14 Third abstract nonsense interlude: The probability monad

The chapter extends the book’s category-theory-inspired view of generic types by showing that a discrete probability distribution type can be treated as a monad. In this framework, ordinary workflows over values can be lifted to workflows over random values, with a unit operation that wraps a single value and a bind operation that threads probabilistic computations together. The core idea is that probability distributions can be manipulated as structured objects rather than by direct calls to random-number APIs.

To make that concrete, the text defines a discrete distribution interface and shows how bind can be implemented by forming a joint distribution from a prior and a likelihood function, then projecting away the variable that is no longer needed. This supports Bayesian-style reasoning, such as combining a prior probability of having a cold with a likelihood of sneezing to obtain a marginal distribution over sneezing outcomes. The chapter also explains that monads imply map and join operations, and demonstrates join by flattening a distribution that produces other distributions, like a coin flip selecting between different dice.

The chapter then argues that discrete distributions also qualify as additive monads: they can be combined, they have an additive zero representing impossibility, and binding zero stays zero. That makes filtering possible in the usual monadic style, even though the presented version is less efficient than the earlier specialized approach. The final reflection is more skeptical: while the abstraction is elegant and useful for principled inference, integer weights and joint-distribution construction create performance and expressiveness limits, and the approach does not handle infinite discrete distributions.

• The discrete distribution type meets the requirements of a monad. We can create a Unit wrapper around a single value, and we can implement Bind.
• Our implementation of Bind interprets the monad passed in as a prior and the function passed in as a likelihood.
• If we want to, we can make the discrete distribution type an additive monad by creating an empty distribution, in which everything is impossible, and make a few small changes to the existing code.
• The tools we’ve developed are a big improvement over methods using Random. We can build software that manipulates finite categorical distributions using principled, accurate Bayesian reasoning. But this is only a tiny part of the world of statistical programming.

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