2
$\begingroup$

In Gaussian clustering (i.e. General Mixture Models) we model the data with some clusters. For example, in the below figure, we have two clusters $C_1, C_2$, each of which are modeled with a Gaussian (Normal) distribution: $p_{C_1} \sim \mathcal{N}(\mu_1, \Sigma_1)$ and $p_{C_2} \sim \mathcal{N}(\mu_2, \Sigma_2)$ with proportions $\pi_1$ and $\pi_2$.

enter image description here

The probability that a given point $x$ belongs to $C_1$ (is generated from $C_1$) is: $$ p(C_1 | x ) = \frac { \pi_1 \mathcal{N}(\mu_1, \Sigma_1)} {\sum_{i=1}^2{ \pi_i \mathcal{N}(\mu_i, \Sigma_i) } } $$ The probability that $x$ belongs to $C_2$ can be similarly calculated.

Now, assume that we have a group of points that are generated (sampled) from another Gaussian distribution $q(x) \sim \mathcal{N}(\mu_q, \Sigma_q)$.

enter image description here

Obviously, we can compute the probability of each point belonging to each cluster. However, we instead want to compute the expected probability of a sample belonging to each cluster. In other words, if we generate infinite samples from the $q(x)$, what is the average (expected) value of $p(C_1 | x )$ for all points?

**P.S. ** I need the solution in closed-form.

$\endgroup$
0
$\begingroup$

Well it's clearly

$$ \int p(x|C_i) q(x)\, dx $$ but wouldn't you rather have $P(C_i | x)$ and the expected values of these, i.e. probabilities of classification?

$\endgroup$

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.