Bayesian Inference for a Binary Outcome Rate
Summary
The article explains how to estimate the probability of a binary outcome using Bayesian inference. It uses coin flips to introduce a Bernoulli likelihood for observations and a Beta distribution as a prior for the unknown success probability. Because these distributions are conjugate, the posterior remains Beta after observing successes and failures. The same structure can describe other two-outcome rates, such as defect prevalence or treatment recovery proportions.
A worked example starts with a prior centered near 0.5 with standard deviation 0.1, then observes 10 heads in 50 flips. The article reports that the posterior mean moves to about 0.3 and its standard deviation falls to about 0.05, illustrating how evidence updates the estimate and reduces uncertainty. The inference assumes independent trials and a fixed underlying probability. The result depends on the chosen prior, and the example does not establish that those assumptions hold for real financial data or other applications.
Key ideas
- A Bernoulli likelihood models each observation as one of two outcomes given a probability parameter.
- A Beta distribution can represent prior uncertainty about that probability.
- With a Beta prior and Bernoulli observations, the posterior is also Beta.
- The example updates a prior near 0.5 using 10 heads in 50 flips and reports a posterior mean near 0.3.
- The model assumes independent trials and an underlying probability that remains fixed.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.