Derive data likelihood for conditional probability for autologistic model

Question

The log conditional probability for the autologistic model is

$\log\Pr(y_i\mid \{y_j : j \neq i\}) = \alpha_iy_i + \sum_j^N\theta_{ij}y_iy_j - \log(1 + \exp(a_i + \sum_j^N\theta_{ij}y_j))$

From Cressie's Statistics for Spatial Data, I follow how the conditional probability follows from the definition of negpotentials, and how that implies that the log likelihood have the form.

$\log \Pr(\mathbf{y}) = \sum_i^N\alpha_iy_i + \mathop{\sum\sum}_{1 \leq i < j \leq N} \theta_{ij}y_iy_j - \sum_{\mathbf{z} \in \zeta}\exp(\sum_i^N\alpha_iz_i + \mathop{\sum\sum}_{1 \leq i < j \leq N} \theta_{ij}z_iz_j)$

where $\zeta$ is the set of all configurations of the N sites.

For my own understanding, I've been trying to find a derivation of the full likelihood from the conditional probabilities.

Answer 1

The autologistic model has an intractable likelihood in the sense that it involves a normalizing constant that cannot be obtained in closed form (this is, you cannot derive the full likelihood and its evaluation is computationally expensive). There is a recent paper that describes this difficulty and proposes an approximation to the maximum likelihood estimator that does not require the evaluation of the likelihood function:

Bee, M., Espa, G., & Giuliani, D. (2015). Approximate maximum likelihood estimation of the autologistic model. Computational Statistics & Data Analysis, 84, 14-26.