You can now write Less Wrong comments that contain polls! John Simon picked up and finished some code I had written back in 2010 but never finished, and our admins Wesley Moore and Matt Fallshaw have deployed it. You can use it right now, so let's give it some testing here in this thread.

The polls work through the existing Markdown comment formatting, similar to the syntax used for links. Full documentation is in the wiki; the short version is that you can write comments like this:

What is your favorite color? [poll]{Red}{Green}{Blue}{Other}

How long has it been your favorite color, in years? [poll:number]

Red is a nice color [poll:Agree....Disagree]

Will your favorite color change? [poll:probability]

To see the results of the poll, you have to vote (you can leave questions blank if you want). The results include a link to the raw poll data, including the usernames of people who submitted votes with the "Vote anonymously" box unchecked. After you submit the comment, if you go back and edit your comment all those poll tags will have turned into [pollid:123]. You can edit the rest of the comment without resetting the poll, but you can't change the options.

It works right now, but it's also new and could be buggy. Let's give it some testing; what have you always wanted to know about Less Wrongers?

Your question is confused. The uniform distribution hypothesis only requires that the (assumed infinite) population picks the answers independently with equal probability. Under this hypothesis, the observed poll answers (for a fixed number of respondents) will follow a multinomial distribution with parameters (0.2, 0.2, 0.2, 0.2, 0.2). A typical realization will not have an equal number of respondents giving each answer, although asymptotically the empirical frequencies will converge to equality.

Anyways, as a Bayesian, the better question is what should my posterior belief about the response probabilities be after running the poll and updating off the answers? The canonical way to do this would be to put a Dirichlet prior over the response probabilities. By the miracle of conjugacy, your posterior distribution will itself by a (generally different) Dirichlet distribution.

By taking the expectation of indicator variables like I{"probability of First Answer under 0.2"} under the posterior, you can figure out what degree of belief you must give to statements like "respondents have an aversion toward First Answer".

That makes sense - I had imagined doing something similar, but I had never heard of Dirichlet priors.