The Art of Statistics: Learning from Data

by David Spiegelhalter

The Bayesian response to this problem is known as multi-level regression and post-stratification (MRP). The basic idea is to break down all possible voters into small ‘cells’, each comprising a highly homogeneous group of people – say living in the same area, with the same age, gender, past voting behaviour, and other measurable characteristics. We can use background demographic data to estimate the number of people in each of these cells, and these are all assumed to have the same probability of voting for a certain party. The problem is working out what this probability is, when our non-random data may mean that we only have a few people, or maybe none at all, in a particular cell. The first step is to build a regression model for the probability of voting a particular way given the characteristics of the cell, so our problem is reduced to estimating the coefficients of the regression equation. But there are still too many coefficients to estimate reliably using the standard methods, and this is where Bayesian ideas come in. Coefficients corresponding to different areas are assumed to be similar, a sort of intermediate point between assuming they are precisely the same, and assuming they are utterly unrelated. Mathematically, this assumption can be shown to be equivalent to assuming all these unknown quantities have been drawn from the same prior distribution, and this enables us to move many individual, rather imprecise, estimates towards each other, which results in sm...

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