Referring to the Markov Random Field Wikipedia page, there is essentially nothing stopping them from writing a Markov network as an exponential family. Now let's say you have a simple Markov network written in canonical form for the probability of being in state $i$

$$P_i = \frac{1}{Z}e^{\sum_kw_kX_k(i)}$$

where $X_k(i)$ is some random variable in state $i$ with $w_k$ weights and $Z$ is just a normalization constant historically referred to as the partition function

$$Z = \sum_ie^{\sum_kw_kX_k(i)}$$

My question is: Is it also permissible to write the exponential family with a negative exponential? Meaning

$$P_i = \frac{1}{Z}e^{-\sum_kw_kX_k(i)}$$

The reason I ask is because in many situations, computation of the partition function is analytically intractable. Yet, in many applications the weights $w$ are necessarily positive by construction. If an inverse exponential family was used instead, it would make the problem more analytically tractable because the sum would be easily calculated.

As an example, say $X$ is a random variable that can take values from $0$ to $\infty$. Then the partition function would diverge in the first case since $e^{w_k}>1$


but converge with a negative exponent since $e^{w_k}<1$ (assuming $w_k\neq0$)

$$Z = \prod_k\frac{1}{1-e^{-w_k}}$$

making the second formulation far superior.


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