Bayesian vs Frequentist Statistics

The frequentist and Bayesian approaches differ in what probability is defined to mean, and that difference cascades into how each performs inference. Frequentists treat model parameters as fixed, unknown constants; probability describes how data would vary across many hypothetical repetitions of an experiment. A frequentist confidence interval, such as a 95% interval for a coefficient, does not say there is a 95% chance the true value lies in that range; it says that if the experiment were repeated many times, 95% of such intervals would contain the true value. Bayesians instead treat parameters as random variables with their own probability distributions, encoding uncertainty directly. They start with a prior distribution reflecting existing belief, combine it with the likelihood of observed data via Bayes' theorem, and produce a posterior distribution. A Bayesian credible interval can be interpreted directly: there is a 95% probability the parameter lies within it, given the model and prior. In machine learning, this distinction shows up concretely: maximum likelihood estimation, common in logistic regression and neural network training, is an estimation method often used in frequentist analyses, while methods like Bayesian neural networks, Gaussian processes, and naive Bayes classifiers apply Bayesian reasoning to produce uncertainty estimates alongside predictions. A common misconception is that one approach is objectively correct; in practice, both are used side by side depending on whether prior knowledge is available and reliable, and whether uncertainty quantification or computational simplicity matters more for the task.