statisticalrethinkin..

114 4. LINEAR MODELSheight140 150 160 170 18030 35 40 45 50 55 60weightFIGURE 4.6. Samples from the quadratic approximate posterior distributionfor the height/weight model, m4.3. Le: Each point is a joint sample fromthe posterior. Estimates of the slope b and the intercept a are strongly negativelycorrelated. Right: One-hundred lines sampled from the posteriordistribution, showing the uncertainty in the regression relationship.e result is shown in the righthand plot in FIGURE 4.6. By plotting multiple regressionlines, sampled from the posterior, it is easy to see both the highly confident aspects of therelationship and the less confident aspects. e cloud of regression lines displays greateruncertainty at extreme values for weight. is is very common. In this case, the lines donot vary a lot, however, since there is so much data. But you’ll encounter other examples inwhich the relationship is much less precise.4.4.3.4. Plotting regression intervals and contours. e cloud of regression lines in FIG-URE 4.6 is an appealing display, because it communicates uncertainty about the relationshipin a way that many people find intuitive. But it’s much more common to see the uncertaintydisplayed by plotting an interval or contour around the MAP regression line. In this section,I’ll walk you through how to compute any arbitrary interval you like, using the underlyingcloud of regression lines embodied in the posterior distribution. en we’ll plot a shadedregion around the MAP line, to display the interval.Here’s how to plot a 95% interval around the regression line. is interval incorporatesuncertainty in both the slope β and intercept α at the same time. To understand how itworks, focus for the moment on a single weight value, say 50 kilograms. You can quicklymake a list of 10-thousand values of µ for an individual who weighs 50 kilograms, by usingyour samples from the posterior:R code4.46mu_at_50