Frequentists and Bayesians

Followers of today's two main traditions of statistical inference are known as frequentist and Bayesian. Ronald Fisher attempted during the 1930s to develop a third school called fiducial inference (1), but this was broadly considered controversial and do not play a major role today. The fundamental difference between frequentists and Bayesians is that they define probability in different ways.

For a frequentist, a probability is an objective measure, a long-run relative frequency. For example, a fair coin has probability 0.5 of heads because, across a very large number of comparable tosses, half are heads, and a drug lowering blood pressure has an objectively measurable average effect among the patients taking the drug. The central limit theorem, which states that as the sample size increases, the distribution of sample means tends to approximate a normal distribution, is essential in frequentist inference.

A Bayesian, on the other hand, sees probability as a subjective measure, a degree of belief. Unlike frequentist analyses, Bayesian analyses can be used to investigate the risk of events that are not occurring repeatedly and therefore have no relative frequency, such as a nuclear war or a climate disaster. Posterior probabilities are derived based on data using Bayes theorem for conditional probabilities of events.

Statistical inference is thus either based solely on data (frequentist) or on both a degree of belief and data (Bayesian).  The primary critique of Bayesian inference is the subjectivity. Different opinions on prior probabilities can lead to different posterior probabilities and conclusions from the analysis.

However, in many cases, frequentist and Bayesian methods leads to similar results, and Bayesian statistics have several advantages. For example, while frequentist methods only give probabilities of data conditional on hypotheses like p-values, and confidence intervals that many find difficult to interpret, Bayesian methods give probabilities of hypotheses and credible intervals. These analysis outcomes may be more in line with what researchers prefer: to be able to say that the probability that the null hypothesis is true is less than 5%, and that there is a probability of 95% that the population value lies within the 95% confidence interval. In fact, it has been suggested this preference explains why frequentist confidence intervals and significance tests researchers are persistently misinterpreted (2).

Frequentist measures like p-values and confidence intervals have dominated scientific publications for a long time, but Bayesian methods are becoming increasingly common also in fields like clinical trials. It is sensible to keep up with developments. In practice, the choice between statistical methods should follow the scientific question, study design, available prior information, and decision context.

References

  1.  Fisher, R. A. (1935). The fiducial argument in statistical inference. Annals of Eugenics. 5 (4): 391–398. doi:10.1111/j.1469-1809.1935.tb02120.x. hdl:2440/15222.
  1. Bland JM, Altman DG. Bayesians and frequentists. BMJ. 1998 Oct 24;317(7166):1151-60. doi: 10.1136/bmj.317.7166.1151. PMID: 9784463; PMCID: PMC1114120.

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