Variance-extreme distributions

Post date: Jul 05, 2021 12:45:9 PM

What distributions will extremise variance within a credal set associated with a Dempster-Shafer structure? Note that we measure variance with the population variance statistic (the one with the division by N rather than N-1) . This statistic is not sensitive to sample size so it characterises the distribution rather than the sample.

Language: It might be reasonable to use the phrase "most dispersed" or "dispersive" to refer to distributions from within some class that maximise variance over that class. Such distributions need not be unique.

Consider Dempster-Shafer structures composed of only two interval focal elements. Such simple structures have basically four possible configurations, as shown in the plots below. They may not intersect at all; they may overlap partially; or one may be nested inside the other, either symmetrically or asymmetrically. As numerical examples, let's use these four Dempster-Shafer structures:

{(0.5,[1, 2]), (0.5,[3, 4])} {(0.5,[1, 3]), (0.5,[2, 4])}

{(0.5,[1, 5]), (0.5,[2, 4])} {(0.5,[1, 6]), (0.5,[2, 4])}

where the probability masses for each interval focal element are equal (given by equal weights). These cases are depicted below as line segments. The associated two- or more-point mass distributions that extremise variance within the respective credal sets implied by each Dempster-Shafer structure are depicted as points. The blue (larger) circles are for the maximum variance; the red (smaller) circles are for the minimum variance. The simple-minded code used to make these plots can be found at Variance-extreme distributions.

In the first case of non-overlapping intervals (upper, left-hand graph), the minimum variance comes from a distribution whose two point masses are as close as possible and the maximum variance comes from one whose masses are as far apart as possible. When they overlap partially as in the second case (upper, right graph), the minimum comes from a Dirac delta distribution at any point at which they overlap which has zero variance.

When one interval is inside the other, it can sometimes happen that the variance-maximising distribution has not two but four point masses. In the third case (lower, left), the distribution with equal mass at the points {1,2,4,5} is one of multiple distributions consistent with the intervals with mass at more than two points that all have the extremal variance 2.5. If we think the focal elements of the Dempster-Shafer structure represent zero-variance intervals, then the distribution with the largest possible variance for this structure is 2.25, which is illustrated by the two-mass distribution {1,4} or the two-mass distribution {2,5}. When the intervals are nested asymmetrically (lower, right), the most dispersed distribution can be difficult to identify by inspection, and you might need to use software to find the extremum and an extremal distribution. In this case, the maximum possible variance for a distribution consistent with these asymmetric intervals is 4.25 such as by the discrete distribution { (0.1, 1), (0.5, 2), (0.4, 6) }.

The focal elements of the two lower Dempster-Shafer structures are

[1, X], [2, 4]

where X is either 5 or 6. The qualitative difference between these two cases in the lower graphs of the figure above depends on the magnitude of X.

We can compare the variances of four- and two-mass distributions for various values of X

four = {1, 2, 4, X}

two = {2, X}

in the graph below where the x-axis is the value of X and y-axis is the population variance of the corresponding discrete distributions, with the four-mass distributions in red and the two-mass distributions in blue. The graph shows that the four-mass distribution will have larger variance than the two-mass distribution until X gets to be larger than about 5.474, after which the two-mass distributions is the most dispersed.

(At least for equally weighted focal elements, it <<is not / does not seem to be>> possible that variance-maximising distributions have more that two point masses unless one interval encloses the other. If the weights are not equal, it may be possible that variance-maximising distributions have more that two point masses even if the intervals are not nested.)

The simple-minded R code used to make these plots is at Variance-extreme distributions. See also the more sophisticated but still brute-force Matlab code for computing variance at Code for computing upper variance of p-box and fuzzy number.