Beliefs about equilibrium paths with overlapping strategies This is a basic question, but I cannot find the answer to it. In short, equilibria are often conditioned on a specific strategy. E.g., given A's strategy, B updates her beliefs to xyz. My problem is that often an action could be a part of the equilibrium path for many different strategies, so how does B know which strategy A is following?

Below is an example that might clarify the question.

Consider a classic signaling game (2 types, 2 actions) à la Spence. Suppose furthermore that there are two equilibria: one pooling, one separating. In the pooling eq., say, both types of senders send "Low". In the separating, the "Strong" type sends High, the Weak type sends Low.

The part that I struggle with is: how does the receiver know what equilibrium path they are in? For example, say the receiver observes "Low". If we are in "pooling world", then the receiver's posteriors are just going to be her priors. If, however, we are in "separating world", then she can update to p(Weak|Low)=1. But observing "Low" alone does not tell the receiver what strategy the sender is following, so how can she possibly update her beliefs? It seems to me that she would need to have beliefs not only about types, but also about strategies being followed. 

Sorry if this is idiotic, but this has puzzled me for a while.
In signaling games, beliefs as well as strategies constitute an equilibrium. In other words, beliefs are a part of the equilibrium object. By deciding which equilibrium to focus on, you also (partially) determines the beliefs consistent with that equilibrium. The key restriction is that beliefs be derived from Bayes' rule using sender's strategy and the prior. 

Consider the following game, where P1 is the sender and the common prior is Type I realizes with probability $p$. 

Separating equilibrium



In a separating equilibrium, different types choose different actions (Type I chooses $A$ and Type II chooses $B$). Then, upon seeing action $A$, P2 (the receiver) updates her belief using Bayes' rule:
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(0)(1-p)}=1
$$
Therefore P2's conditional belief at node $x$ is $1$. Similarly for her belief at the information set $\{x',y'\}$

Pooling equilibrium



Now, consider a pooling equilibrium, where both types choose $A$. Then, upon seeing action $A$, P2's belief is calculated as 
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(1)(1-p)}=p
$$
which is the same as her prior. This makes sense because P1's action provides no additional information in a pooling equilibrium. (P2's off-equilibrium belief can be arbitrary, unless you want to apply refinements such as consistency of the sequential equilibrium).



The bottom line is: you cannot talk about an equilibrium separately from the beliefs. By talking about a particular equilibrium, you also implicitly restrict the set of beliefs consistent with the said equilibrium.