## Saturday, July 20, 2013

### Decisions from posterior distributions: Tail probability or highest density interval?

How should we decide whether a parameter's posterior distribution "rejects" a particular value such as zero? Should we consider the percentage of the distribution above/below the value? Should we consider the relation of the highest density interval (HDI) to the value? Here are some examples to explain why I think it makes more sense to use the HDI.

Here are the two decision rules being compared. First, the tail-probability decision rule: If there is less than 2.5% of the distribution on either side of the value, then reject the value. This is tantamount to using a 95% equal-tailed credibility interval: Values outside the 95% equal-tailed credibility interval are "rejected." Second, the HDI decision rule: Values outside the 95% HDI are "rejected." (Of course, I like to enhance the decision rule with a ROPE, to allow acceptance decisions and to provide a buffer against false alarms -- but that's a separate discusssion.)

The two histograms below represent the MCMC results for two hypothetical parameters.
The upper panel shows a posterior distribution for which the parameter value of zero falls will within the 95% HDI, but 2.0% (<2.5%) of the distribution falls below the value zero. Do we "reject" zero or not? I think it would be wrong to reject zero, because the probability density (i.e., credibility) of zero is high. Zero is clearly among the most credible values of the parameter, even though less than 2.5% of the distribution falls below zero.

The lower panel shows a posterior distribution for which the parameter value of zero falls well outside the 95% HDI (thus, even with of a modest ROPE, e.g. from -1 to +1, zero would still fall outside the 95% HDI), but a full 3.0% (>2.5%) of the distribution falls below zero. Do we "reject" zero or not? If we use a tail-probability decision rule, we do not reject zero. But clearly zero is not among the most credible values of the parameter, in that zero has low probability density.

Proponents of using equal-tailed credibility intervals will argue that percentiles of distributions are invariant under transformations of the parameter, but HDIs are not. True enough, but I think that most parameters are specifically scaled to be meaningful, and we want to know about credibility (probability density) on the meaningful scale, not on a meaningless transformed scale. But I am not saying that one decision rule is "correct" and the other is "wrong." The decision rules are merely rules with differing interpretations and characteristics; I am showing examples that convince me that a more useful, intuitive rule is using the HDI not the equal-tailed interval.

Then why do the plots in DBDA (like those above) bother to display the percentage of the distribution below/above the comparison value? Primarily merely as an additional descriptive statistic, but also to inform those who wish to think about tail probabilities for a decision rule that I eschew.

For another example, see this previous post.