Let $X$ be a discrete random variable distributed according to $p_x(\cdot)$. Let $Y_1, Y_2, \ldots Y_N$ be discrete random variables that depend on $X$. Suppose that $Y_1, Y_2, \ldots, Y_N$ are independent and identically distributed conditioned on $X$, such as $$p_{Y_n\mid X}(y\mid x) = P_{Y\mid X}(y\mid x)$$ for $n = 1,2, \ldots, N$.
How could I apply Baye's rule to calculate $p_{X\mid Y_1, \ldots, Y_N}(x\mid y_1, \ldots, y_N)$ in terms of the given values $p_X(\cdot)$ and $p_{Y\mid X}(\cdot\mid\cdot)$?
I have been struggling with this application of Baye's rule and I would really appreciate any help.
I'm going to abuse notation by eliminating subscripts on $P$; that shouldn't cause any confusion. We have $$P(x|y_1,\cdots,y_n) = P(x , y_1 ,\cdots , y_n) / P(y_1 , \cdots , y_n).$$
As the $Y_i$'s are independent, the denominator is just the product of the Bayesian priors $P(y_i)$. Again, by independence of the $Y_i$'s, the numerator is $P(x) P(y_1|x)\cdots P(y_n|x)$. So you can write $P(x|y_1,\ldots,y_n)$ in the suggestive form $$P_X(x) \prod_{i=1}^n \frac{P_{Y|X}(y_i|x)}{P_Y(y_i)}.$$ In particular, you seem to need a prior distribution on $Y$ as well as on $X$.