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I am currently debugging a hand-coded implementation of a hidden Markov model, and as part of this, am scrutinising whether I have appropriately specified the log-likelihood computation algebraically. In particular, I would like to check whether my interpretations of "log-likelihood" in this context are appropriate, and what general conventions there are for reporting "log-likelihood", if any.

Queries.

1. If I were to report the joint log-likelihood of the hidden Markov model using state vectors $\mathbf{y}$ and observation vectors $\mathbf{x}$ am I correct in understanding that there is neither the need for a forward algorithm nor a backward algorithm?

For clarity, I am taking the meaning of "forward" and "backward" algorithm to refer to a procedure for efficient exact inference in a tree-structured probabilistic graphical model, involving sum-product message passing/belief propagation. And my query amounts to whether the following computation requires sum-product message passing, or is rather, merely a task of "plugging-in" $N$ values $\mathbf{x}$ and $\mathbf{y}$.

$$l_1(\Theta) = \prod^N_{n=1} p(\mathbf{x}^{(n)}, \mathbf{y}^{(n)}; \Theta)$$

My current position on this is that there is no need for sum-product message passing, given that there is no marginalisation involved, but for debugging purposes, I am in need of absolute clarity on this.

2. As a follow-up, if I were to report the log-likelihood of the hidden Markov model using state vectors $\mathbf{y}$ and observation vectors $\mathbf{x}$, am I correct in understanding that reporting the following log-likelihood computation DOES require a forward algorithm/sum-product message passing algorithm?

$$l_2(\Theta) = \prod^N_{n=1} p(\mathbf{x}^{(n)}; \Theta)$$

My current position on this is that there is a need for sum-product message passing, given that marginalisations are required. And that the above computation involves the marignalisations $\sum_{y_1} \dots \sum_{y_{t_n}} p(\mathbf{x}, \mathbf{y}; \Theta)$, whereby the number of marginalisations corresponds to the number of components in the state vector $\mathbf{y} = (y_1, ..., y_{T_n})$

3. Are there general conventions governing whether the phrase "log-likelihood" in context of a hidden Markov model refers to $l_1(\Theta)$ or $l_2(\Theta)$ in the notation above, or is it assumed that it is the responsibility of the relevant author to clarify this?

Perhaps this might be viewed as unnecessary hair splitting, but I am trying to reconcile the potential ambiguities in the meanings above in context of classical parametric statistics.

Because in frequentist parametric statistics, we normally say that the log-likelihood function $l(\Theta)$ is a function of the parameters $\Theta$, evaluated at the observed values of the data say $X_1 = x_1, ..., X_N = x_n$. However in the HMM case, our data consists of two sets of random variables $\mathbf{x} = (X_1, ..., X_{T_n})$ and $\mathbf{y} = (Y_1, ..., Y_{T_n})$, and we can choose to report either $l_1(\Theta)$ or $l_2(\Theta)$.

Further context.

I have a hidden Markov model, consisting of an initial distribution $\boldsymbol{\pi} \in \mathbb{R}^M$, a state transition matrix $\mathbf{A} \in \mathbb{R}^{M \times M}$, and an emission matrix $\mathbf{B} \in \mathbb{R}^{M \times V}$. That these parameters $\Theta = \{\boldsymbol{\pi}, \mathbf{A}, \mathbf{B} \}$ are all matrices implies that the hidden Markov model has discrete states, observations; and also time is discrete.

My training data set consists of $N$ observation-state vector pairs $\{(\mathbf{x}^{(n)}, \mathbf{y}^{(n)}) \}^N_{n=1}$, where both observation vector $\mathbf{x} = (x_1, ..., x_{T_n})$ and state vector $\mathbf{y} = (y_1, ..., y_{T_n})$ both have $T_n$ components, corresponding to a variable length state or observation sequence. Where this variability in sequence length is represented through indexing of the instance $n$.

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  1. 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.)
  2. Yes, now that you’ve marginalized out the latent variable, you should use the forward algorithm.
  3. The natural way to remedy these two notions is to remember that the likelihood is a function of the parameters, given the data: $L(\theta \mid \{Z\})$. So it naturally covers both, based on the data you have. Whether it’s the joint likelihood or the marginal likelihood depends on what the data is—do you have observations, or observations and states? It’s typical not to have supervision about the states. (One exception is part-of-speech tagging, where you ascribe a meaning to each state. In this case, you’d want to provide supervision to ensure the states match your intended meaning.) A text like SLP (the Jurafsky and Martin book) talks about the ‘likelihood’ for an HMM as, using your notation, $l_2$.
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  • $\begingroup$ Thank you for supplying clarity on this. And my application in this context is a toy parts-of-speech tagging context whereby I can choose to report $l_1(\Theta | \mathbf{X}, \mathbf{Y})$ or $l_2(\Theta | \mathbf{X})$, given supervised word-POS tag training data. You've prompted that my notation for likelihood functions could be more explicit. $\endgroup$ – microhaus Apr 17 at 17:21

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