# Dispersion measure - probability density function

I am wondering whether someone has a tip on this potentially very basic question. i have done some grid-based bayesian analysis and ended up with a non-standard discrete posterior density function over an interval of 100 observational values (per graph) below and as shown in the vector format per below. I can easily find the corresponding value of the distribution in R (which.max) but am struggling to find a way to calculate any measure of dispersion of the probability distribution. Ideally - i would like to have a quick way of calculating in R that gives me the answer that x% of the density lies between e.g. the index values of 20 and 80; any tips very welcome...

  [1] 4.615412e-02 7.685555e-02 7.936106e-02 9.355243e-02 9.390333e-02 9.415255e-02 8.812764e-02 8.094180e-02
[9] 7.111677e-02 6.096649e-02 5.067963e-02 4.113239e-02 3.252751e-02 2.514941e-02 1.900486e-02 1.405895e-02
[17] 1.018350e-02 7.230007e-03 5.033092e-03 3.437899e-03 2.305284e-03 1.518504e-03 9.831581e-04 6.261243e-04
[25] 3.925111e-04 2.424243e-04 1.476553e-04 8.878700e-05 5.277273e-05 3.104750e-05 1.810726e-05 1.048550e-05

• – Tim
Sep 28 '20 at 18:17

If you want the smallest possible range with $x \%$ probability for a unimodal distribution (like the one you plotted), then I recommend the HPDinterval function in the R coda package. See the vignette here. There are other ways to construct intervals (e.g., centered quantiles through the quantile function) but this HPD interval seems to be what you want.
What the HPD interval is essentially doing is taking a horizontal line from far above, lowering it incrementally and stopping once the area under the posterior density (ignoring the line) is equal to $x/100$. Then it considers the points of intersection of the horizontal line and the posterior density curve and returns the interval with endpoints the same as that of the line segment formed by the intersection. Mathematically this captures the smallest interval of the posterior which has exactly $x \%$ probability.