prase comments on Bayesian Flame - Less Wrong

37 Post author: cousin_it 26 July 2009 04:49PM

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Comment author: prase 27 July 2009 03:26:28PM *  0 points [-]

Maybe the difference lies in the format of answers?

  • We know: set of n outputs of a random number generator with normal distribution. Say {3.2, 4.5, 8.1}.
  • We don't know: mean m and variance v.
  • Your proposed answer: m = 5.26, v = 6.44.
  • A Bayesian's answer: a probability distribution P(m) of the mean and another distribution Q(v) of the variance.

How does a frequentist get them? If he hasn't them, what's his confidence in m = 5.26 and v = 6.44? What if the set contains only one number - what is the frequentist's estimate for v? Note that a Bayesian has no problem even if the data set is empty, he only rests with his priors. If the data set is large, Bayesian's answer will inevitably converge at delta-function around the frequentist's estimate, no matter what the priors are.

Comment author: cousin_it 27 July 2009 03:36:43PM *  1 point [-]

http://www.xuru.org/st/DS.asp

50% confidence interval for mean: 4.07 to 6.46, stddev: 2.15 to 4.74

90% confidence interval for mean: 0.98 to 9.55, stddev: 1.46 to 11.20

If there's only one sample, the calculation fails due to division by n-1, so the frequentist says "no answer". The Bayesian says the same if he used the improper prior Cyan mentioned.

Comment author: prase 27 July 2009 03:59:26PM *  0 points [-]

Hm, should I understand it that the frequentist assumes normal distribution of the mean value with peak at the estimated 5.26?

If so, then frequentism = bayes + flat prior.

Improper priors are however quite tricky, they may lead to paradoxes such as the two-envelope paradox.

Comment author: cousin_it 27 July 2009 04:02:42PM *  0 points [-]

The prior for variance that matches the frequentist conclusion isn't flat. And even if it were, a flat prior for variance implies a non-flat prior for standard deviation and vice versa. :-)

Comment author: prase 27 July 2009 04:48:39PM 0 points [-]

Of course, I meant flat distribution of the mean. The variance cannot be negative at least.

Comment author: Cyan 27 July 2009 03:46:27PM 0 points [-]

Using the flat improper prior I was talking about before, when there's only one data point the posterior distribution is improper, so the Bayesian answer is the same as the frequentist's.

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