In various places it says that the likelihood (e.g. in the Bayes formula) is "proportional to a probablility".
For example https://alexanderetz.com/2015/04/15/understanding-bayes-a-look-at-the-likelihood/ "likelihood is proportional to a probability"
A similar statement is in the book Think Bayes, "likelihood doesn't need to compute a probability, it only has to compute something proportional to probability"
Lastly, in a crossvalidated discussion https://stats.stackexchange.com/questions/2641/whatisthedifferencebetweenlikelihoodand%20probability there is a more specific statement "the likelihood function is proportional to the probability of the observed data." I am not sure if that is correct or not.
Notation, $P(A|B) = P(B|A) P(A) / P(B)$. The likelihood is $P(B|A)$.
The question: If I am understanding however, the likelihood is proportional to a different probability for each value of $A$. That is, there is no single constant $c$ such that the likelihood is equal to some version of the probability $P(B)$, i.e. this statement is false $$ P(B) = c \cdot P(B|A) $$ Rather, it would have to be $$ P(B) = c(A) \cdot P(B|A) $$ meaning that the constant of proportionality $c$ varies with each choice of $A$.
Is this true? If so, it seems to me that saying likelihood is proportional to a probability is vacuous. You can always make something "proportional to" something else if the proportionality is a function rather than a constant.
This question is somewhat related to these previous questions about likelihood, which I have read, however I believe this is a different question (albeit maybe one that could be "derived from" the answers to other questions, buy someone smarter than me!)
When is likelihood also a probability distribution?
EDIT: Maybe another way of asking the question is this: if the likelihood is proportional to a probability, which probability is it proportional to? Is it the probability $P(B|A)$ regarded as a function of B with A held fixed, or is it the marginal probability P(B), or is it some other (generic) probability?
EDIT: here is an attempt at an example:
A Normal distribution is parameterized by mean and variance. Let's pick something simpler,a PMF that is parameterized only by a single parameter $M$. It assigns probabilities to integer values,
and the parameter translates the PMF on the integer axis.
Here is the PMF for the parameter value M=5:
$$
\begin{align*}
& P(X=5|M=5) = .5 \\
& P(X=6|M=5) = .3 \\
& P(X=7|M=5) = .2 \\
\end{align*}
$$
And the PMF for parameter value M=6:
$$
\begin{align*}
& P(X=5|M=6) = 0 \\
& P(X=6|M=6) = .5 \\
& P(X=7|M=6) = .3 \\
& P(X=8|M=6) = .2 \\
\end{align*}
$$
More generically, the PMF is like P(X=M)=0.5, P(X=M+1)=.3, P(X=M+2)=.2, zero otherwise.
Now consider the likelihood form of this, where the data is given as the fixed value X=5 and the parameter $M$ is what varies: $$ \begin{align*} & P(X=5|M=3) = .2 \\ & P(X=5|M=4) = .3 \\ & P(X=5|M=5) = .5 \\ \end{align*} $$
So in the PMF case the shape is (.5,.3,.2), whereas in the likelihood case the shape is (.2,.3,.5). There is no single constant that can make these equal.
What is my mistake?