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I am trying to understand proper notation for functions related to state transition models (e.g., a HMM). There are two indexing variables $t$ (time step) and $i$ (state). Such that it matters, in my scenario, the variable we are conditioning on may be taken to be a realized (observed) instance of a random variable: $P(X=x|Y=y)$, not $P(X=x|Y)$. Below is an example, an entropy function over the transition model, but the equation could be anything including a sum over states and probability conditioned on previously observed states or similar:

$$H_t = -\sum P(x_{t}|x_{t-1})log_{2}P(x_{t}|x_{t-1})$$

I would call this correct but incomplete/informal, right? Importantly, it lacks the index variable for summation. I get a little confused when I attempt to add this in. Should it look like this:

$$H_t = -\sum_{i=1}^{n}P(x_{t,i}|x_{t-1})log_{2}P(x_{t,i}|x_{t-1})$$

This feels off to me in that both $x_{t}$ and $x_{t-1}$ have $n$ states. The index $i$ affixed to $x_{t}$ (i.e., $x_{t,i}$) provides a specificity greater than that for $x_{t-1}$, which lacks a state value designation. Given that $x_{t-1}$ is a realized (random) variable and $x_{t}$ is an (unrealized) random variable, shouldn't there be some notation to indicate the value of $i$ is realized (even more specific) with respect to $x_{t-1}$? If so, how? Surely not $x_{t-1,1}$ or $x_{t-1,j}$. Perhaps this is where the $X=x$ grammar is useful but, that seems only to displace the problem, ex.:

$$H_t = -\sum_{i=1}^{n}P(X_{t}=x_{i}|X_{t-1}=x)log_{2}P(X_{t}=x_{i}|X_{t-1}=x)$$

Clearly there is something I am missing. Please send thoughts. Math notation is not my strength, where I have seemingly or obviously stated something incorrectly or oddly, please let me know in the comments and I will respond and make edits.

FYI, here is somewhat related to this question.

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1 Answer 1

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$P(x_t|x_{t-1})$ is an abuse of notation to make it shorter. The unambiguous version is as you have noted: $P(X_t=x_t|X_{t-1}=x_{t-1})$. In graphical models, such as HMM, $x_t$ generally denotes the latent variable, i.e. the state, at time $t$. So, if there are $n$ states, $x_t$ can be one of $1,2,...,n$. Index $i$ can be used to denote one of them, e.g. $x_t=i$ means we're in state $i$ at time $t$.

It seems like $H_t$ is the conditional entropy of $X_t$ given a specific value of $X_{t-1}$, i.e. $H(X_t|X_{t-1}=x_{t-1})$. Then, the summation converts into the unambiguous form: $$H_t=\sum_{i=1}^n P(X_t=i|X_{t-1}=x_{t-1})\log_2P(X_t=i|X_{t-1}=x_{t-1})$$

Note that $H_t$ will be a function of $x_{t-1}$.

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  • $\begingroup$ Very clear. Thank you! $\endgroup$
    – jbjo
    Aug 4, 2019 at 19:54
  • $\begingroup$ I like this answer but can we change the summation index to $x_t$: $$H_{t}=\sum_{x_{t}=1}^{n}P(X_{t}=x_{t}|X_{t−1}=x_{t−1})log_{2}P(X_{t}=x_{t}|X_{t−1}=x_{t−1})$$ In the answer, $j$ is an index value, taken from 1 to $n$ on the $n$ states of $X_t$. $P(X_t=j)$ implies that the RHS value in this grammar is an address value, not the value taken by $X_t$. I had thought common usage was the opposite, eg, $P(X=3)$ is the probability $X$ takes the value 3. Am I thinking about this incorrectly? Is "3" both a value and an address on the number line? Can you clarify this for me? $\endgroup$
    – jbjo
    Aug 4, 2019 at 23:52
  • $\begingroup$ correction, in the answer, the index variable is $i$, not $j$... $\endgroup$
    – jbjo
    Aug 5, 2019 at 2:26

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