FAQ

<<Anyone have pithy answers?  Feel free to add more questions if appropriate.  When you think you've supplied the full answer to the question, then please make the question black rather than red.  Continue to use red for things that need attention, and blue for things that you invite comments or elaborations on.>>

• What are the differences between a fuzzy number, possibility distribution, possibility structure, and consonant structure?

As we are using these phrases in the F&PIG project, they are essentially synonyms.  The variation has persisted from early indecision and on-going disagreements.  With respect to the unified uncertain numbers which is a primary focus of the project, there are no differences at all.  (In future discussions of "hybrid numbers", it may be useful to make distinctions.)

• Are possibility distributions used to model uncertainty about constants or random variables?

Both!  <<But don't we need some considerations?  For instance, we talk about generic intervals and zero-variance intervals.  The latter are models of constants for which we therefore an upper bound on variance better than a generic interval would suggest.  Are there not similar considerations when using a possibility distribution to model a constant?>> 

• What is the credal set corresponding to a possibility distribution?

All distributions consistent with the nested focal elements defining the possibility distribution.  <<Nice to have pictures of the Medusa of CDFs or PDFs.>>

• What scalars (Dirac delta distributions) are in the credal set corresponding to a possibility distribution?

Only values that are in the core of the possibility distribution. 

• What distributions are excluded by a possibility structure?

The green CDF is in the p-box given by the cumulative possibility and necessity distribution but NOT in the credal set of the possibility distribution itself because, e.g., in the alpha=80%-cut there is 0 probability mass which is less than the required 1-alpha = 20%.

• What does a possibility distribution imply about the moments of the underlying random variable?

So far: The same as if I constructed a p-box from the possibility distribution and queried it for its implied moment bounds.

The mean of a consonant structure can be computed as the interval whose left bound is the mean of the left bounds of the focal elements, and whose right bound is the mean of the right bounds.  It is obviously best possible, and it is equivalent to the interval mean that the corresponding p-box would yield.

Conjecture:  Consonant structures generally give tighter bounds on variance than p-boxes to which they are converted do.  See Moment meditations and Code for computing upper variance of p-box and fuzzy number.  Consider the consonant structure formed by the following nested focal sets:

[1, 3]
[1.1, 2.9]
[1.19999, 2.8]
[1.3, 2.70001]
[1.39999, 2.60001]
[1.5, 2.5]
[1.6, 2.4]
[1.69999, 2.3]
[1.8, 2.20001]
[1.89999, 2.10001]

Because the focal elements are nested, the lower bound on variance is zero.  Marco's code at EXACT upper bound variance  gives 0.3850 as the upper bound.  The mass in each interval focal element doesn't have to be concentrated at a point of course.  It can be distributed across a focal element, which can influence the possible variance.  See also the post at Variance-extreme distributions.>>

(Note that we always compute the population variance, which is the one computed with n in the denominator rather than n-1.  This is appropriate because we have the whole distribution rather than samples.  The distribution is described by focal elements, but those are not samples in the statistical sense.  If they were merely samples, then characterizing the corresponding distribution is a more complex procedure.  Discrepancies and numerical confusion about variance is often traced to confusion between population and sample versions of variance.  You can compute the population variance from the more common sample variance var using (n-1)*var(x)/n.)

Another conjecture:  When moments are obtained from the edges of a p-box (as opposed to being implied by ancillary information such as moments carried in calculations that created the p-box), converting a p-box to a consonant structure [fuzzy number], moment information is retained by the transformation.  

This conjecture seems to be false, at least with our current ability to compute relevant moments.  Both transformations between consonant structures and traditional p-boxes lose information about the underlying moments.  Here is a numerical example:

a = makeconsonantfrommoments(0,100,20,1)   # mean = 20,                             variance = 1
b = makepbox.consonant(a)
moments.consonant(a)                                                      # mean = [18.01, 22.29],    variance = [0, 44.28]
b                                                                                                       # mean = [18.01, 22.29],    variance = [0, 44.28]

c = mmms(0,100,20,1)                                                        # mean = 20,                            variance = 1
d = makeconsonant.pbox(c)
forgetmoments(c)                                                                 # mean = [18.30, 22.30],    variance = [0, 76]
moments.consonant(d)                                                      # mean = [17.44, 22.68],   variance = [0, 30.92]

This conclusion needs to be reviewed after the new moment algorithms are introduced and the Chebyshev consonant structure is revised not to depend on the inequality with the absolute value.

• What does a possibility distribution imply about the p-box for the random variable?

We can compute a cumulative possibility and necessity distribution.

Cum. Pos. Fun. is given by cumulative maximum of membership function.Cum. Nec. Fun. is given by 1-complementary cumulative maximum of membership function.

• What does a p-box imply about the possibility distribution?

• What do the first couple of moments imply about the possibility distribution?

Not much, but you can use the Markov and Chebychev inequality to construct some crude possibility distributions.  See the Dubois et al. paper https://www.academia.edu/download/51452062/Probability-Possibility_Transformations_20170120-10435-598qmf.pdf.  See also Moments.

• If A and B are such possibility distributions / consonant confidence structures, how can we compute a confidence structure for A + B?

  • What dependence assumption (if any) was used?

  • How might such an assumption be changed or relaxed?

See Hose and Hanss, 2019 (MSSP) and Gray et al., 2021 (ISIPTA)

• How do other operations besides work with possibility distributions?

It's all about knowing a joint possibility distribution! Then, use the implicit variant of the Extension Principle. *poof*

• How does an inequality work with possibility distributions?

You still look at the joint. Subtract one from the other and evaluate, e.g., Pos( X-Y < 0) and Nec( X-Y < 0) to evaluate X < Y.

• Is a merged number just the tuple of a p-box and a possibility distribution on the same quantity?

Preliminary group consensus: YES

• Is the credal set from a merged number simply the intersection of the credal sets from the p-box and the possibility distribution?

Preliminary group consensus: YES

• Does arithmetic with merged numbers produce more informative results than either p-boxes or possibility distributions?

Surely.

• Can we improve this by doing arithmetic with part p-box and part possibility distribution?

Need to investigate. Perhaps not very cost-efficient.

• How are merged numbers represented in the computer? 

• How should merged numbers be displayed?

• Do merged numbers obviate Kaufmann's hybrid numbers or some kinds of polymorphically uncertain structures?

• How does the Hose-Hanss interpretation of possibility distributions differ from that of Kaufmann & Gupta, from that of Dubois?  How do they differ from fuzzy numbers?

• Do the Hose / Troffaes  / Jost / Fisher operations resolve neatly as dependence assumptions?

• Can you sample from a possibility distribution (in the sense similar to how you can sample from a random variable's distribution)?  If so, what would a sample look like? How can you use this for solving the extension principle?

Like how a set is a random sample of a p-box,  a random alpha-cut can be considered a "sample" of a possibility distribution. This follows from the random-set interpretaton of possibility thoery.
Advantages: rigorous
Disadvantages: repeated variables, code-intrusive.

A different way is "interpolating" by sampling (uniform, lhs, grid, etc) on the x-axis and "carrying along" the membership values.
(Dis-)Advantagas opposite of previous.

• Do we think the level-wise operations suggested by Kaufmann and Gupta are not useful for anything?

In short: No

But: 

It corresponds to solving Zadeh's origional extension principle, so for classical possibility theory, it's ok. For imprecise probabilities more care must be taken. A possibility distribution can be correctly propagated through a univariate operation this way. For multivariate functions, extra steps are required depending on dependence assumptions (a rescaling of alpha-levels).

• Can you estimate a possibility distribution from empirical information?  If so, what information is usable to estimate a possibility distribution?

• Must possibility distributions from empirical data, or is their calculus independent of evidence?

• Given random sample data, how can we construct a consonant confidence structure for

  • the mean, standard deviation, and distribution of observable values assuming the distribution is stationary and normal,

  • the binomial rate assuming the distribution is a stationary Bernoulli, and

  • random sample data from stationary but unknown distribution (the nonparametric case)?

• Can these structures always be expressed as possibility distributions?

• Is there a role for t-norms or t-conorms in the arithmetic for possibility distributions?

It probably depends on what you're using possibility distributions for. For imprecise probabilities it's likely that only t-norms which are copulas make sense, those which have a probabilistic interpretation. T-norms and t-conorms have been studied for many years in fuzzy arithmetic, but when they are used in binary operators don't seem to be be consistent with probability theory (see Williamson https://www.sciencedirect.com/science/article/abs/pii/016501149190157L ). Copulas however may play an important role in creating dependent multivariate possibility distributions from their lower dimensional marginals (with a random set dependence interpretation), and thus may be used in a dependent possibilistic arithmetic in a similar way to copulas in p-box arithmetic (see Gray et al, ISIPTA 2021).