Talk:Rule of succession

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[edit] the correct formula of Euler integral

Michael, the formula

\int_0^1 p^s(1-p)^{n-s}\,dp={(s+1)!(n-s+1)! \over (n+2)!}

should be corrected as

\int_0^1 p^s(1-p)^{n-s}\,dp={(s)!(n-s)! \over (n+1)!}

May I correct this for you? -- Jung dalglish 2 July 2005 08:35 (UTC)


Go ahead -- I'll check it closely later. Michael Hardy 3 July 2005 00:17 (UTC)

I changed it. - April 2006, after some idiot had reverted Jung's correction

[edit] Notation

I'm studying this stuff in school and I didn't know the Cartesian Product symbol... Couldn't we put a side-bar in that lists the mathematical notation used and link it to the "Table of Mathematical Symbols" or something along those lines?

I think it would be helpful for math pages in general. Many other articles for different subjects have sidebars like that.

[edit] Why does Laplace use the expected value (mean) of the posterior?

Why does Laplace use the expected value (mean) of the posterior? This is not obvious to me, but it's crucial. Could someone who knows add something to the entry about the justification for this? Alex Holcombe 11:13, 14 October 2007 (UTC)

When the parameter whose posterior probability distribution we're finding is itself a probability, then its expected value is the marginal probability, conditional on current data, of the event in question---in this case, tomorrow's sunrise. That is the content of the law of total probability. Michael Hardy 16:06, 14 October 2007 (UTC)
I think I'm interested in a different kind of answer. As I understand it, the expected value minimizes the *average* difference between the estimate and the actual unknown parameter value. But in other contexts, Bayesians may minimize the average sum of squared difference, and in maximum likelihood the mode of the posterior distribution is used. Did Laplace have any particular reason for choosing the mean? Is this somehow embedded in the law of total probability? Sorry if this is a naive question. Alex Holcombe 10:23, 15 October 2007 (UTC)
No, it does not. The expected value does not minimize that average distance. The median does. The expeected values minimizes the average square of the distance. Besides, it's not clear that estimation is what's being done here. Michael Hardy (talk) 19:07, 7 January 2008 (UTC)

[edit] The probability that the sun will rise tomorrow

I want to drop that part of the article. It is already contained in the article about the sunrise problem, and it only furthers misunderstandings of the rule of succession - including the reference to "Bayesian philosophy".

The rule of succession is a mathematical theorem. Problems arise only when the conditions are ignored under which the theorem holds or in some cases, if certain issues should be treated as probabilities. -- Zz (talk) 13:50, 7 January 2008 (UTC)

Reverted.
Saying "the rule of succession is a theorem" and, therefore, that "the article should contain nothing but the theorem" is a cop-out. The rule of succession is a calculation. The important question is whether (or when) the calculation is appropriate.
It is useful to include the sunrise question here, because (i) it helps make the discussion much more concrete; (ii) it is the original connection in which Laplace presented the calculation.
Of course, the rule of succession has many other applications -- eg in symbol probability estimates for information coding, and/or more widely in probabilistic modelling on the back of rather limited quantities of data.
But the general principle, and the general controversy (or at least debatability of the appropriateness of such estimates/calculations) is well reflected in the sunrise problem.
If there is a separate article on the sunrise problem, then arguably it ought to be turned into a redirect here. Jheald (talk) 16:46, 15 January 2008 (UTC)
I wrote here. No objection came. I changed the article accordingly. You revert without waiting for the disccussion of your objection. That is not the best style.
The rule of succession is a theorem. It is a mathematical fact. It can be proved. Laplace did not grab it from thin air. As every theorem, it holds only if the conditions are fulfilled. For a reasonable discussion of the objections raised, see for instance Jaynes' book of probability theory. The examples there are illuminating.
As for the the "sunrise problem", there is no deeper connection other than it was an example, and a famously misleading one as that. Laplace saw no sunrise problem in the first place. Instead, he gave an example with a twofold purpose: 1. if it were probability, you would calculate it this way, 2. seeing the causal connection is much more rewarding than using the approach of probabnility. And most people missed the point. Anyway, if you think that the articles should be one, propose to merge them. -- Zz (talk) 13:31, 18 January 2008 (UTC)