I am reading a bit about Bayesian analysis, but I cannot understand the difference between the classic quantile-based intervals and the Highest Posterior Density Intervals. What is the difference between the two? I have simulated and plotted some data, the Q95% and the 95% HPDI seem to be similar but not identical. Are there situations where they differ to a larger amount? Thanks a lot
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2 Answers
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For unimodal, more-or-less symmetric distributions, HPD- and quantile-based credible intervals won't be too different. But consider a bimodal posterior distribution with well-separated modes: the HPD-based credible region will be two disjoint intervals whereas the central quantile-based credible region is a single interval by construction.
From a decision theory perspective, the two different kinds of intervals correspond to two different loss functions. The big difference is that the HPD corresponds to a loss function that includes a penalty for the length of the credible region(s). (um, no, as guest implicitly points out, it's) that if the interval fails to cover the true value, the loss function for the quantile-based credible interval penalizes you for how wrong you are whereas in the loss function for the HPD interval, "a miss is as good as a mile".
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5$\begingroup$ You could have also mentioned one notable inconvenient of the HPD interval : its lack of invariance under reparametrization: if $[a(x), b(x)]$ is the HPD credibility interval for $\theta$ then, given a monotone function $f$, the interval $[f(a(x)), f(b(x))]$ is $\textit{not}$ the HPD credibility interval for $f(\theta)$. $\endgroup$ Commented Mar 13, 2012 at 20:25
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$\begingroup$ Good point. I think the decision theory approach illuminates this phenomenon too -- the two HPD intervals correspond to two different loss functions, one of which penalizes the volume of the region in $\theta$-space the other of which penalizes the volume of the region in $f(\theta)$-space. $\endgroup$– CyanCommented Mar 13, 2012 at 21:47
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$\begingroup$ I personally think allowing a Highest Posterior Density credible interval which is not a single interval is confusing, and potentially a misleading use of language. In my view it should simply be the narrowest single interval covering the specified measure of the posterior distribution. $\endgroup$– HenryCommented Mar 14, 2012 at 1:57
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$\begingroup$ @Cyan>> do you mean that the HPD and the quantile-based interval can be considered as "lowest posterior loss region" for some loss function ? I investigated the case of the intrinsic discrepancy loss for a simple example in my paper sciencedirect.com/science/article/pii/S0378375812000870 $\endgroup$ Commented Mar 14, 2012 at 10:40
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$\begingroup$ @Stéphane Laurent Yup, that's exactly what I mean. $\endgroup$– CyanCommented Mar 15, 2012 at 7:01
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A simple example would be if you bought a light-bulb with a lifetime which was exponentially distributed with a mean of 1000 days.
With a 95% credible region: would you tend to see it as likely to last for between 25 and 3689 days (based on the quantiles at 0.025 and 0.975), or would you see it as likely to last fewer than 2996 days (based on the Highest Density Interval)?
In other words, would it surprise you if it died almost as soon as you bought it, even though the mode of the distribution is at zero?
