# Potts model and model selection

Suppose to have two vectors of random variables $\mathbf{x}=(x_1,\dots,x_n)^{\prime}$ and $\mathbf{y}=(y_i,\dots,y_n)^{\prime}$, observed over a two dimensional space domain.

Let $\boldsymbol{\xi}_{i}=(\xi_{i,1},\dots \xi_{i,K})^{\prime}$ be a vector of dichotomous variables, i.e. $\xi_{i,k} \in \{0,1\}$, and assume that $\boldsymbol{\xi} = (\boldsymbol{\xi}_{1},\dots , \boldsymbol{\xi}_{n})$ follows a Potts model: $$f(\boldsymbol{\xi}|\rho)= \frac{\exp( \rho/2 \sum_{i=1}^n\sum_{j \in \mathbb{C}_i}\boldsymbol{\xi}_{i}^{\prime}\boldsymbol{\xi}_{j})}{W(\rho)}$$

where $\mathbb{C}_i$ is the set of indices of the $i-th$ observation's neighbours, i.e. if $j \in \mathbb{C}_i$ then the observations $i$ and $j$ are ''close'' to each other, and $W(\rho)$ is the intractable normalization constant.

The model is concluded assuming that $$f(\mathbf{x},\mathbf{y}|\boldsymbol{\xi},\{\theta_{k} \}_{k=1}^K) = \prod_{i=1}^n\prod_{k=1}^Kf(x_i,y_i|\theta_{k})^{\xi_{i,k}}$$

where $\theta_k$ is a vector of parameters. The model is estimated under a Bayesian framework and $W(\rho)$ is computed using path sampling.

In this model we have to make two decisions: 1) The number of colors $K$ of the Potts model 2) how close should be $i$ and $j$ to be considered neighbours

I have samples from the posterior distribution for models under different values of $K$ and different neighbour structures, but I am not sure how to perform a model selection. A computation of the BIC and AIC is not feasible since we do not know in closed form the likelihood $f(\mathbf{x},\mathbf{y}|\{\theta_{k} \}_{k=1}^K)$.

EDIT: I specify how I computed the normalization constant and that I have already posterior samples

• Given that the likelihood is intractable, I would suggest using an ABC approach. As we did for instance in a paper selecting the neighbourhood structure of an Ising model. – Xi'an Nov 25 '15 at 9:35
• @Xi'an I edited the post and added some new information. I am not sure how to implement ABC since the Gibbs random field is latent and non-observable, hence I cannot compute the sufficient statistics $S(\boldsymbol{\xi})$. I should probably compute some statistic of the data, $\mathbf{x}$ and $\mathbf{y}$, to be used in the ABC algorithm, but It is not an easy choice. $x_i$ is a circular variable and $y_i$ a linear one, and they are correlated. I am trying to think about a meaningful summary statistic. – niandra82 Nov 25 '15 at 10:27