The guide
These models write down how the data could have been generated, then run that story backwards to infer what is hidden. The hidden part can be the next state of a chain, the state behind an observation, the cluster a point came from, or the weights of a regression.
Markov chains and hidden Markov models handle sequences. The forward algorithm and Viterbi are both dynamic programming over a chain, and that pattern returns in speech recognition, CRFs and alignment.
Gaussian mixtures bring in the EM algorithm: guess the hidden assignments, refit the model as if the guess were right, and repeat. Bayesian linear regression shows what you gain by keeping a full distribution over the weights instead of one best guess: a measure of how sure the model is.