Some disagree with Laplace's socalled "Rule of succession".
See : http://en.wikipedia.org/wiki/Rule_of_succession
Not sure if this is what you wanted. But some say he is " getting away with it " (or was).
In fact it is said that he gets away with it because he did other good math.
Some people use his rule and others sometimes argue it should not be used in THAT CASE for which he just did.
As for " when to use the rule " there is also no consensus. Although the extreme opponents never use it ofcourse.
I came up with this so fast because my mentor is an opponent of this.
To justify his opinion somewhat here is one of his more or less formal arguments :
Suppose a 6sided dice gets an equal amount of 1,2,3,4,5,6 after 60 spins.
So 10 times 1. 10 times 2. etc.
1) By the rule of succession the probability for (1 or 2) is 21/62.
2) By the rule of succession for the probability of (3 or 4) we have 21/62.
3) By the rule of succession for the probability of (5 or 6) we have 21/62.
So the total probability for (1 or 2 or 3 or 4 or 5 or 6 ) is 21/62 + 21/62 + 21/62 = 63/62.
63/62 ?? That is not 1.
Of course one can argue that the addition for independant probabilities A and B is no longer A + B. But that is just silly.
Notice that if we used to sum all the probabilities for each value of 1,2,... then we got 66/62 ; yet another number !
However that means we used Probability(1 or 2) = Probability(1) + Probability(2) = 11/62 + 11/62 = 22/62.
That is inconsistant with the successes of (1 or 2) +1 !
Another often defense is that the Rule of succession ONLY applies to TWO possible outcomes.
However 2 is a divisor of 6.
In other words the researcher could not know that the " success " is composed of little " successes ".
So if succes is defined as (1,2,3) and failure as (4,5,6) then it seems logical to assume that TWO possible outcomes are actually collections of Higher possible outcomes.
SO the rule P(A+B) = P(A)+P(B) should apply. Thus follows from S = succes is defined as
S = s1+s2+s3.
In other words the successes and their probabilities follow the rule S = s1 + s2 + s3 <=> P(A+B+C)= P(A)+P(B)+P(C) seems almost trivially true.
Resume : If a success depends on other success factors ( as often in statistics ! ) then using the rule of succession is disagreeing with P(A+B) = P(A)+P(B) for independant probabilities !!
Saying the rule of succession only works for binary possibilities is thus only consistant if you say " only P(A+B) exists " and thus not P(A) or P(B) ... Unless P(B) = 1 - P(A).
The rule of succession has thus issues with the rule of addition for independant probabilities , the possibility of subsuccesses or successfactors , the rule of total probability = 1.
The logic is quite funny :
suppose we have a coin
we get 10 heads and 10 tails.
now succession gives 11/22. Whereas ordinary gives 10/20 or 1/2.
They are equal but 11/22 just looks silly.
Suppose we get 5 heads and 15 tails.
heads : 6/22 , tails : 16/22
and then someone mentions the probability of heads is 1/4. And then he justifies this 6/22 by saying that if we take more coin tosses with prob 1/4 for heads our 6/22 finally becomes 1/4 in the limit.
LOL so you have to admit 6/22 is wrong to finally arrive at a justification for it ??
Did we admit 6/22 is wrong ? Yes because you assumed 1/4 ! AND your limits gives 1/4.
Why did you not take 1/4 then from the beginning ???
I call the rule of succession : " Orwellian statistics "
For those who are stubborn I continue
Let P mean probability and S succes.
a dice has 6 sides.
but we have P(1) and P(NOT 1).
P(NOT 1) = P(2)+P(3)+P(4)+P(5)+P(6) and/or S(NOT 1) = S(2)+S(3)+S(4)+S(5)+S(6).
( P and S are independant(!) probabilities and successes ! )
If you agree with any of those 2 you get the paradoxes from above.
If you disagree with both , you will have a hard time doing statistics !
One more time, a dice has 6 possibilities.
but 1 or not 1 is a binary choice.
So in fact the rule of succession for binary possibilities does apply afterall !
similarly "2" or "not 2" is a binary choice. Notice "not 2" CONTAINS 1. So we need
P(1) = 11/62. P(NOT 1) = 51/62
P(NOT 1) = 51/62 = P(2)+P(3)+P(4)+P(5)+P(6)
Now by symmetry we know P(2)=(P(2)+P(3)+P(4)+P(5)+P(6))/5 = 51/310.
But also by symmetry P(1) = P(2). But 51/310 =/= 11/62 !
So the rule of succession is not consistant with
1) the rule of addition for independant probabilities
2) the possibility of subsuccesses or successfactors
3) the rule of total probability = 1.
4) symmetry for equal probabilities.
Note the quote is the way I remember it. Not an actual quote. I believe it was posted on sci.math. see also http://en.wikipedia.org/wiki/Orwellian