Bayesian Bootstrap (Rubin)
Also known as: Bayesian Bootstrap (Rubin), Rubin bootstrap, Dirichlet-weighted bootstrap
The Bayesian Bootstrap, introduced by Donald B. Rubin in 1981, is a resampling method that produces a Bayesian counterpart to the frequentist bootstrap by assigning each observation a random weight drawn from a Dirichlet distribution. It yields a full posterior distribution for a statistic and allows prior information to be incorporated.
Key highlights
- Produces a coherent Bayesian posterior distribution for a statistic, not just a point estimate.
- Makes no normality assumption and works with continuous or categorical data.
- Allows prior information to be incorporated and gives stable posteriors in moderately small samples.
Intuition
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How it works
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When to use it
Use the Bayesian Bootstrap when observations are independent and you want a posterior distribution for a descriptive or predictive statistic without assuming normality. It suits continuous or categorical data and is helpful in small samples where a coherent posterior is valued, provided there are at least about 10 observations and ideally B ≥ 1000 replications. It is not appropriate for very small samples (below about 20), where the Dirichlet weights become unstable.
Strengths & limitations
- Produces a coherent Bayesian posterior distribution for a statistic, not just a point estimate.
- Makes no normality assumption and works with continuous or categorical data.
- Allows prior information to be incorporated and gives stable posteriors in moderately small samples.
- With fewer than about 20 observations the Dirichlet weights become unstable and the standard bootstrap is preferable.
- Below about 10 observations the posterior distribution is unreliable.
- Requires independent observations and a sufficient number of replications (B ≥ 1000) for accurate posteriors.
Common pitfalls
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Applications
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Frequently asked
How is the Bayesian Bootstrap different from the ordinary bootstrap?
The ordinary bootstrap redraws observations with replacement, giving each a random integer count, while the Bayesian Bootstrap keeps all observations and assigns them continuous Dirichlet weights that sum to one. The result is interpreted as a posterior distribution rather than a sampling distribution.
How many replications should I run?
At least about 1000 (B ≥ 1000) weight draws are recommended so that the posterior distribution is smooth and its quantiles, used for credible intervals, are stable.
Can I use it on small samples?
It needs at least about 10 observations, but below roughly 20 the Dirichlet weights become unstable. In that case the standard bootstrap is preferred, and below about 10 observations a permutation test is the safer choice.
Does it require the data to be normally distributed?
No. The Bayesian Bootstrap makes no normality assumption and works with continuous or categorical data; it only requires that the observations be independent.
Sources
- 1.Rubin, D. B. (1981). The Bayesian Bootstrap. The Annals of Statistics, 9(1), 130-134.
- 2.Lo, A. Y. (1987). A Large Sample Study of the Bayesian Bootstrap. The Annals of Statistics, 15(1), 360-375.
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ScholarGate. (2026, June 1). Bayesian Bootstrap. ScholarGate. https://scholargate.app/statistics/bayesian-bootstrap