does the pair of $\langle\cdot\rangle$ mean some special relationship in Bayes' rule? something like inner in Linear Algebra?

Question

This CMU textbook (P.8) uses this notation

$$P(D = \langle α_1,α_0\rangle\mid θ)$$

to denote the data likelihood $P(D\mid θ)$

does the pair of $\langle\cdot\rangle$ mean some special relationship? like inner product in Linear Algebra?

Answer 1

Based on reading that page, it's clear to me that the notation can be read as a notation for an ordered pair, i.e. a sequence of two values, in order, so for example $\langle 0.25,0.75 \rangle \neq \langle 0.75,0.25 \rangle$. This is a common use of the angle brackets. $\alpha_1$ is the number of 1's in the $D$ above, and $\alpha_0$ is the number of 0's in $D$, as the text states.

$D= \{1,1,0,1,0\}$ appears to be a multiset, since it has multiple instances of the same value, but since given its purpose, it doesn't matter, in the end, what order the 1's and 0's are in.

Since $D$ is also equal to $\langle \alpha_1, \alpha_0 \rangle$, we should read this ordered pair notation as a way of summarizing the contents of the multiset: $\{1,1,0,1,0\}=\langle \alpha_1, \alpha_0 \rangle$, where, again, $\alpha_i$ is the count of the number of $i$'s in the multiset. This is consistent with the paragraph that defines the $\alpha_i$'s.

(The notation can't mean inner product, by the way, although you didn't suggest that it should. The $\alpha_i$'s are defined as integers, so an inner product would just be regular multiplication, but at no point does it make sense for the $\alpha_i$'s to be multiplied.)