Alan,
Thanks for responding to my post, As you point out, normalization does
occur across a series of simulated variables in sequence, But are we
talking about the simulated instantiation of causes normalizing or the
causes themselves normalizing. When we speak of causes theoretically, we
refer to operations that extend across some range, X1+X2=Y, whether the
values of X1 and X2 are low, medium or high. This relationship is
conjunctive and uniform, The distribution of such abstract constructs is
uniform, since the operations apply equally at the extremes as in the middle
ranges. The inferences we make from such abstract values have generality
based on laws, This why they tend to serve as the backbone of substantive
theories,
The picture is clouded, however, when model such lawful operations by
simulating instantiations. The instantiation process brings added
constraints (central limit) that are potentially unrelated to the causes
being modeled. To understand this it is good to consider when constriction
would NOT be the case, Let's say that we model a sequence of causes where
y1=x1+x2, y2=y1+x3, y3=y2+x4, etc. Throw in some constants that will make
the scales of y(n) comparable to each x(n). If we simulate this with random
numbers, then the instantiation of the values will be constricted by the
central limit, The distributions of y(n) will tend to be more normal as we
progress, even if we start with perfectly uniform x1 and x2. All this we
have said before,
Now consider a real life model. Let's say that the causal relations are
y(n)=x(n)+x(n+1), as before, But in this case all levels of the ranges of
the subsequent y variables are equally represented, that is at each level of
the sequence, the causes are uniformly distributed, In this model, we
remove the normalizing constriction and allow nature to replenish the ranges
of each variable in the sequence, Such replenishment allows us to make
conjunctive statements,
In nature, we could have either the constrained or replenished model. The
constrained instantiation would lead to mixed conjunctive/disjunctive models
while the replenished model would allow for the simple conjunctive model.
As you suggest, perhaps the range of potential values simply is constricted
by nature as a causal sequence progresses, Such causation would be
nonreplenished, But this kind of nonreplinished modeling would require that
we begin with the prime causes of the variables, so that we can factor in
the history of their normalizations, We would not be able to arbitrarily
start measuring causes at some point in a sequence, None the less, all of
science does begin at some arbitrary starting point, Otherwise we would have
to model the whole history of our variables. At worst, this might require
going back at least to the most recent big bang. At best, we would have to
be very explicit about the recent history of constraints on the variation of
all relevant causes.
One might argue that we should just let the empirical distributions tell us
if a constricting model is called for. The problem with this is that many
factors irrelevant to the potentially uniform relationships between
variables can cause the constriction. One of these factors contributing to
the constrained instantiation may simply be that it is inconvenient for us
to find or generate the extremes, Thus our science becomes confounded by
irrelevant factors. This is why I have advocated sampling by potential.
But this brings up the question of just what the potential is? Is the
distribution of a variable at some point of a sequence uniform or normal by
POTENTIAL? To answer this we must ask another question.
A conceptual test of the potential distribution of causes in sequence, is to
pick out different points along a sequence of causes and ask for each one
"Does the function or law I am assuming (y=x1+x2) hold for the pairing of
both extreme and moderate values of x1 and x2?" If so, the relationship is
conjunctive and should be sampled as uniform,
If, on the other hand, we say "No, the combination of causes is different in
the extremes versus midranges of effect," then we are modeling a disjunctive
cause, Among other cases, this would occur when constraining instantiations
to normal distributed causes (since random pairings of extreme x1 and
extreme x2 (conjunctive causation) would be very rare and most extreme y
values would be determined by EITHER extreme x1 OR extreme x2 but not both,
i.e. by disjunction).
If the extremes of y are unlikely to be a combination of both x1 and x2,
then we have a mixed conjunctive/disjunctive model, In other words,
extremes of y do not really equal extreme x1+ extreme x2 because such
combinations are not allowed adequate representation in our calculations.
We should be explicit about what type of inferences we are willing to make
at each point in a causal sequence, This will require our being explicit
and proactive when simulating the distributions of variables along the
causal sequence,
Bill Chambers
-----Original Message-----
From: alan penn <[log in to unmask]>
To: simsoc <[log in to unmask]>
Date: Tuesday, September 14, 1999 4:54 AM
Subject: Re: Disjunctive and Conjunctive causes in simulations
>William Chambers asked an interesting question to which I havent seen any
>responses yet - perhaps they were off line. For my part I think that this
>may not be thought of as a problem for simulators since the real world
>systems they are often trying to simulate show exactly the tendency to
>normalisation that you describe. We carry out simulations with a series of
>apparently independent uniform 'causes' and find stability emerging and
>count this as success because it seems lifelike. Any views?
>
>Alan
>
>> Dear List Members: I am developing a family of statistical methods
>>that allows us to infer causation from continuous linear variables:
>>http://www.wynja.com/chambers/regression.html I have applied the
>>methods of corresponding correlations and corresponding regressions to
>>real data but most of my work is based on simulations, I have some
>>general questions about how experts conduct simulations. The methods I
>>am using assume that the causes are uniformly distributed. If we create
>>a series of such causes, the subsequent causes become progressively
>>normally distributed. Consider the following model: y1=x1+x2, where x1
>>and x2 are uniformly distributed, y2=y1+x3 y3=y2+x4 Notice that half
>>of the causes (yn) tend to be progressively more normally distributed. As
>>the model progresses, the distributions become thinner in the extremes,
>>This gives more weight to midrange variables, because they are more
>>frequently instantiated, The problem is worse when we create the
>>dependent variables directly from series of normally distributed causes,
>> The dependent variable that is generated from normally distributed
>>causes tends to be determined in the extremes by either one cause or the
>>other (disjunctive causation), The combination of two extreme values of
>>x1 and x2 (conjunctive cause) is very rare when the causes are normal
>>because extremes of x1 or x2 are rare, even on their own. Their
>>combination is even more rare. The upshot is that as we go from uniform
>>to normal distributions, the causal model in the extremes of y(n) becomes
>>progressively disjunctive while that in the mid range of y(n) stays
>>conjunctive, Have any of you dealt with this problem before? How
>>do you keep causal (x) distributions uniform in sequences? Thanks,
>>William Chambers
>
>
>________________________________________________________
>Alan Penn, Reader in Architectural and Urban Computing
>Director, VR Centre for the Built Environment
>The Bartlett School of Architecture and Planning
>1-19 Torrington Place (Room 335)
>University College London, Gower Street, London WC1E 6BT
>tel. (+44) 020 7504 5919 fax. (+44) 020 7916 1887
>mobile. (+44) 0411 696875
>email. [log in to unmask]
>www. http://www.vr.ucl.ac.uk/
>________________________________________________________
>
>
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