statisticalrethinkin..

2.1. PROBABILITY IS JUST COUNTING 31proportions of water—from 0 to 1—could produce the data. e number of ways that eachpossible proportion could produce the data will provide a relative measure of its plausibility.And that’s Bayesian inference.2.1.1. Enumerate the possibilities. To begin, we’ll simplify the analysis by considering only11 different proportions of water: 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1. is abbreviatedlist will get the lesson across. en in the next major section, you’ll see how togeneralize to an infinite number of possibilities.is leads us to a forced choice: what do we think are the relative plausibilities of eachof these 11 proportions, before we’ve seen any data? e easiest answer, and the most conventional,is to answer that we want to count all of them equally, until we have a good reasonnot to. 33 We do already have good reason not to: since you are probably reading this bookon dry land, there is obviously some land. So the proportion of water is not 1. And sinceyou know there are oceans on planet Earth, the proportion is not 0. But we’ll work with theequal counting approach, because it is foundational and easiest to understand. Later in thechapter, we’ll repeat the analysis from a different beginning, so you can see its impact. Fornow, think of this as announcing:“I do not know which of these possibilities is correct. So until I have goodreason to do otherwise, I’ll keep track of all of the possibilities equally.”As you’ll see later, if ever you receive information that contradicts this egalitarian bookkeeping,it’s very easy to incorporate it.So begin by stating that there’s at least one way that each proportion could be the correctone. So we assign a count of “1” to each possibility. Let’s start organizing this information ina table:proportion 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ways 1 1 1 1 1 1 1 1 1 1 1Going forward, we’ll keep counting the ways things can happen, updating this table at eachstep.2.1.2. How many ways can the data happen? Building on top of this meager beginning,consider only the first datum: W (water). How many ways can each of the possible proportionsgenerate this observation? To answer this question, it helps to remember that globetosses, like coin flips, are not really random. e physics are deterministic. It’s just that wedon’t know enough about the initial conditions and cannot measure them with sufficientprecision in order to predict the outcome. Probability indicates ignorance of an observer—whether machine or person—not true indeterminacy. e “randomness” of the globe orcoin resides in us and the machines we build, never in the globe or coin itself.So consider the globe toss in detail. When you pick up the inflatable globe, you holdit in a particular way. ere are very many ways to hold it. en when you toss it in theair, there are very many ways to impart momentum to it. en as the globe briefly defiesgravity, it interacts with air currents and particles in the air. ere are many possibilitieshere as well. And when you catch the globe, there are again many ways to grasp and holdit, each leading to either “water” or “land.” All of these possibilities form a garden of forkingpaths, each producing a different perfectly deterministic series of events, producing aperfectly deterministic outcome.But since we do not know enough to identify the one unique path that did happen, wespeak of the globe toss as “random.” What’s important is the relative numbers of paths that