I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being picked and generate a sample, $s_0$. What is the probability that it was generated from the distribution $N_1$?
I think this is related to the Bayes theorem. I have tried the following. I set a variable, $X$, which takes the index of the chosen distribution. Let there be a variable $s$ representing a sample from any of the $d$ distributions. I know :
- $P(X=1) = \cdots = P(X=d) = \frac{1}{d}$.
- $P(s|X=k) \sim N_k(\mu_k, \sigma_k^2)$ for all $k \in [1, \cdots d]$.
- $\displaystyle P (X=1 | s=s_0) = \frac{\frac{1}{d}\cdot P(s=s_0|X=1)}{\sum_{i=1}^d \frac1d \cdot P(s=s_0 | X=i)} = \frac{P(s=s_0|X=1)}{\sum_{i=1}^d P(s=s_0 | X=i)} $
However, I don't know the probability of a single sample and area under the PDF curve evaluated at a single point is 0. How do I proceed?