Yes, to evaluate the joint probability of the observations and states (i.e., the data are fully observed), you don’t need the forward algorithm’s dynamic programming. (If you want, you can think of it as a special case of sum-product where there’s nothing to sum.)
Yes, now that you’ve marginalized out the latent variable, you should use the forward algorithm....
This playlist is a great explanation and is based on the paper by Rabiner mentioned in the answer above. -
This above playlist is a 12 series lecture which begins with explanation of Markov Chains/ observable Markov Models and then moves on to HMMs
Expanding on my comment: the answer is clearer when you realize that you can't ignore the observed variables. They affect each model differently. As it turns out, the MEMM is not I-equivalent to the linear-chain CRF and HMM.
As a recap, the HMM looks like this:
y1 -> y2 -> y3 -> ... -> yn
| | | |
v v v v
In a Hidden Markov Model, the Viterbi algorithm is the right way to find the highest-probability sequence of hidden states $\bf x$, given your sequence of observations $\bf y$. It will find an exact solution (not an approximation) in time $O(TK^2)$ instead of the $O(K^T)$ that you'd suffer with the brute force solution.
Given that it eliminates sequences ...
An excellent, approachable review that unifies these approaches is A Unifying Review of Linear Gaussian Models by Roweis and Ghahramani, published in Neural Computation. The first two sentences of the abstract are
Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models ...