Re: [Phys-l] What's wrong with P(x|y)? (was Re: Symbol for uncertainty)

[John Denker <jsd@av8n.com>]

On 05/11/2011 02:54 AM, Brian Blais wrote:
> I totally agree with the random variable business being horrible,

:-)

>  but
> what do you find problematic with conditional probability notation
> P(x|y)?  Is there an alternative that is clearer?  Other than
> students finding conditional probabilities unintuitive, I've never
> heard a big criticism of it or the notation.

Consider the following contrast:

A)	sqrt is a function
	If x=4 then sqrt(x) is 2.
	So far, so good.


B)	P(x|y) must be a function of x and y.
	Similarly P(y|x) must be some /other/ function of x and y.
	How do we interpret P(4|9)?
	Is that x=4 and y=9, or the other way around?

	This is fixable by writing P(x=4|y=9) as distinct from
	P(y=4|x=9) ... but this is a rather dramatic departure
	from the usual notation for a function that we all
	learned in grade school.  It's an abuse of the notation.
	It uses the name of the variable to tell us the shape
	of the function.
	If you were writing a computer program, you would have
	to write two different functions with different names,
	e.g. X_given_Y(4, 9) and Y_given_X(4, 9).  The code to
	implement one function might well be wildly different
	from the code to implement the other.

A')	It must be emphasized that the sqrt function does not
	suffer from any such problem.  You can draw a graph of 
	the sqrt function without knowing the name of the variable.
	You can write the code to implement sqrt without knowing
	the name of the variable(s) to which the function will
	be applied.

========

Quagmires of this sort are not unique to probability theory.
Something similar is a big problem in thermodynamics.

A)	E is a function of state
	T is a function of state
	P is a function of state
	So far, so good.

B	You may be tempted to write
	E = E(T)		for a heat capacity experiment
	E = E(P)		for a compression experiment
	E = E(4)		is this P=4 or T=4?
	Big mess.

	This is fixable by writing E(T=4) or E(P=4).

	In a computer program you could wrote some code for 
	E(state) and then write some more code to find the state 
	as a function of P and yet more code to find the state 
	as a function of T.


A good discussion of such issues can be found in the remarkable
book 
   Sussman and Wisdom, 
   _Structure and Interpretation of Classical Mechanics_

They have some pithy things to say about inconsistent notation,
starting on the first page of the preface:
  http://mitpress.mit.edu/sicm/book-Z-H-5.html

and then (unlike almost everybody else) they don't surrender.
They rederive the whole subject without using the usual hand-
waving and dirty tricks such as "proof by pun" et cetera.

Among other things, this work serves as an existence proof, 
proving that we are not obliged to use self-inconsistent 
notation and other dirty tricks.