Re: how to interpret a conditional probability

Takashi Yamauchi wrote:
"David Jones" <dajxxx@xxxxxxxxx> wrote in message
news:44fc55d4$1@xxxxxxxxxxxxxxxxx
Takashi Yamauchi wrote:
I wonder if you could help me and point me to somewhere.

this is a simple question (I think), but not for me.

Say, you estimated a conditional probability

P(theta_1 | x) = 0.54,

THETA = {theta_1, theta_2}, wehre P(theta_1)=P(theta_2)=0.5.

So, Theta is dichotomous (yes, no), x is data/observation, and the
prior

probabilities are 0.5. This conditional probability means that
after

observation x, the probability of P(theta_1 | x) improved only
slightly

(compared to the prior P(theta_1) = 0.5).

How do you evaluate this conditional probability? Any index to say

something statistically? For example, you can caculate the
posterior
odds for theta_1;

0.54/(1-0.54) or take the logorithm of the odds. Then, how do you
evaluate

these statistically? Is there any way to evluate how much 0.54 is

informative (statistically)? where should I look?

Thank you

takashi

One approach would be to extend the discussion to a fully defined
decision problem. The "optimal" decision depends on the probability
distribution, as does the expected cost. Thus you could measure how
useful the observation x is, by comparing the expected costs before
and after the observation. This is on the basis that "money" is
easier to understand/compare than probabilities or odds.

David Jones


Thank you. Can I use the Bayes factor? such that
P(x | theta_1) / P(x | theta_2) = Bayes factor

If so, how do I interpret the Bayes factor? Do you know of a good
paper/book that discusses the interpretation of Bayes factor?

takashi yamauchi

I have two thoughts on this question: (i) it has nothing to do with
your original question; (ii) it has everything to do with your
original question. This may be because you need to decide exactly what
you question is, but also, quite a lot, because of my limited
familiarity with all this.

A suggestion for a good source of information about (fairly current)
Bayesian ideas is the book:

Bernardo JM and Smith AFM (2000) Bayesian Theory. Wiley, Chichester

David Jones


.

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