The data $X$ is modeled as a random variable. There exists two events $A_1$ and $A_2$.
I want to determine the posterior probability of an event $A_1$ given the data $x$, that is, $P(A_1|X = x)$.
I am given that the event $A_i$ is defined as $A_i = \{x \sim \mathcal{N}(0, \sigma_i^2)\}$.
Now, I know by the Bayes' formula, I need to compute the following:
$$ P(A_1|X=x) = \frac{p_{x|A_1}(x|A_1)\cdot P(A_1)}{p_{x|A_1}(x|A_1)\cdot P(A_1) + p_{x|A_2}(x|A_2)\cdot P(A_2)} $$
However, I am confused with the given statement saying "the event $A_i$ is defined ~" with a Gaussian density of a random variable $x$.
In particular, I am not sure if I use the Gaussian function (exponential function) for $p_{x|A_1}(x|A_1)$, or for $P(A_1)$.
Does the given statement saying "the event $A_1$ is defined as ~" same as "if an event $A_1$ already happened, the random variable $x$ takes the following distribution"?