The nonparametric bootstrap was my first love. I was lost in a muddy swamp of zs, ts and ps when I first saw her. Conceptually beautiful, simple to implement, easy to understand (I thought back then, at least). And when she whispered in my ear, “I make no assumptions regarding the underlying distribution”, I was in love. This love lasted roughly a year, but the more I learned about statistical modeling, especially the Bayesian kind, the more suspect I found the bootstrap. It is most often explained as a procedure, not a model, but what are you actually assuming when you “sample with replacement”? And what is the underlying model?
Still, the bootstrap produces something that looks very much like draws from a posterior and there are papers comparing the bootstrap to Bayesian models (for example, Alfaro et al., 2003). Some also wonder which alternative is more appropriate: Bayes or bootstrap? But these are not opposing alternatives, because the nonparametric bootstrap is a Bayesian model.
In this post I will show how the classical nonparametric bootstrap of Efron (1979) can be viewed as a Bayesian model. I will start by introducing the socalled Bayesian bootstrap and then I will show three ways the classical bootstrap can be considered a special case of the Bayesian bootstrap. So basically this post is just a rehash of Rubin’s The Bayesian Bootstrap from 1981. Some points before we start:

Just because the bootstrap is a Bayesian model doesn’t mean it’s not also a frequentist model. It’s just different points of view.

Just because it’s Bayesian doesn’t necessarily mean it’s any good. “We used a Bayesian model” is as much a quality assurance as “we used probability to calculate something”. However, writing out a statistical method as a Bayesian model can help you understand when that method could work well and how it can be made better (it sure helps me!).

Just because the bootstrap is sometimes presented as making almost no assumptions, doesn’t mean it does. Both the classical nonparametric bootstrap and the Bayesian bootstrap make very strong assumptions which can be pretty sensible and/or weird depending on the situation.