Original article (link) posted: 24/09/2005
Gilboa, Lieberman and Schmeidler "Empirical Similarity"
Although the above paper was supposed to be talked about, Professor Gilboa mainly explained the following paper.
Billot, Gilboa, Samet and Schmeidler (2005) "Probabilities as similarity-weighted frequencies" Econometrica, 73
The above two papers consider a decision rule when a decision maker who has data on past outcomes is asked to express her beliefs by assigning probabilities to certain possible states. As the original database becomes large, empirical frequency may not help for her to make a decision at all. Instead, she may assign a higher weight to more similar case in evaluating the probability of a state.
Billot et.al show that if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case. However, their axiomatization does not suggest a particular similarity function, or even a particular form of the function. Gilboa et.al develop tools of statistical inference for parametric estimation of the similarity function, assuming that such a function governs the data generating process.
Notice that the axioms in the papers cannot be consistent with the situation where the range of belief becomes smaller as the number of observation increases or a decision maker cares about a trend of outcomes.
Presentation by Professor Gilboa was very clear and he was quite good at using power point (!!). But it was difficult to understand the material. I might need to study decision theory at least little bit... (It might be good to read his book "A Theory of Case-Based Decisions". The Japanese translated edition is also available.)