I think the quickest answer is: the formula computes the test statistic as a difference of log values, rather than log of ratio of values. By not normalizing, the entropy is multiplied by a factor (sum of the counts) vs normalized. So you do end up with a statistic N times larger when counts are N times larger.
On Thu, Jun 20, 2013 at 9:52 AM, Dan Filimon <[email protected]> wrote: > My understanding: > > Yes, the log-likelihood ratio (-2 log lambda) follows a chi-squared > distribution with 1 degree of freedom in the 2x2 table case. > A ~A > B > ~B > > We're testing to see if p(A | B) = p(A | ~B). That's the null hypothesis. I > compute the LLR. The larger that is, the more unlikely the null hypothesis > is to be true. > I can then look at a table with df=1. And I'd get p, the probability of > seeing that result or something worse (the upper tail). > So, the probability of them being similar is 1 - p (which is exactly the > CDF for that value of X). > > Now, my question is: in the contingency table case, why would I normalize? > It's a ratio already, isn't it? > > > On Thu, Jun 20, 2013 at 11:03 AM, Sean Owen <[email protected]> wrote: > >> someone can check my facts here, but the log-likelihood ratio follows >> a chi-square distribution. You can figure an actual probability from >> that in the usual way, from its CDF. You would need to tweak the code >> you see in the project to compute an actual LLR by normalizing the >> input. >> >> You could use 1-p then as a similarity metric. >> >> This also isn't how the test statistic is turned into a similarity >> metric in the project now. But 1-p sounds nicer. Maybe the historical >> reason was speed, or, ignorance. >> >> On Thu, Jun 20, 2013 at 8:53 AM, Dan Filimon >> <[email protected]> wrote: >> > When computing item-item similarity using the log-likelihood similarity >> > [1], can I simply apply a sigmoid do the resulting values to get the >> > probability that two items are similar? >> > >> > Is there any other processing I need to do? >> > >> > Thanks! >> > >> > [1] http://tdunning.blogspot.ro/2008/03/surprise-and-coincidence.html >>
