# Maximum Likelihood Inference

The authors propose a covariate-dependent consecutive-ratio Markov random field (CR-MRF) that jointly models multivariate ordinal ratings, show that comparison models such as Bradley-Terry and Plackett-Luce arise as restrictions of it under independence and an exponential-race argument, and develop maximum likelihood inference with an importance-sampling estimator of the intractable normalizer; in simulations and on MovieLens, modeling ordinal scores directly yields more accurate and better-calibrated out-of-sample pairwise preference predictions than fitting comparison models to derived win-loss data.

The authors propose a covariate-dependent consecutive-ratio Markov random field (CR-MRF) that jointly models multivariate ordinal ratings, show that comparison models such as Bradley-Terry and Plackett-Luce arise as restrictions of it under independence and an exponential-race argument, and develop maximum likelihood inference with an importance-sampling estimator of the intractable normalizer; in simulations and on MovieLens, modeling ordinal scores directly yields more accurate and better-calibrated out-of-sample pairwise preference predictions than fitting comparison models to derived win-loss data.