Bayesian and frequentist perspectives
Both schools of statistics answer useful questions. Understanding their differences helps you choose methods and read research wisely.
Different questions
Frequentist methods ask: if the hypothesis were true, how surprising are these data? Bayesian methods ask: given these data, how probable is the hypothesis? A p-value answers the first; a posterior probability answers the second. Both are legitimate, but they are not interchangeable.
Intervals compared
A 95% confidence interval comes from a procedure that captures the true value 95% of the time over repeated samples. A 95% credible interval states a 95% posterior probability for this data set. With weak priors and large samples, the two often give similar numbers, though their interpretations differ.
Hypothesis comparison
Instead of p-values, Bayesians often use Bayes factors, the ratio of how well two hypotheses predict the data. A Bayes factor of 10 means the data are ten times more likely under one hypothesis than the other. Unlike a p-value, a Bayes factor can provide evidence for the null hypothesis, not just against it.
Objections and responses
Critics say priors make Bayesian results subjective. Bayesians reply that every analysis involves choices, such as which model to use, and that stating priors makes assumptions transparent. Critics of frequentist methods point to the widespread misinterpretation of p-values. In practice, many statisticians use both approaches pragmatically.
Bayesian thinking in everyday life
Updating beliefs proportionally to evidence is good reasoning in general. Extraordinary claims require extraordinary evidence because their prior probability is low. A single surprising study should shift your beliefs somewhat, not completely, especially when the prior evidence points the other way.
Computation
Simple models have formula-based posteriors, but realistic models require simulation. Markov chain Monte Carlo methods draw thousands of samples from the posterior, letting analysts summarise any quantity of interest. Software packages have made Bayesian methods practical for researchers in medicine, ecology, and machine learning.
Worked example: interpreting a trial
A trial reports a treatment effect with \(p = 0.04\) and a Bayesian posterior probability of benefit of 0.93 using a sceptical prior. The frequentist result says the data would be unusual if there were no effect; the Bayesian result says that, given a cautious prior, benefit is quite likely but not certain. Together they give a fuller picture.
Common mistakes
Interpreting a p-value as a posterior probability. Claiming Bayesian results are 'objective' regardless of the prior. Dismissing either school entirely. Ignoring sensitivity analyses that show how conclusions change with different priors.
Key takeaways
Frequentist and Bayesian methods answer different questions: data given hypothesis versus hypothesis given data. Confidence and credible intervals differ in interpretation. Bayes factors can support the null. Priors should be transparent and tested for sensitivity. Good statisticians understand and use both.
Flashcard
How surprising are the data if H0 is true?
Check your understanding
Which method can provide evidence in favour of the null hypothesis?