting limit is a “point-mass” in GGX. That is to say s · P → δp as |s| → ∞, where δ: GX → GGX is the unit, p the real probability (which we suppose here is in the support of P), |s| the size of s. To prove this, one can see readily that:

log(s · f_i/s · f_j) ∼ |s| × KL(p, p_j) ≥ 0 (3)

assuming p=p_i is the real probability, where KL is the relative entropy of p and p_j (aka the Kullback-Leibler divergence).

I imagine that the above justifies the update rule (1) as a good one, as it does eventually find the solution; it also shows that KL is an natural tool to assess convergence of this update rule.

So my question is, is that kind of view elaborated in the compsci/maths literature?

You could try to request it from here.

Jeremy Gibbons announced on the categories list that he found a copy.