Bayesian statistics · BEGINNER LESSON
Prior probability
Prior probability is the probability distribution used before new study data are seen to represent the strength of belief in a hypothesis or parameter value. In Bayesian statistics, it is combined with the data to form a posterior distribution.
What it is
Prior probability is the “before-data” part of Bayesian statistics. It is not just one number in many uses; it is often a probability distribution that shows how strongly different values of a parameter or different explanations seem plausible before a new study is analyzed. In this framework, probability is treated as a degree of belief rather than only a long-run frequency from repeated trials. The parameter may be something like a treatment effect or a mortality rate.
[1] Bayesian Statistics for Clinical Research - PMC - NIH [2] [PDF] MS&E 226: Fundamentals of Data Science [3] Introduction to Bayesian Statistics - PMC
How it works
Bayesian analysis combines the prior with the likelihood, which is the probability of the observed data under each possible hypothesis or parameter value. The result is the posterior distribution, meaning the updated uncertainty after the new data are included. If the prior is informative, it can influence the answer strongly; if it is flat or non-informative, the data drive the result more. Historical studies, meta-analyses, or expert judgment can all be used to build a prior, but they should match the new study closely.
[1] Bayesian Statistics for Clinical Research - PMC - NIH [2] [PDF] MS&E 226: Fundamentals of Data Science [3] Introduction to Bayesian Statistics - PMC
Limitations and cautions
A prior can be poorly matched to reality if old data are not similar to the new situation, which can distort results. Strong priors can protect against overconfidence, but they can also hide truly new effects. Flat priors are often called non-informative, but they are not always truly neutral. Some priors can even be improper, meaning they are not valid probability distributions.
[2] [PDF] MS&E 226: Fundamentals of Data Science [3] Introduction to Bayesian Statistics - PMC [1] Bayesian Statistics for Clinical Research - PMC - NIH
Why it matters
Prior probability matters because Bayesian methods make uncertainty explicit before and after the data are observed. This helps when earlier evidence exists and when the analysis needs to show how much of the final answer comes from previous knowledge versus the current study. Bayesian results are often summarized with a credible interval, which describes where parameter values are believed to lie given the prior and the data.
[1] Bayesian Statistics for Clinical Research - PMC - NIH [3] Introduction to Bayesian Statistics - PMC [2] [PDF] MS&E 226: Fundamentals of Data Science
SEE THE IDEA
Bayesian update with prior probability
Belief before the new study; may come from earlier trials or a meta-analysis.
How well each possible value explains the observed data.
Updated belief after combining prior and likelihood.
Prior distribution + likelihood from the study data -> posterior distribution.
The prior is the starting belief; the data update it into a posterior.
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.