Comment on Less Wrong Polls in Comments

Sun, 30 Sep 2012 15:34:09 GMT

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".

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