Integrating Insight and Uncertainty: Bayesian modeling for data-rich livestock systems.

Bayesian statistical modeling provides a principled framework for integrating biological knowledge, process understanding, and observed data in animal science research. By treating all unobserved quantities as random variables and explicitly defining probability distributions for both processes and observations, Bayesian methods allow researchers to quantify uncertainty, incorporate prior information, and evaluate full distributions of plausible outcomes. This manuscript introduces the foundational concepts of Bayes' theorem, probability, probability distributions, and Markov chain Monte Carlo, and demonstrates how these components form coherent, process-based statistical models. Two case studies illustrate the application of Bayesian regression to questions central to beef cattle research: the influence of day-to-day variation in dry matter intake on average daily gain, and the prediction of intake from biometric variables collected using precision livestock technologies. Together, these examples highlight how Bayesian approaches enhance model transparency, improve interpretation, and support decision-making in complex biological systems. This framework equips animal scientists with the tools needed to ask clearer questions, build biologically grounded models, and leverage emerging data streams to advance research and management.