This paper presents a fully Bayesian treatment of Probabilistic Matrix Factorization (PMF) using Markov Chain Monte Carlo (MCMC) methods. The authors address the limitations of traditional MAP estimation, which can lead to overfitting, by automatically controlling model capacity through integration over all model parameters and hyperparameters. They demonstrate that Bayesian PMF models can be efficiently trained on large datasets, such as the Netflix dataset with over 100 million movie ratings, achieving significantly higher prediction accuracy compared to MAP-trained PMF models. The paper also discusses the advantages of Bayesian models in handling uncertainty and provides experimental results showing that Bayesian PMF models outperform both linear and logistic PMF models, even at higher feature dimensions. The authors conclude that MCMC methods, despite their computational demands, are effective for large-scale applications and provide a more accurate representation of the posterior distribution compared to variational methods.This paper presents a fully Bayesian treatment of Probabilistic Matrix Factorization (PMF) using Markov Chain Monte Carlo (MCMC) methods. The authors address the limitations of traditional MAP estimation, which can lead to overfitting, by automatically controlling model capacity through integration over all model parameters and hyperparameters. They demonstrate that Bayesian PMF models can be efficiently trained on large datasets, such as the Netflix dataset with over 100 million movie ratings, achieving significantly higher prediction accuracy compared to MAP-trained PMF models. The paper also discusses the advantages of Bayesian models in handling uncertainty and provides experimental results showing that Bayesian PMF models outperform both linear and logistic PMF models, even at higher feature dimensions. The authors conclude that MCMC methods, despite their computational demands, are effective for large-scale applications and provide a more accurate representation of the posterior distribution compared to variational methods.