# Re: Programming is the Engineering Discipline of the Science that is Mathematics

Date: 9 Jun 2006 18:04:47 -0700

Message-ID: <1149901487.382879.226140_at_m38g2000cwc.googlegroups.com>

Keith H Duggar wrote:

*> vc wrote:
**> > Erwin wrote:
*

> > > > This is why PT is a /generalization/ of logic. It

*> > > > reduces to logic when applied to truth-valued
**> > > > statements. Just as gamma reduces to factorial for
**> > > > natural arguments. (Again no quibbles about offset by
**> > > > 1 etc).
**> > > >
**> > >
**> > > You mean like :
**> > >
**> > > AND (p1, p2) === p1*p2
**> >
**> > P(p1 and p2) is not equal P(p1)*P(p1) in general, so no
**> > such 'generalization' is possible.
**>
**> Wow! vc is going off the VI deep end at the moment. "no such
**> 'generalization' is possible"? Saying that PT is not a
**> generalization is one thing; but, none possible??
**>
*

Recall that the OP claimed that PT is a logic 'generalization' in the sense that 'probability depends only on the constituent probabilities'. He failed to prove the bizzare assertion and refused/or was unable to solve the trivial puzzle that disproves his statement.

> Erwin, what vc was referring to is that

*>
**> P(AB) = P(A|B)P(B) -or-
**> P(AB) = P(B|A)P(A)
**>
*

Note, that I said nothing about conditional probabilities. I merely requested to compute the P(A and B) probability in terms of P(A) and P(B) which was promised by the OP (see above).

> where | means given and AB is short for "A and B". This is

*> called the product rule. Something that vc seems not to know
**> (given his questions in the other post) is that in the limit
**> of true (0) and false (1) the conditional probability
**> product rule reduces to the logical conjunction truth
**> table. Here is the proof
**>
**> g : P(A) = 0
**> p : P(AB) = P(B|A)P(A)
**> u : P(AB) = 0
*

Unfortunately, it's no proof but just mindless playing with formulas. The conditional probability is *defined* as

P(B|A) def P(A and B)/P(A)

the requirement for such definition being that P(A) <>0, naturally. The definition can be found in any introductory PT textbook.

*>
**> g : P(B) = 0
*

> p : P(AB) = P(A|B)P(B)

*> u : P(AB) = 0
**>
*

See above.

*> g : P(A) = 1
*

> g : P(B) = 1

*> s : P(~B) = 0
**> m : P(A) = P(AB) + P(A~B)
**> p : P(A) = P(AB) + P(A|~B)P(~B)
**> u : P(A) = P(AB)
**> c : P(AB) = P(A)
**> u : P(AB) = 1
*

This is even funnier. First, we do not know what P(A|B) is (see above) and second the question is what kind of events might A and B be if the probability of either is one ? What about P(A or B) given the respective probabilities are one ? Is it two by any chance ? (I asked the same question in another message).

*>
**> thus
**>
*

> P(A) : P(B) : P(AB)

*> 0 : 0 : 0
**> 0 : 1 : 0
**> 1 : 0 : 0
**> 1 : 1 : 1
**>
*

Unfortunately, there can be no 'thus'.

> descriptions

*> g : given
**> p : product rule
**> s : sum rule
**> m : marginalization (derived from sum rule)
**> u : substitution
**>
**> -- Keith --
*

Received on Sat Jun 10 2006 - 03:04:47 CEST