See below for minutes from some previous meetings.
The next meeting will be on 7 February 2024.
Meetings are held via Zoom and are open to everyone
https://liverpool-ac-uk.zoom.us/j/7555947629 Ander's Zoom
https://liverpool-ac-uk.zoom.us/j/92826249395?pwd=Y3lhUmpGVmlicWhoUVpMT2FzdFJldz09 Old Zoom
Alternate Wednesdays
18:00 EEST (Helskinki, Sophia, Athens)
17:00 CEST (Stockholm, Berlin, Paris)
16:00 BST (London)
12:00 BRST (Rio de Janeiro)
11:00 EST (New York)
09:00 MDT (Albuquerque, El Paso)
08:00 PDT (Los Angeles)
hosted by Ander Gray (Ander.Gray@liverpool.ac.uk)
Everyone is welcome to attend
Minutes
10 January 2023
Airport book. Peter Hristov and colleagues is planning an "airport book" on uncertainty, that is a simplified and very high-level introduction targeting business people who pick up something interesting to read on long flights. It would be nice to have some text or examples ready to contribute to their effort if we want. It might also be useful to create an airport book for our own outreach efforts.
Hybrid versus union. We're distinguishing the unified numbers that combine probabilistic, constraint-analytic and possibilistic methods to whittle down credal sets using complementary approaches from hybrid numbers that combine probabilistic with fuzzy information but assiduously keep the two dimensions separate like the real and imaginary parts of complex numbers. We're first focusing on the unified numbers in F&PIG, but we note that there's a lot of literature on hybrid numbers from Moller, Beer, Kreinovich and others.
Resources to be developed. Our working drafts at present consist of these Google pages, Ander's Julia code, Scott's R code, Marco's algorithms for computing variance of fuzzy structures, and the draft paper on the unified numbers. Marco's code is in Python, as it much of the interval statistics algorithms, so it probably makes sense to develop a parallel Python implementation of the unified numbers. We would like to compare the results of the various implementations quantitatively as mutual checks.
What needs to be done. The R code has not yet implemented the independent operations, ('sigma convolutions') or transformations for non-monotone unary operations (abs, cos, etc.), and its algorithm to convert Dempster-Shafer structures to fuzzy possibilistic structures is completely wrong. Ander feels the Julia code for this conversion may also have problems. It seems that both codes still need to develop methods and algorithms that allow cross strengthening of moments and the p-box edges and the fuzzy membership function and a function that can 'forget' consonant part of the unified number which will be essential in testing and learning about the arithmetic of unified numbers. It would also be useful to have agreement on conventions for visually displaying unified numbers, how stochastic mixtures would work and how the unified numbers should handle various aggregation operations.
Criteria to define the DSS to FN constructor. There are various criteria on which to compare uncertain numbers. For p-boxes, for instance, the breadth (area between the two edges) might be useful to argue that one p-box is better (tighter) than another. But this is not a perfect criterion. For example, if a box has much smaller tails, an analyst might prefer it even if it has a slightly larger breadth. There does not yet seem to be agreement about what criterion or criteria can be most useful to compare fuzzy/possibilistic structure. Unless credal sets are nested, it may not be sensible to talk about one credal set being smaller than another.
Neither uniqueness nor optimality. It appears that the construction of fuzzy/possibilistic structures from arbitrary Dempster-Shafer structures on the real line is not actually 'optimal' but could theoretically admit multiple solutions. This may represent flexibility that could be useful in responding to real-world challenges, but it is also concerning to will surely be troubling to reviewers
For next time. Ander will try to tighten the Julia code, particularly the conversion from DSSs to fuzzy. Marco will undertake to integrate the code(s) for computing variance.
Tom will explore whether and how the general Chebyshev inequality (without the absolute value function) can be integrated into the algorithms that construct fuzzy/possibilistic structures from moment information. Scott will develop five or so exemplar problems along the lines of the TNO RPrepro reliability analysis challenge problems that could be useful as benchmark problems illustrating calculations with the unifed numbers.
Additional notes from the Zoom meetings on 7 & 14 August 2020
***Information that can be extracted from possibility distributions
bounds on expectation
core (singleton point masses)
support
lower bound (1-alpha) on an alpha-cut
upper bound [0,alpha] on double tail risks of the sublevel set at alpha {X: pi(X) <= alpha} = complement of alpha cut
conservative bounds on p-box, the bounding distributions of which are in the credal set
***How to specify possibility distributions
possibility density
mode
range
***Information from p-boxes
bounds on quantiles
bounds on tail risks
core
support
bounds on moments
family
crappy bounds on internal risks, Pr(a<X<b)
***How to specify p-box
range
moments
constraints on quantiles or (exceedance) probabilities
shape (family, unimodality, symmetry, positivity, etc.)
mode
bounds on probability densities
parametric distribution families with interval parameters
data (fitting, predictive distributions from c-boxes)
***Extra
inference with c-boxes
***What p-boxes cannot do
reveal tight bounds on the probability of internal risks Pr(a < X < b)
retain mode and modality information