Alzheimer's disease (AD) is an incurable neurodegenerative disorder which affects neurons, reducing their function and causing their death. AD is the most common form of dementia and it manifests with difficulties in remembering, thinking and performing everyday activities. Data science can play an important role in enhancing our understanding of this disease and it can help to characterise the pathological evolution of the biomedical parameters in AD patients, compared to healthy elderly subjects. We propose here the latent slope-intercept model, a Bayesian latent variable model for longitudinal data analysis, inspired by the Probabilistic Principal Component Analysis technique. The model was derived analytically, implemented in Python and then applied to clinical scores and brain imaging data coming from Alzheimer's patients.