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Stream: theory: categorical probability

Topic: Jacobs' "Updating"

Ben Sprott (Jun 18 2021 at 16:35):

I am trying to understand how to update a prior distribution with a multiset of new data. I am reading Jacobs' text on probability because that is how I am understanding the subject, via monads like multiset and distribution.

Looking at Definition 2.3.1, I see he is updating a distribution with a predicate. The formula is:

$\omega |_p = \sum_{x} \frac{ \omega (x) \cdot p(x) } {\omega \models p} | x \rangle$

Predicates on set $X$ are defined by maps $p: X \rightarrow [0,1]$. I want to be able to take a multiset and find an update to a distribution, and I don't see how this definition can be used to do that.

A simple example is that I have a coin which, I believe, has a uniform bias. I then flip the coin ten times and get some multiset for the result, like $\{ H,T,H,H,H,H,T,H,H,T \}$. I don't see how I can interpret the multiset as the predicate? Is the predicate just the proportions: $\frac{3}{10} |T \rangle + \frac{7}{10} |H \rangle$

Can someone explain what I am not understanding?

Spencer Breiner (Jun 21 2021 at 18:52):

It looks to me like you are missing a distribution over the bias of the coin, say a state $b_0:1\to B$.

Let's set $X_k=2^k$ to be the state spaces for $k$-many flips. There is a channel $f_k:B\to X_k$ giving the distribution over $k$-flips expected from any given bias. You can compose that with the state $b_0$ to get the expectation from your particular coin (given your uncertainty about its bias).

Now we can think of your H/T sequence as a $X_10\to [0,1]$, but (I think) you need to invert the channel (as discussed in Jacobs) in order to update your bias estimate $b_0\mapsto b_1$.