I found that in the forward algorithm used in the CRF(and perhaps also in the HMM) the mechanism applied is almost the same as that in the variable elimination(VE) except that the emission probabilities do not exist in the VE. Am I right? What is the relationship between the variable elimination and the forward algorithm?
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asked Jul 14, 2019 at 7:37
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$\begingroup$ I learned that CRF is also an emplement of the sum-product algorithm, and hence they are just the same. $\endgroup$– Lerner ZhangCommented Oct 19, 2019 at 6:33
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You're right, the forward algorithm is exactly the variable elimination algorithm.
Here's an excerpt from Koller and Frideman's book Probabilistic Graphical Models:
Variants of the variable elimination algorithm were invented independently in multiple communities. [...] Another early variant is the forward-backward algorithm, which performs inference in hidden Markov models (Rabiner and Juang 1986). [...]
(page 337, Section 9.8 Relevant Literature, Chapter 9: Variable Elimination)
