statisticalrethinkin..

11 Counting and Classification[Summarize Spelke experiments on counting? Goal is to provide intuition for why uncertaintyaround a count scales with the count.]11.1. BinomialA filtered exponentiale binomial density is, like you saw early in this book:y ∼ Binomial(n, p),where y is a count (zero or a positive whole number), p is the probability any particular “trial”is a success, and n is the number of trials.11.1.1. e importance of log-odds. How do we convert the parameters of the binomialdistribution into an additive function of parameters and predictor variables? What is a goodlink? Several different strategies are available, but the best isn’t to just replace, for example,the probability p with an additive model. Instead, we define the log-odds of the event as anadditive model. e log-odds of any event E is just:plog1 − p .is is simply the logarithm of the odds. Why do you want to define the density through thelog-odds? Why not the plain odds, or even just the probability p itself? A very basic answer,but one that is unsatisfying for most, is that log-odds are a natural parameter of the binomialdistribution, much like τ but not σ, is a natural parameter of the normal. But so what? Whatdoes using log-odds give us that plain probability or regular odds do not?Log-odds have several pragmatic advantages. First, unlike the probability p, the logoddsare continuous. ey can take any value between −∞ and +∞. A probability like p isin contrast bounded between zero and one. Having a continuous scale to build our modelsupon will make them much easier to work with, as you’ll see.Second, unlike the plain non-log odds, p/(1 − p), the log-odds are symmetrical: eyaccelerate at the same rate in both directions, as p increases above one-half or as it decreasesbelow one-half. e same is absolutely not true of the regular odds. is will be easier toappreciate with a graph. In FIGURE 11.1, I show the contrast in symmetry between log-oddsand ordinary odds. On the le, log-odds of an event are plotted against the probability ofthe event. e horizontal dashed line shows the log-odds value at which the event is equallylikely to happen as not, p = 0.5. Notice that this occurs where the log-odds are equal tozero. Furthermore, the shape of the log-odds curve is perfectly symmetrical on both sides ofzero, as if reflected in a mirror. In contrast, the plot on the right shows that ordinary odds are277