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I was reading the rstanarm documentation and came across this about its use of 90% intervals as the default. I was hoping someone might be able to provide some clarification.

Default 90% intervals We default to reporting 90% intervals rather than 95% intervals for several >reasons:Computational stability: 90% intervals are more stable than 95% intervals (for which each end relies on only 2.5% of the posterior draws).

What exactly does this mean? Does this mean that the 90% intervals will be more reliable if your MCMC algorithm performs less than ideally? Or that the 90% interval will be more similar for different priors?

Relation to Type-S errors (Gelman and Carlin, 2014): 95% of the mass in a 90% >central interval is above the lower value (and 95% is below the upper value). For >a parameter θ, it is therefore easy to see if the posterior probability that θ>0 (or θ<0) is larger or smaller than 95%.

Why is this not true of other intervals?

BKV
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1 Answers1

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For simplicity, suppose we're talking about centered credibility intervals rather than HPDs. So the 90% interval is the interval from the 5th percentile to the 95th, and the 95% interval is from the 2.5th to the 97.5th.

If the total length of all your chains combined is $n$, you have $.05n$ samples in the left tail from the 5th percentile, and $.025n$ samples in the left tail from the 2.5th percentile. If $n=4000$, you have 200 samples to estimate the left end of the 90% interval, and only 100 samples for the 95% interval. The latter is small enough that you might be a little worried about whether the second decimal point of your estimate is correct. But if you don't actually care about the second decimal point, then you might as well just use the 90% interval. (And if you really care about the second decimal point, then presumably you really want the 99% interval, in which case you need $n=20000$ just to get 100 samples for the endpoint.)

Dan Hicks
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