Humanist Discussion Group, Vol. 13, No. 391.
Centre for Computing in the Humanities, King's College London
<http://www.princeton.edu/~mccarty/humanist/>
<http://www.kcl.ac.uk/humanities/cch/humanist/>
Date: Thu, 10 Feb 2000 07:51:55 +0000
From: "Osher Doctorow" <osher@ix.netcom.com>
Subject: two alternatives in decision-making
Dear Willard McCarty:
I'd like to entitle this contribution Computer Testing of Two Alternative
THeories in Decision-Making. I'm going to translate everything into
ordinary Humanist English here, under the plausible conjecture that if it
can't be expressed that way, it's probably not real.
Microsoft decision-making algorithms and programs are based on Bayesian
conditional probability algorithms. For the non-specialist, a simple
illustration of conditional probability is the probability that it will
rain given that it is snowing, which is calculated as the probability of
both rain and snow divided by the probability of snow. This is
symbolically written P(rain given snow) or more simply P(rain/snow), where
/ means given. Notice that this is undefined when the probability of rain,
symbolically P(rain), is 0, since you can't divide by 0 in mathematics.
Logic-based probability (LBP), which the author has developed since 1980,
assigns a probability to the logical conditional (which in unrelated to
conditional probability) if a then b, which can be written a arrow b or
a-->b or with the above example if snow then rain, i.e., snow---> rain. So
we calculate the probability that if it snows then it rains. It turns out
purely mathematically that this probability is the probability of rain and
snow minus the probability of snow plus 1, and it is denoted P(snow -->
rain) = P(snow and rain) - P(snow) + 1 where P( ) is probability of.
If you compare the two, Bayesian P(rain/snow) and LBP P(snow --> rain),
they only differ algebraically by replacing division in the former by
subtraction in the latter, noting that the occurrence of 1 insures that the
result is a singule probability in P(snow --> rain). The Bayesian
P(rain/snow) does not contain a 1 in its formula, and therefore turns out
not to be a single probability but a ratio of probabilities, which has the
disadvantage of being undefined when the denominator P(snow) is 0. Of
course, the actual computed results for the two alternative types will be
very different in most situations, because the results of division and
subtraction differ enormously mathematically.
Now comes the computer. Microsoft wants very much to market its
(statistical) decision algorithms, based on Bayesian conditional
probability, also called Bayesian probability or conditional
probability. So it does just that. The author, on the other hand,
develops a computer algorithm using LBP probability. Which do you think
works much better? The answer is LBP.
Why does LBP work much better than Microsoft's Bayesian version? After
all, they only differ algebraically by changing division to
subtraction. But that is precisely the point. Since you cannot divide by
0 in Microsoft's division-based algorithm, you miss very rare events
(which, believe it or not, are assigned probability 0 under certain general
assumptions in probability and statistics - technically continuous
probability distributions if you want to know). So Microsoft cannot deal
with very rare events (genius, great inventions, catastrophes, strokes of
enormous luck, etc.). This does not happen with LBP, because it has no
division. It also turns out that there are some very common events which
have probability 0, believe it or not, namely, what are called lower
dimensional objects in ordinary 3-dimensional space under the above
techincal continuous probability assumption. For example, a line or a
piece (segment) of a line in 3 dimensional space has dimension 1 because it
only has length, and a plane or planar object like a rectangle has
dimension 2 because it only has length and width, and a single point in 3
dimensional space has dimension 0 because it has neither length nor width
nor depth. All of these objects have probability 0. Since it can be shown
that the surface of any 3-dimensional object (for example, a person's skin,
or the surface of the earth) is 2-dimensional, it also has probability
0. Therefore, Microsoft's program cannot deal with the surface of the
earth, the surface of a person's body (the skin), the surface of a person's
internal organs, the surface of a cell, etc.
The most remarkable result of all this is that Microsoft's programs only
give comparatively trivial decisions. After all, if you cannot handle
rare or unexpected events, and you cannot deal with the surfaces of
physical objects, and you cannot deal with centers of physical objects
(which are points of dimension 0), etc., what else is there?
Well, there is still enough left to make somewhat obscure
predictions. After all, there really is such a thing as conditional
probability, the probability of rain given snow for example. So Microsoft
can compute (it theory, anyway) the probability of rain given snow, and if
you translate that into business or economic or political or other
decisions, it can compute various probabilities that tend to be on the
mediocre level of conceptual interest and importance.
The really conceptually interesting and important decisions come from LBP
probability algorithms. For example, LBP can tell you what decisions to
make if a rare meteor hits a spacecraft, if a politician is assassinated,
if a star is born, and so on ad infinitum. One important qualification
should, however, be noted. Microsoft's Bayesian probability ratios tell
you the probability of rain if the event of Snow is fixed. In other words,
no pun intended, if you freeze the event that It is Snowing at a moment in
time, and then asked what the probability of rain occurring is for that
level of fixed Snowing, Microsoft's Bayesian probability ratio will give
you an accurate result. LBP's probability P(snow --> rain) gives you
something different entirely. Instead of telling you the probability that
it will rain for a fixed or frozen level of snow, it tells you the probable
influence which snow will have on rain. On a scale from 0 to 1, which is
the standard probability scale, an LBP probability of snow --> rain of
value 1 means that snow has maximum possible influence on rain, and so
on. In analyzing decisions, it obviously is much more important to know
how much one event influences another than to know what happens to one
event when a second event is fixed or frozen, but there are conceivable
scenarios (like simple gambling with cards) where both are useful to know.
In the above situations, computer algorithms and the results of computer
programs even in thought experiments or Gedanken experiments helped to
clarify the nature and the preferences of and between two competing
probability types. Hopefully, readers will be interested in following up
or adding to these examples.
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