statisticalrethinkin..

4.3. A GAUSSIAN MODEL OF HEIGHT 93database query.4.3.2. e model. Our goal is to model these values using a Gaussian distribution. First, goahead and plot the distribution of heights, with dens(d2$height). ese data look ratherGaussian in shape, as is typical of height data. is may be because height is a sum of manysmall growth factors. As you saw at the start of the chapter, a distribution of sums tends toconverge to a Gaussian distribution. Whatever the reason, adult heights are nearly alwaysapproximately normal.So it’s reasonable for the moment to adopt the stance that the model’s likelihood shouldbe Gaussian. But be careful about choosing the Gaussian distribution only when the plottedoutcome variable looks Gaussian to you. Gawking at the raw data, to try to decide how tomodel them, is usually not a good idea. e data could be a mixture of different Gaussiandistributions, for example, and in that case you won’t be able to detect the underlying normalityjust by eyeballing the outcome distribution. Furthermore, as mentioned earlier inthis chapter, the empirical distribution needn’t be actually Gaussian in order to justify usinga Gaussian likelihood.So which Gaussian distribution? ere are an infinite number of them, with an infinitenumber of different means and standard deviations. We’re ready to write down the generalmodel and compute the plausibility of each combination of µ and σ. To define the heightsas normally distributed with a mean µ and standard deviation σ, we write:h i ∼ Normal(µ, σ).In many books you’ll see the same model written as h i ∼ N (µ, σ), which means the samething. e symbol h refers to the list of heights, and the subscript i means each individualelement of this list. It is conventional to use i because it stands for index. e index i takes onrow numbers, and so in this example can take any value from 1 to 352 (the number of heightsin d2$height). As such, the model above is saying that all the golem knows about eachheight measurement is defined by the same normal distribution, with mean µ and standarddeviation σ. Before long, those little i’s are going to show up on the righthand side of themodel definition, and you’ll be able to see why we must bother with them. So don’t ignorethe i, even if it seems like useless ornamentation right now.Rethinking: Independent and identically distributed. e short model above is sometimes describedas assuming that the values h i are independent and identically distributed, which may be abbreviatedi.i.d., iid, or IID. You might even see the same model written:h iiid∼ Normal(µ, σ).“iid” indicates that each value h i has the same probability function, independent of the other h valuesand using the same parameters. A moment’s reflection tells us that this is hardly ever true, in a physicalsense. Whether measuring the same distance repeatedly or studying a population of heights, it is hardto argue that every measurement is independent of the others. For example, heights within familiesare correlated because of alleles shared through recent shared ancestry.e i.i.d. assumption doesn’t have to seem awkward, however, as long as you remember thatprobability is inside the golem, not outside in the world. e i.i.d. assumption is about how the golemrepresents its uncertainty. It is an epistemological assumption. It is not a physical assumption aboutthe world, an ontological one, unless you insist that it is. E. T. Jaynes (1922–1998) called this the mindprojection fallacy, the mistake of confusing epistemological claims with ontological claims. 57