Statistics / Bayesian inference · BEGINNER LESSON
Posterior probability
Posterior probability is the probability of a hypothesis or treatment effect after the data have been observed. In Bayesian inference, it updates a prior belief with the observed data to give a data-based probability statement.
What it is
Posterior probability is the updated probability of a hypothesis after observing data. In the sources, it is contrasted with a p value: a p value asks how unusual the data would be if there were no treatment effect, while posterior probability asks how likely there is a treatment effect after seeing the data. This is a Bayesian probability statement, meaning the hypothesis itself is assigned probability, not just the data under a fixed assumption. [0,1,2]
[1] Figure 2 [2] Détente: A Practical Understanding of P values and Bayesian ... - PMC [3] Lecture 16: Bayesian inference (v3) Ramesh Johari ramesh ...
How it is built
Bayesian inference starts with a prior distribution, which summarizes the investigator’s belief before collecting data. The observed data contribute a likelihood, the probability of the data under each possible parameter value. Bayes’ rule combines prior and likelihood to produce the posterior distribution; the posterior probability is then the probability mass in the region of interest, such as values indicating a treatment effect. [0,2]
[1] Figure 2 [3] Lecture 16: Bayesian inference (v3) Ramesh Johari ramesh ...
What it measures in practice
Posterior probability measures how strongly the updated evidence supports a hypothesis, such as the chance that a biased coin was drawn or that a treatment effect exists. The sources note that it can be more intuitive than a p value and can be used to make direct probability statements like “there is a 50% chance” or “there is a 95% chance” about a parameter or effect. [0,1,2]
[1] Figure 2 [2] Détente: A Practical Understanding of P values and Bayesian ... - PMC [3] Lecture 16: Bayesian inference (v3) Ramesh Johari ramesh ...
Limitations and interpretation
Posterior probability depends on the chosen prior, so different prior assumptions can change the result. It also depends on sample size and on how common the observed evidence is under both the null hypothesis and the alternative; for the same p value, the posterior probability of a treatment effect can get smaller in larger studies. The sources also warn that very large studies can produce statistically significant p values for effects that are clinically unimportant. [0,1,2]
[1] Figure 2 [2] Détente: A Practical Understanding of P values and Bayesian ... - PMC [3] Lecture 16: Bayesian inference (v3) Ramesh Johari ramesh ...
SEE THE IDEA
How posterior probability is formed
Belief about the parameter before the study begins
Trial results or coin flips that provide evidence
How probable the data are under each possible hypothesis
Updated probability of the hypothesis after combining prior and data
Prior distribution and observed data are combined through the likelihood to produce the posterior probability.
Bayesian updating turns prior belief plus data into an updated probability statement about the hypothesis.
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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.