This is a question on my assignment, I have arrived at an answer but I'm starting to second-guess myself. The grammar in the question is a bit wonky, so I'll type out the question and my interpretation of it as well.
The original statement: If for some feature value $x$ we have $p(x|w_1)=p(x|w_2)$, where $w_1$ and $w_2$ are classes. This condition doesn't give any information about the state of nature. Then, based on what is the decision made, i.e., to which class does $x$ belong? The options given are the posterior probabilities, the prior probabilities, the evidence, and none of the above.
Here's my interpretation of the question:
Suppose that for some feature variable $x$ I know that the likelihood for classifying it into 2 classes is equal i.e. $p(x|w_1)=p(x|w_2)$ for some unknown classes $w_1$ and $w_2$. This is all I know. I don't know anything about $w_1$ and $w_2$, I don't know $pdf(x)$, the probability distribution function/evidence. I don't know the prior probabilities for the problem i.e. $p(w_1)$ and $p(w_2)$. How would I go about classifying $x$?
I know that if the posterior probabilities are unequal, then I'd classify $x$ as belonging to the class where the posterior probability is greater i.e. $p(w_1|x)>p(w_2|x)$ implies that $x$ belongs to $w_1$. This means that on the right side of Bayes' theorem, $\frac{p(x|w_1)p(w_1)}{p(x)}$ is greater than $\frac{p(x|w_2)p(w_2)}{p(x)}$. This means that I'll only need to consider the numerator. Since I don't know anything about the prior probabilities, I'll have to look at the likelihood ratio, $\frac{p(x|w_1)}{p(x|w_2)}$. Since both prior probabilities are equal, would I end up having to use the posterior probabilities to classify $x$?
To me, the answer to this question seems pretty straightforward. Since I only know that $p(x|w_1)=p(x|w_2)$, I'll have to consider the LHS of Bayes' Theorem. Here, the ratio $p(w_1|x)$ and $p(w_2|x)$ is equal to $\frac{p(w_1)}{p(w_2)}$. Since I don't know the RHS, I'll have to look at the LHS to classify $x$. However, there's a tiny part of me that's doubting myself, because the grammar of the question is wonky. Am I right for going with option A, the posterior probabilities, or should I change it to B, the prior probabilities?
Bayes theorem...
$$P(w_i|x)=\frac{P(x|w_i)P(w_i)}{P(x)}$$
We certainly don't use $P(x)$ as the above is proportional to
$$=P(x|w_i)P(w_i)$$
Ok then we have
$$P(w_i|x)=CP(w_i)$$
so it is proportional to
$$=P(w_i)$$
Hence we use the prior to make the decision. The above is proportional to the Posterior, so it is correct answer too the way you formulated the question (and in fact always the answer), but what we need is the prior and it will determine the posterior in this case. I think they want to hear prior, because otherwise the question doesn't make a lot of sense.