Pavitra comments on Taking Ideas Seriously - Less Wrong

51 Post author: Will_Newsome 13 August 2010 04:50PM

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Comment author: Pavitra 29 August 2010 09:26:24PM *  0 points [-]

I'm not convinced that there's a meaningful difference between prior distributions and prior probabilities.

Going back to the beans problem, we have this:

50% mixed bag.

  • 50% draw white bean

  • 50% draw black bean

50% unmixed bag.

  • 100% draw white bean

This can easily be "flattened" into a single, more complex, probability distribution:

25% draw white bean from mixed bag.

25% draw black bean from mixed bag.

50% draw white bean from unmixed bag.

If we wish to consider multiple draws, we can again flatten the total event into a single distribution:

1/8 mixed bag, black and black

1/8 mixed bag, black and white

1/8 mixed bag, white and black

1/8 mixed bag, white and white

1/2 unmixed bag, white and white

Translating the "what is that number" question into this situation, we can ask: what do we mean when we say that we are 5/8 sure that we will draw two white beans? I would say that it is a confidence; the "event" that has 5/8 probability is a partial event, a lossy description of the total event.

Comment author: Perplexed 29 August 2010 09:47:17PM *  3 points [-]

I'm not convinced that there's a meaningful difference between prior distributions and prior probabilities.

There isn't when you have only two competing hypotheses. Add a third hypothesis and you really do have to work with distributions. Chapter 4 of Jaynes explains this wonderfully. It is a long chapter, but fully worth the effort.

But the issue is also nicely captured by your own analysis. As you show, any possible linear combination of the two hypotheses can be characterized by a single parameter, which is itself the probability that the next ball will be white. But when you have three hypotheses, you have two degrees of freedom. A single probability number no longer captures all there is to be said about what you know.

Comment author: Pavitra 29 August 2010 09:50:27PM 0 points [-]

In retrospect, it's obvious that "probability" should refer to a real scalar on the interval [0,1].

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