# Difference between class conditional distribution and likelihood in the context of Mixture of Gaussians?

I am reading some Mahchine learning lecture notes, and the writer is introducing Maximum Likelihood (ML) method of parameter estimation of $\theta$ as $\text{argmax}_{\theta}Pr_{\theta}(x|H)$ where $H$ is a hypothesis and $Pr(x|H)$ is the likelihood of the observation.

Now later on, when he introduced the mixture of Gaussians, He used Bayes theorem to define the probability of a class $c_k$ as $Pr(c_k|x)=kPr(x|c_k)Pr(c_k)$ where the probability of each class $Pr(c_k)$ is multinomial, and $Pr(x|c_k)$ is Gaussian.

Now here is where I am getting confused When it came to applying ML to learn the parameters of the posterior, namely $\pi_k=Pr(c_k)$ and the covariance matrix $\Sigma$ and means $\mu_1, ..., \mu_k$ for each distribution, We'll call all these parameters $\Theta$, then he tried to maximize:

$$Pr(X|c_k)Pr(c_k)= \Pi_n \Pi_j [\pi_jN(x_n|\pi_j,\Sigma)]^{y_j}$$

Where $y_j$ is $1$ is $x_n$ belongs to class $j$ and $0$ otherwise.

However, shouldn't he have maximized just $Pr(X|c_k)$ which is the likelihood here?

If the author's intent was to find the maximum likelihood estimate of the parameters $\Theta$, then yes, the relevant optimization problem is $$\max_\Theta \text{Pr}(X | c_k).$$ But it looks like what's going on here is actually an example of maximum a posteriori (MAP) estimation, which produces an estimate of the parameters by maximizing the posterior distribution $\text{Pr}(c_k|X)$. As you point out, the posterior distribution is proportional to the product of the likelihood and the prior distribution, $\text{Pr}(X|c_k)\text{Pr}(c_k)$, so it suffices to find the value of $\Theta$ that maximizes this product.