In Part 1, Example1, I raised the question: What is the
probability that my tax return will be audited?
The point of this example is that PT (standard probability
theory) cannot answer questions of this type. My brief analysis in
Part 1 was not as complete as it should be. A more complete analysis
follows. The analysis involves straightforward computation of bounds
on the fraction of tax returns that are audited.
Suppose that I start with the knowledge that 1% of tax returns
in the United States are audited. This implies that, picked at
random, a resident of the United States can expect that his/her tax
return will be audited with probability 0.01. But I know much more
about myself than my being a resident of the United States. For
example, I reside in the Bay area. What, then, is the probability
that I will be audited?
Suppose that the population of the United States is N1; that
of the Bay area is N2; and that the fractions of tax returns that are
audited in the United States and the Bay area are, respectively, r1
and r2. Now, if N2 is less than or equal to min( N1r1, N1(1 - r1) ),
then all that can be said about r2 -- given r1 -- is that it lies
between 0 and 1. Thus, the knowledge that I live in the Bay area
provides no information about the probability that my tax return will
be audited.
Next, suppose that N2 is less than or equal to N1r1. Then the
upper bound on r2 remains equal to 1 but the lower bound may increase
from 0 to max(0, N2 - N1(1 - r1)) / N2. Dually, if N2 is greater than
or equal to N1r1, and is less than or equal to N1(1 - r1), then the
lower bound remains equal to 0 while the upper bound may decrease from
1 to (N2 - N1r1) / N2. Finally, if N2 is greater than or equal to
max( N1r1, N1(1 - r1) ), then the upper and lower bounds on r2 are,
respectively, N1r1/N2 and ( N2 - N1(1 - r1) ) / N2.
What we see, then, is that if N2 is not known, the knowledge
of r1 provides no information about r2. If N2 is known, then the
bounds on r2 may be tighter than 0 and 1.
Suppose that the IRS can provide me with the values of N2 and
r2. But I know that I live in Berkeley, so that the same problem
arises again. In the absence of information about the population of
Berkeley, the probability that I will be audited remains bounded by 0
and 1. And so on.
This conclusion may appear to be counterintuitive because
usually we have in hand some information that goes beyond what is
assumed above. However, in general the information that we have is
based on perceptions rather than measurements. PT, unlike PT++, does
not have a capability to process perception-based information. This
is the reason why PT++ is needed to come up with an answer -- but not
a precise answer -- to the question: What is the probability that my
tax return will be audited.
A closely related problem which does not involve probabilities
is the following.
Consider a function, y=f(x), defined on an interval, say [0,
10], which takes values in the interval [0, 1]. Suppose that I am
given the average value, a, of f over [0, 10], and am asked: What is
the value of f at x=3? Clearly, all I can say is that the value is
between 0 and 1.
Next, assume that I am given the average value of f over the
interval [2, 4], and am asked the same question. Again, all I can say
is that the value is between 0 and 1. As the length of the interval
decreases, the answer remains the same so long as the interval
contains the point x=3 and its length is not zero. As in previous
example, additional information does not improve my ability to
estimate f(3).
The reason why this conclusion appears to be somewhat
counterintuitive is that usually there is a tacit assumption that f is
a smooth function, In this case, in the limit the average value will
converge to f(3). How to express this precisely is a problem that
some of the members of the BISC Group may be able to solve. Note that
the answer will depend on the way in which smoothness is defined.
Warm regards to all and
best wishes for the New Year,
Lotfi Zadeh
------------------------------------------------------
Lotfi A. Zadeh
Professor in the Graduate School and Director,
Berkeley Initiative in Soft Computing (BISC)
CS Division, Department of EECS
University of California
Berkeley, CA 94720-1776
Tel/office: (510) 642-4959 Fax/office: (510) 642-1712
Tel/home: (510) 526-2569 Fax/home: (510) 526-2433
email: zadeh@cs.berkeley.edu
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