Netflix Customer-Product Rating Example(Netflix Customer-Product Rating) o 480189-by-17770 customer-product rating matrix X o X is incomplete: 98.82% of values missing However o pairwise comparison graph G=(V, E)is very dense o only 0. 22% edges are missed, almost a complete graph o rank aggregation may be carried out without estimating missing values o imbalanced: number of raters on e ee varies Caveat: we are not trying to solve the Netflix prize problem
Netflix Customer-Product Rating
Netflix example continued Shakespeare in Love The first order statistics mean 4.5 score for each product, is often 另 inadequate because of the following 2.5 most customers would rate 2 40 just a very small portion of the products Dune different products might have different raters Whence mean scores involve noise due to arbitrary individual rating scales(right figure) 40 6o How about high order statistics?
Netflix example continued • The first order statistics, mean score for each product, is often inadequate because of the following: – most customers would rate just a very small portion of the products – different products might have different raters, whence mean scores involve noise due to arbitrary individual rating scales (right figure) How about high order statistics?
From 1st order to 2nd order Pairwise Ranking o Linear Model: average score difference between product i and i over all customers who have rated both of them ∑(Xk-x) #k: Xki, Xk exist] Invariant up to translation o Log-linear Model: when all the scores are positive, the logarithmic average score ratIo, ∑k(ogXk-logX) #k: Xki, Xk exist] Invariant up to a multiplicative constant
From 1st order to 2nd order: Pairwise Ranking
Pairwise ranking Continued o Linear Probability Model: the probability that product is preferred to i in excess of a purely random choice, 你=P{k:>×; Invariant up to monotone transformation o Bradley-Terry Model: logarithmic odd ratio(logit) Prk: Xki> Xki l - log Prk: Xkj< Xkil Invariant up to monotone transformation
Pairwise Ranking Continued
Another view on Pagerank Define pairwise ranking 无法显示该图片 Where p is the pagerank markov matrix Claim: if P is a reversible markov chain i e 无法显示该图片。 Then 囚无法显示该图片
Another View on Pagerank Define pairwise ranking: Where P is the Pagerank Markov matrix. Claim: if P is a reversible Markov chain, i.e. Then