statisticalrethinkin..

2.2. COLOMBO’S FIRST BAYESIAN MODEL 37one possibility (or none) can be correct, so don’t think that just because these numbers arenow probabilities that they define frequencies of any events. Probabilities change when ourinformation changes; frequencies do not. And that’s because probabilities are just a shortcutto counting up possibilities.If you find all of this rather deflationary—probability theory seems kinda simple minded—good. Probability theory is just a mechanism for keeping track of the ways that things canhappen, according to our assumptions. It isn’t the mind of angels or2.2. Colombo’s first Bayesian modelYou can solve any problem in probability by just counting the numbers of ways each possibilitycould produce each event of interest. But that’s tedious. So instead we nearly alwayswork with standardized counts, probabilities, and decompose the counts into componentprobabilities that represent the different elements in our counting in the previous section.is decomposition comprises a statistical model of one sort.Consider four different kinds of things we counted in the previous section.(1) e initial number of ways each possibility could be correct.(2) e number of ways each possibility could produce an individual W or L observation.(3) e accumulated number of ways each possibility could produce the entire data.(4) e total number of ways to produce the data, across all possibilities.Each of these counts has a direct analog in Bayesian data analysis.[older content follows, still needs revision]To get the logic moving, we need to make assumptions, and these assumptions constitutethe model. Designing a simple Bayesian model benefits from a design loop with three steps.(1) Data story: Motivate the model by narrating how the data might arise(2) Update: Educate your model by feeding it the data(3) Evaluate: All statistical models require supervision, leading possibly to model revisione next sections walk through these steps, in the context of the globe tossing evidence.Rethinking: e logic of plausibility. Logic is usually identified with Aristotle and rules for consistentreasoning about true and false statements. e logic applied here is a continuous version that dealsin plausibility of each statement. e American physicist Richard T. Cox showed that if one wishesto reason with plausibility and still be consistent with Aristotelian logic, then one is forced to useprobability theory to perform the reasoning. 34 e consequence of this fact is that Bayesian analysis,which obeys the laws of probability, is a consistent system for plausible reasoning.2.2.1. A data story. Bayesian data analysis usually means producing a story for how the datacame to be. is story may be descriptive, specifying associations that can be used to predictoutcomes, given observations. Or it may be causal, a theory of how some events produceother events. Typically, any story you intend to be causal may also be descriptive. But manydescriptive stories are hard to interpret causally. But all data stories are complete, in the sensethat they are sufficient for specifying an algorithm for simulating new data. In later sectionsof this chapter, you’ll see examples of doing just that, as simulating new data is a necessarypart of model criticism.