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Published in:
2022 | OriginalPaper | Chapter
14. Probability Distributions
Author : Peter McCullagh
Published in: Ten Projects in Applied Statistics
Publisher: Springer International Publishing
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Abstract
To the uninitiated, a stochastic model may seem to emerge from nowhere with little explanation. This chapter attempts to address the information gap by arguing that exchangeability is the key concept that justifies essentially all statistical models.
In its simplest form, exchangeability is the statement that a sample of values taken in one order has the same distribution as the sample taken in a different order. Depending on the context, the statement may be reasonable or it may be unreasonable. If the claim is unreasonable, it must be possible to demonstrate that by pointing to some aspect of the baseline sample configuration that makes the permuted sample different from the original. The general idea is that every sample comes with a baseline covariate configuration, and exchangeability is a verifiable consequence of congruent sample configurations. Stationarity and isotropy are manifestations of exchangeability, albeit with different groups.
This chapter discusses the implications of various forms of exchangeability in regression models, in block designs, in time series, and so on. The overarching idea is that two samples having the same baseline covariate configuration must have the same response distribution. Various examples of models having this property are discussed.
Treatment is modelled as a group action on probability distributions. To each control distribution P
θ, the group element g associates a treatment distribution P
gθ. Thus, if g is the identity element, the treatment and control distributions are equal. Essentially all generalized linear models with a treatment effect are of this form, with the treatment group being the additive group of real numbers. For example, hazard multiplication is a group action on survival distributions.