Confusing definition of Potts model

Question

I came across some Markov random field models and noticed something that didn't make sense to me. One of these models for a set of latent variables $\{z_{i}\}$ is the following:

$$p(z) = C(\beta)^{-1}\exp{\bigg(\sum_{i}^{N} \alpha_{i}(z_{i}) + \beta \sum_{i \sim j}w_{ij}f(z_{i},z_{j})\bigg)}$$

where $C(\beta)$ is a normalization constant. This model describes the joint distribution of $z$. Each $z_{i}$ will take one of several categories (e.g. $\{1,2,\dots, K\}$). The sum with $i \sim j$ refers to summing over all neighbors of $i$. When we drop the first term of the exponential and set $f(z_{i},z_{j}) = I(z_{i}=z_{j})$ and $w_{ij} = 1$, we obtain Potts model. However, I have seen Potts model being described as:

$$p(z) = C(\beta)^{-1}\exp{\bigg(\beta \sum_{i \sim j}I(z_{i}=z_{j})\bigg)}$$

I don't think it makes much sense as it is because the equation above is saying that the joint distribution of $z$ depends only on the sum of the neighbors of $z_i$. Others define the previous equation as $p(z_{i})$ and it certainly looks correct but they still refer to it as a joint distribution. Maybe, the authors forgot to add a sum over $i$ in the following way:

$$p(z) = C(\beta)^{-1}\exp{\bigg(\sum_{i}^{N} \beta \sum_{i \sim j}I(z_{i}=z_{j})\bigg)}$$

Does anyone know the correct definition of Potts model?

Answer 1

I found a sane definition of Potts model.

$$\pi(x) = \displaystyle \frac{1}{c(\beta)}\exp{\bigg(\beta \sum_{s,s'} I(x_{s},x_{s'})\bigg)}$$

where $\sum_{s,s'}$ is over all neighbouring sites $s$ and $s'$.

The source explains:

Notice: A pair $(s, s_0)$ of neighbouring sites with identical labels contributes to $p()$ by the amount $\exp(β)$. If the labels are different, the contribution is $\exp(0) = 1$.