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

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?

## 1 Answer

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.)

• so, the notation $\langle \cdot \rangle$ denotes just an ordered pair here, right? – JJJohn Nov 9 '19 at 8:22
• That's my reading. The notation itself just means ordered pair, nothing more. However, the ordered pair is being used to summarize a multiset. I don't see any other interpretation that makes sense of what's on that page. I can't see how any other interpretation makes sense of what's on the page. – Mars Nov 10 '19 at 17:49