Bayesian vs frequentist inference

There are two great traditions for turning data into conclusions, and they disagree about something fundamental: what 'probability' is allowed to describe. A frequentist reserves probability for the long-run frequency of repeatable events, so a fixed unknown like a drug's true effect has no probability — it just is some number. A Bayesian allows probability to describe a degree of belief, so it is perfectly fine to put a distribution on that unknown and update it. From this one philosophical fork, two whole toolkits grow.

Watch how each treats the unknown theta and the data. The frequentist holds theta fixed and treats the data as random; inference asks 'over many hypothetical repetitions, how would my estimator behave?' — giving maximum-likelihood estimates, p-values, and confidence intervals, none of which assign probability to theta itself. The Bayesian holds the observed data fixed and treats theta as the random unknown; inference is just the posterior, from which credible intervals and predictions fall out directly. A vivid contrast: asked for a 95 percent interval, the frequentist guarantees the PROCEDURE covers theta 95 percent of the time across repetitions, while the Bayesian asserts a 0.95 probability that theta is in THIS interval.

It is tempting but wrong to crown a winner. Frequentist methods need no prior and come with clean repeated-sampling guarantees, but their outputs (especially p-values and confidence intervals) are routinely misinterpreted as Bayesian statements. Bayesian methods give directly interpretable answers and naturally fold in prior knowledge and uncertainty, but they require you to state a prior and usually demand heavy computation. In practice the two often agree numerically — with lots of data and a mild prior, the posterior mean sits near the maximum-likelihood estimate and a credible interval nearly matches a confidence interval — and good practitioners borrow freely from both.