Exemplar problems

From vacuity

1) Characterise probability P, assuming nothing about the probability.

2) Characterise positive constant Q, assuming no empirical information about it.

3) Characterise beta distribution R, assuming no empirical information about it.

We believe that these are essentially intervals, whether expressed as p-boxes, possibility distributions, or DSS's.  

1) P = [0,1] = fn(0,1) = minmax(0,1) = DSS{1, [0,1]}, the dunno interval.

2) Q = [0, ∞] = fn(0,Inf) = minmax(0, Inf) = DSS{1, [0,∞]} = positive = nonnegative(?). 

3) R = [0,1] = fn(0,1); but this is not quite equal to beta([0,∞],[0,∞]) = {H(0), H(1), [0, 1], [0, 1/4], beta}, where H denotes the Heaviside function, and beta denotes the class of all beta distributions. There does not seem to be a way to express the restriction to beta distributions in Dempster-Shafer theory.

From moments and order statistics

1) Characterise distribution M1 given its minimum is 0, its maximum is 1, and its mode is 0.1.

2) Characterise distribution M2 given its minimum is 12, its maximum is 18, and its median is 13. 

3) Characterise distribution M3 given its minimum is 1, its maximum is 5, its mean is 2, and its standard deviation is 0.5.

4) Characterise distribution M4 assuming it is unimodal and given its minimum is 0, its maximum is 9, its mean is 3, and its standard deviation is 1.

1) We are unsure about the relationship between the mode (maximum of the density) and a possibility distribution

2) This could be the interval [12, 18] for all alpha < 1, and the singleton {13} for alpha = 1. Adding more quantile information would further constrain the credal set (see From coverages)

3) From Markov/Chebyshev style inequalities, but are these distinct to the p-box case?

4) Unsure how to include unimodality

From random data

Characterise constant W from independent repeated direct measurements Wi = dput(signif(rlnorm(10,4,0.01),3)):{54.6, 54.7, 55.4, 54.3, 54.4, 55.9, 55, 54.2, 55.1, 54.9}.

Characterise constant X from independent repeated direct but imprecise measurements Xi = o='['; c=','; q=']'; n=12; w=0.4;r= rlnorm(n,2.5,0.01); L = signif(r-runif(n,0,w),3); R = signif(r+runif(n,0,w),3); for(i in 1:n) cat(paste(o,L[i],c,R[i],q,c,sep='')):{[12.1,12.4],[12,12.2],[12.1,12.2],[12,12],[12.4,12.4],[12.2,12.5], [12.1,12.4],[12.2,12.4],[12.1,12.3],[12.1,12.2],[12,12.1],[12,12.1]}.

Characterise distribution Y from some random observations Yi = dput(signif(rexp(13,1/4.1),2)):{4.6, 0.67, 0.13, 1.2, 0.36, 2.8, 4.1, 2.8, 0.21, 11, 2.5, 8.7, 3.5}.

Characterise distribution Z ~ exponential(mean=m) from some random observations Zi = dput(signif(rexp(9,1/2.67),2)):{{0.37, 0.14, 0.22, 0.34, 0.07, 0.35, 0.91, 0.23, 1.1}, where m is a constant but otherwise unknown.

Characterise constant m from some random observations Zi ~ exponential(mean=m) = dput(signif(rexp(10,1/10.3),2)):{{3.7, 6, 9.2, 1.9, 9.7, 25, 6.4, 30, 2.7, 1.5}.

Characterise V ~ exponential(mean=m) from some random observations Zi = dput(signif(rexp(15,rnorm(15,1/0.1)),2)):{0.054, 0.032, 0.33, 0.013, 0.041, 0.017, 0.18, 0.019, 0.032, 0.061, 0.099, 0.21, 0.33, 0.26, 0.019}, without assuming that m is a constant.

From coverages

Characterise distribution C1 assuming that 100% of its mass lies within the range [0,10], and at least 50% of its mass lies within the range [1,4].

Characterise distribution C2 assuming that 100% of its mass lies within the range [0,100], at least 75% of its mass is within the range [1,4], and at least 50% of its mass is within [1,2].

Characterise distribution C3 assuming that 100% of its mass lies within the range [0,10], and at least 40% of its mass lies within the range [1,3] and at least 45% of its mass is within [2,4].

Characterise distribution C4 assuming that 100% of its mass lies within the range [0,10], and at least 40% of its mass lies within the range [1,2] and at least 45% of its mass is within [3,4].

From density bounds

Characterise distribution D whose density is bounded below and above by the red and blue curves depicted in this figure:

From a model

Characterise B1 = sqrt(A), where A has been characterised by the unified number {p-box = mmms(min=4, max=100, mean=10, standard deviation=20); Hose structure = consonant(range=[2, 90], core=[5,15])}.

Characterise B2 ~ A - ln A.

Characterise B3 ~ A + B, where B is independent of A and is characterised by the unified number {p-box = exponential(mean=[50,55], Hose structure = consonant(range=[0, 250], core=[100, 200])}.

Characterise B4 ~ A + B, but without making any assumption about the dependence between A and B.

For a variety of related uncertainty representation challenges, see the https://sites.google.com/view/exemplarproblems/exemplar-problems, especially the list of "Links to related challenge problems" near the bottom of the page.