statisticalrethinkin..

4.3. A GAUSSIAN MODEL OF HEIGHT 91Compare to the calculations in earlier chapters.4.3. A Gaussian model of heightLet’s build a linear regression model now. Well, it’ll be a “regression” once we have apredictor variable in it. For now, we’ll get the scaffold in place and construct the predictorvariable in the next section.We’ll work through this material by using real sets of data. In this case, we want a singlemeasurement variable to model as a Gaussian distribution. ere will be two parametersdescribing the distribution’s shape, the mean µ and the standard deviation σ. Bayesian updatingwill allow us to consider every possible combination of values for µ and σ and to scoreeach combination by its relative plausibility, in light of the data. ese relative plausibilitiesare the posterior probabilities of each combination of values µ, σ.Another way to say the above is this. ere are an infinite number of possible Gaussiandistributions. Some have small means. Others have large means. Some are wide, with a largeσ. Others are narrow. We want our Bayesian machine to consider every possible distribution,each defined by a combination of µ and σ, and rank them by posterior plausibility. Posteriorplausibility provides a measure of the logical compatibility of each possible distribution withthe data and model.In practice we’ll use approximations to the formal analysis. So we won’t really considerevery possible value of µ and σ. But that won’t cost us anything in most cases. Instead thething to worry about is keeping in mind that the “estimate” here will be the entire posteriordistribution, not any point within it. And as a result, the posterior distribution will be adistribution of Gaussian distributions. Yes, a distribution of distributions. If that doesn’tmake sense yet, then that just means you are being honest with yourself. Hold on, workhard, and it will make plenty of sense before long.4.3.1. e data. e data contained in data(Howell1) are partial census data for the Dobearea !Kung San, compiled from interviews conducted by Nancy Howell in the late 1960’s. 56For the non-anthropologists reading along, the !Kung San are the most famous foragingpopulation of the 20th century, largely because of detailed quantitative studies by people likeHowell.Load the data and place them into a convenient object with:library(rethinking)data(Howell1)d