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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

15.5 Beta

Beta distribution The continuous Beta distribution is the posterior probability distribution for the parameter θ, where one has observed h successes in a given N events, and each event is assumed to have a θ probability of success.

P ( θ | h , N ) = ( N + 1 ) · ( N h ) θ h ( 1 − θ ) N − h

Although it may look like a binomial, the Beta distribution is used to find the best estimate for the parameter θ where the number of successes and events, h and N are given.

The posterior density for θ given 3 heads and 9 tails overlaid with dashed vertical quartile lines: 1% at 0.07, 5% at 0.11, 25% at 0.20, 50% at 0.28, 75% at 0.36, 95% at 0.49 and 99% at 0.59.
Figure C.5. Posterior probability distribution for the θ values of the bent coin - the probability that the coin will land heads. The distribution is shown for data 3 heads and 9 tails. The various quartiles are shown in the plot.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.