[EM] Summable opinion space discovery.

[Kristofer Munsterhjelm]

Dan Bishop wrote:
> Kristofer Munsterhjelm wrote:
>> While trying to find a solution to another problem, I discovered 
>> something that might be used to opinion space from ballots (and the 
>> candidates' position in that space) in a summable manner.
>>
>> Consider a rating- or approval matrix m, where m{voter_1, A} is 
>> voter_1's rating or approval (0 or 1) of candidate A. Say there are v 
>> voters and c candidates. Each candidate can now be assigned a 
>> v-dimensional point according to the ratings by every voter, and we 
>> can calculate ballot similarity between two candidates by determining 
>> the distance (according to some metric) between the two candidates' 
>> assigned points.
>>
>> "m" itself is obviously not summable, because it depends on the number 
>> of voters v. However, we might build a distance matrix q where q{A, B} 
>> is the distance between A and B according to the metric in question. 
>> As long as the metric itself is summable, q will be.
> http://wiki.electorama.com/wiki/Candidate_correlation

Yes, I considered correlation in the other problem of mine as well. The 
problem is that correlation itself is not a metric, since every 
"distance" ranges from 0% to 100% (or in the case of Pearson 
correlation, from -1 to 1), and thus the synthetic coordinate algorithms 
wouldn't work.

>> Perhaps there is a more direct way of building opinion space (by using 
>> SVD or PCA), but I never got that to work. In any case, this is a 
>> rough idea, so don't pick too much on the details :-)
>>
> I've considered an approach like that before. The problem is that 
> knowing a voter chose C>>B>A doesn't tell you whether the spectrum is 
> A------------C-v----------B or A-B----------C------------v, so it's 
> rather difficult to build the opinion space.

One way of getting around that would be to say that if the "C >> B > A" 
vote is the only one in question, then it doesn't matter (you know you 
want C instead of B), and if there are other votes, then the sum 
constraints would favor one rather than the other. For instance, if 
there are others who vote A > C > B and there is just one dimension, 
then it's the former explanation rather than the latter (and those other 
voters are either slightly to the right of A - enough to be closer to A 
than to C - or to the left of A).

However, I see your point. We don't actually have any evidence that in 
any case of ambiguity, the ambiguity doesn't matter. That question would 
depend on the metric, and also for what purpose we're going to use the 
opinion space diagram.