In my self-study, I consider a Gaussian mixture distribution:
$$p(x)= p(k=1) N(x|\mu_1,\sigma^2_1) + p(k=0) N(x|\mu_0,\sigma^2_0)$$
where $p(k=1)+p(k=0)=\pi_1+\pi_0=1$. I am now asked to do three things:
Write down the likelihood of the observations as a product over $n$ observations
Write down the likelihood as a product over the likelihoods for ${K_0}$ and $K_1$, where $K$ is the set of indices for $k=1$ and $k=0$, respectively.
Compute the log-likelihood and maximize for $\mu_0$ and $\sigma_0$.
I am not really sure what I am asked to do. I believe the likelihood is given by:
$$p(x|\pi_0, \pi_1, \mu_0, \mu_1, \sigma_0^2, \sigma_1^2) = \prod_{i=1}^n \bigg[ \pi_1 N(x_i|\mu_1,\sigma^2_1) + \pi_0 N(x_i|\mu_0,\sigma^2_0) \bigg]$$
So the log-likelihood is the sum of the logarithm of the sum in the parentheses. This seems correct. However no closed form solution exists of the derivative nor maximizer.
First, I am not sure which of no. 1 or 2 my likelihood expression solves. I think no. 1 but then, second, I am not sure what I am asked to do in no. 2. I suppose the solution to 2 is easier to maximize in 3. Third, it seems there are two different expressions for the likelihood then, but shouldn't there be just one?
Note: I was thinking for no. 2 along the lines of
$$p(x|\pi_0, \pi_1, \mu_0, \mu_1, \sigma_0^2, \sigma_1^2) = \prod_{i=1}^n \bigg[(\pi_1N(x_i|\mu_1,\sigma^2_1))^{k_i} (\pi_0N(x_i|\mu_0,\sigma^2_0))^{1-k_i} \bigg]$$
but got stuck.