We talk a lot about making decisions based on data but we need to be careful about how hard and fast those decisions are. Our decisions are only as good as our data and our analysis. Data is always a sample of the full scope of reality and analytics is always an interpretation of that sample. We need to be cognizant of the differences between Opinions, Facts and Conclusions. And, just as important, we need to recognize the relationship between our judgement and our ego: all disagreements are personal to some degree.
In this paper we describe the concept of physical impossibility as an alternative to the specification of fault models. These axioms can be used to exclude impossible diagnoses similar to fault models. We show for Horn clause theories while the complexity of finding a first diagnosis is worst-case exponential for fault models, it is polynomial for physical impossibility axioms. Even for the case of finding all diagnoses using physical impossibility axioms instead of fault models is more efficient, although both are exponential in the worst case. These results are used for a polynomial diagnosis and measurement strategy which finds a final sufficient diagnosis.
The Myerson-Satterthwaite theorem is a foundational impossibility result in mechanism design which states that no mechanism can be Bayes-Nash incentive compatible, individually rational, and not run a deficit. It holds universally for priors that are continuous, gapless, and overlapping. Using automated mechanism design, we investigate how often the impossibility occurs over discrete valuation domains. While the impossibility appears to hold generally for settings with large numbers of possible valuations (approaching the continuous case), domains with realistic valuation structure circumvent the impossibility with surprising frequency. Even if the impossibility applies, the amount of subsidy required to achieve individual rationality and incentive compatibility is relatively small, even over large unstructured domains.
Although the study of clustering is centered around an intuitively compelling goal, it has been very difficult to develop a unified framework for reasoning about it at a technical level, and profoundly diverseapproaches to clustering abound in the research community. Here we suggest a formal perspective on the difficulty in finding such a unification, in the form of an impossibility theorem: fora set of three simple properties, we show that there is no clustering function satisfying all three.
Arrow's Impossibility Theorem is one of the landmark results in social choice theory. Over the years since the theorem was proved in 1950, quite a few alternative proofs have been put forward. In this paper, we propose yet another alternative proof of the theorem. The basic idea is to use induction to reduce the theorem to the base case with 3 alternatives and 2 agents and then use computers to verify the base case. This turns out to be an effective approach for proving other impossibility theorems such as Sen's and Muller-Satterthwaite's theorems as well. Furthermore, we believe this new proof opens an exciting prospect of using computers to discover similar impossibility or even possibility results.