I have a question about Bayesian updating.
I have a problem where I have an event $A$ with three possible outcomes: $(A(1),A(2),A(3))$. I need to estimate the probability of each outcome.
I am able to define the prior probability of each outcome by using literature, let's say $$P(A(1))=0.8,P(A(2))=0.1,P(A(3))=0.1$$
Then, I can find a pdf for each outcomes by performing some experiments. Each outcome $(A(1), A(2), A(3))$ follows a Gaussian distribution with mean $(m_1, m_2,m_3)$ and standard deviation $(s_1, s_2,s_3)$.
Now, can I update the probability of each outcome by knowing their prior probability and their pdf?
Can I use the pdf as a likelihood and omit the normalizing constant of the traditional Bayes formula?