statisticalrethinkin..

98 4. LINEAR MODELSere’s no trouble with the mean, µ. For a Gaussian likelihood and a Gaussian prior on µ, theposterior distribution is always Gaussian as well, regardless of sample size. It is the standard deviationσ that causes problems. So if you care about σ—oen people do not—you do need to be careful ofabusing the quadratic approximation.e deep reasons for the posterior of σ tending to have a long righthand tail are complex. But auseful way to conceive of the problem is that variances must be positive. As a result, there must bemore uncertainty about how big the variance (or standard deviation) is than about how small it is.For example, if the variance is estimated to be near zero, then you know for sure that it can’t be muchsmaller. But it could be a lot bigger.Let’s quickly analyze only 20 of the heights from the height data to reveal this issue. To sample20 random heights from the original list:R code4.21d3