0
$\begingroup$

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?

Thanks.

$\endgroup$
0
$\begingroup$

I don't think you specified a prior on the probability of outcomes, you specified them as a constant.

I think what you're saying is you have a mixture distribution with 3 components. You sample $A \sim [.8,.1,.1]$ and the distribution on the data would be, conditional on the sampled $A$:

\begin{equation} \begin{split} X | m,s,A & \sim \sum_{i=1}^3\textrm{I}_{(A=A_i)}\textrm{N}(m_i,s_i)\\ A | \pi & \sim \textrm{Multinoulli}(\pi) \end{split} \end{equation}

Where $\pi$ is the vector of probabilities. Note that if you integrate out $A$, you get:

\begin{equation} X | m,s,\pi \sim \sum_{i=1}^3\pi_i\textrm{N}(m_i,s_i)\\ \end{equation}

Which is just a fully specified mixture distribution. In this setting $\pi$ is a parameter, not something included in the posterior. The only thing included in the posterior in this situation would be $A$.

What you could do instead is to place a hyper-prior on $\pi\sim\textrm{Dirichlet}(\alpha)$. This gives us the model:

\begin{equation} \begin{split} X | m,s,A & \sim \sum_{i=1}^3\textrm{I}_{(A=A_i)}\textrm{N}(m_i,s_i)\\ A | \pi & \sim \textrm{Multinoulli}(\pi)\\ \pi | \alpha &\sim \textrm{Dirichlet}(\alpha) \end{split} \end{equation}

Which makes $\alpha$ the parameter at the deepest level of the hierarchy, and $\pi$ a part of the posterior distribution. Implementation of this requires Gibbs sampling, by sampling $A$ at each iteration and updating $\pi$ using the conjugacy of the Dirichlet to the Multinoulli/Multinomial.

$\endgroup$
  • $\begingroup$ Thank you for the reply. It is very interesting and stimulating, and I really would like to code it in Matlab. However, my Bayesian and statistical background is not that strong. I do not get the reason why A|π needs to follow a multinoulli if I know from experiments that A follows a Gaussian mixture model (a gaussian distribution for each Ai). Can you kindly explain me this step? Regarding the Gibbs sampling, I should start sampling A from the Gaussian, and then move to assess π by using the sampled A as input to the Dirichlet, is that correct? Thank you!! $\endgroup$ – Mat Feb 5 '18 at 15:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.