13,331 research outputs found
Probabilistic Models over Ordered Partitions with Application in Learning to Rank
This paper addresses the general problem of modelling and learning rank data
with ties. We propose a probabilistic generative model, that models the process
as permutations over partitions. This results in super-exponential
combinatorial state space with unknown numbers of partitions and unknown
ordering among them. We approach the problem from the discrete choice theory,
where subsets are chosen in a stagewise manner, reducing the state space per
each stage significantly. Further, we show that with suitable parameterisation,
we can still learn the models in linear time. We evaluate the proposed models
on the problem of learning to rank with the data from the recently held Yahoo!
challenge, and demonstrate that the models are competitive against well-known
rivals.Comment: 19 pages, 2 figure
Mathematical Economics: A Reader
This paper is modeled as a hypothetical first lecture in a graduate Microeconomics or Mathematical Economics Course. We start with a detailed scrutiny of the notion of a utility function to motivate and describe the common patterns across Mathematical concepts and results that are used by economists. In the process we arrive at a classification of mathematical terms which is used to state mathematical results in economics. The usefulness of the classification scheme is illustrated with the help of a discussion of fixed-point theorems and Arrow's impossibility theorem. Several appendices provide a step-wise description of some mathematical concepts often used by economists and a few useful results in microeconomics.Mathematics, Set theory, Utility function, Arrow's impossibility theorem
A graph interpretation of the least squares ranking method
The paper aims at analyzing the least squares ranking method for generalized
tournaments with possible missing and multiple paired comparisons. The
bilateral relationships may reflect the outcomes of a sport competition,
product comparisons, or evaluation of political candidates and policies. It is
shown that the rating vector can be obtained as a limit point of an iterative
process based on the scores in almost all cases. The calculation is interpreted
on an undirected graph with loops attached to some nodes, revealing that the
procedure takes into account not only the given object's results but also the
strength of objects compared with it. We explore the connection between this
method and another procedure defined for ranking the nodes in a digraph, the
positional power measure. The decomposition of the least squares solution
offers a number of ways to modify the method
- âŚ