Direct comparison
Bayesian vs. Frequentist Statistics
A 95% CI and credible interval can look alike yet mean different things. A worked p-value vs. posterior probability example shows why they can disagree.
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How do Frequentist, Bayesian compare side by side?
The table below compares Frequentist, Bayesian across 9 procurement-relevant dimensions, from what the unknown parameter is treated as through where each is the default in practice.
Side-by-side comparison
| Dimension | Frequentist | Bayesian |
|---|---|---|
| What the unknown parameter is treated as | A fixed, unknown constant -- it either has a value or it doesn't; there's no probability distribution over it. | A quantity with a probability distribution, updated as data arrive. |
| What produces the answer | The sampling distribution of the estimator under (hypothetical) repeated sampling. | A prior distribution combined with the observed data via Bayes' theorem, producing a posterior distribution. |
| What a 95% interval means | 95% of intervals built this way, across repeated samples, would contain the true parameter -- not a 95% probability for this specific interval. | A 95% probability the parameter lies in this specific interval, given the data and the stated prior. |
| What a p-value answers | The probability of seeing data this extreme, or more, if the null hypothesis were exactly true. | Not used directly -- replaced by a posterior probability for each hypothesis, or a Bayes factor comparing them. |
| Does the method require a prior | No. | Yes -- every Bayesian analysis states one explicitly, even a deliberately weak or "flat" one. |
| Can it directly state "the null is X% likely" | No -- a p-value is not, and was never designed to be, the probability that the null hypothesis is true. | Yes -- a posterior probability is exactly that statement, conditional on the prior used to get there. |
| Behavior as sample size grows, true effect fixed and small | The p-value keeps shrinking toward 0 as n grows, even for a practically trivial true effect. | Depends on the prior on the null; under some reasonable priors, evidence against a sharp null can grow more slowly than the p-value suggests, or even favor the null more at very large n (Lindley's paradox). |
| Computation | Closed-form for standard tests (z, t, chi-square, F). | Closed-form for conjugate models (e.g. beta-binomial); Markov Chain Monte Carlo (MCMC) simulation for most realistic models. |
| Where each is the default in practice | Confirmatory trial primary endpoints, most journal Results-section reporting, standard hypothesis tests taught in introductory statistics. | Adaptive trial designs, formally updating across a sequence of studies, small or rare-population samples where a weak prior still adds real information. |
Common questions
Common questions about Frequentist vs Bayesian
If a 95% CI and a 95% credible interval come out numerically similar, does the difference stop mattering?
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No. In the worked example on this page, a frequentist 95% CI of [50.2%, 79.8%] and a Bayesian 95% credible interval of [49.4%, 77.9%] are close, because a weak prior was used on a moderate-sized sample -- but they still answer different questions. State which one you calculated and what it actually claims. A reader who assumes a confidence interval means "there's a 95% chance the true value is in this range" is describing the Bayesian credible interval, even when the number they're looking at is frequentist.
Why can a p-value near 0.05 and a Bayesian posterior probability disagree so much?
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Because a p-value only asks how surprising the data would be if the null hypothesis were exactly true -- it never weighs that against how surprising the same data would be under a realistic alternative. A Bayesian analysis compares both, using a stated prior for the size of a plausible effect. The practical result, shown in the worked example on this page, is that a "just significant" p-value close to 0.05 is often much weaker evidence against the null than the 0.05 threshold is conventionally read as implying.
Which framework should a research team use?
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It's usually a decision made per analysis, not a permanent philosophical commitment. Frequentist methods remain the default where a fixed, pre-specified, easily audited primary analysis is expected -- most confirmatory trial primary endpoints, for instance (see our clinical-trial-specific comparison, linked below, for that regulatory context). Bayesian methods earn their added complexity where formally updating on accumulating data, incorporating genuine prior information, or reasoning carefully about a small or rare-population sample matters more than procedural simplicity.
Does using a Bayesian method make an analysis more subjective than a frequentist one?
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It makes an existing source of judgment explicit rather than adding a new one. A frequentist analysis also involves real judgment calls -- which test to run, what significance threshold to use, whether and how to correct for multiple comparisons -- they're simply not labeled a "prior." A Bayesian prior has the advantage of being stated, justified, and testable via a sensitivity analysis across a range of plausible priors (as the worked example does with three different prior widths), which a frequentist analysis's implicit choices usually are not.
