Dear All:
There are few, if any, concepts that are as fundamental as
the concept of causality. Perceptions of causal dependencies govern our
behavior and underlie our decisions. Identification of causal
dependencies plays a pivotal role in the realms of law, medicine,
economics, data mining, AI and many other fields. Indeed, a case could
be made for requiring most students in law, medicine and economics to
take a course in which causality is an object of substantive attention.
Our discussions of causality, with insightful contributions
by prominent members of the legal profession, have clarified many
issues. But a question that remains unsettled is: Does there exist a
formalized, bivalent-logic-based, theory of causality which is capable
of dealing with realistic problems exemplified by the raincoats example?
In earlier messages, I suggested that existing theories do
not have this capability. The principal source of difficulty is that
when we activate an event A0, call it the nominal event, and observe a
consequent event B, it is almost always the case that B is a consequence
of a multiplicity of other events, call them a network of collateral
events, N(A1, A2, �). What this implies is that we cannot answer the
question: Was B caused by A0? with a categorical yes or no. In other
words, we have to assume that causality and related concepts such as
responsibility and propensity are a matter of degree�with the degree of
causal dependence representing our perception of materiality of the role
of A0 in causing B. In turn, this implies that the assumption that
causality is a matter of degree should be the point of departure in any
formalized theory of causality which aspires to provide a body of
operational concepts and techniques for dealing with causal dependencies
in realistic settings.
If we accept this postulate as a basic premise, the next
question is: How can the degree of materiality be assessed?
Unfortunately, there is no simple or obvious answer to this question.
Furthermore, it should be noted that existence of the network of
collateral events raises serious questions regarding the validity of use
of counterfactual conditionals in causal reasoning.
In the case of the raincoats example, the assumption is that
we are dealing with a single experiment and have a single datapoint:
(increase in advertising: 20%; increase in sales: 10%) Furthermore, I do
not have a model of the network of collateral events. The only other
information that I have is (a) world knowledge, e.g., rainy weather
increases demand for raincoats; and (b) case-based knowledge, that is,
knowledge about other experiments which in some sense are similar to my
experiment. Both (a) and (b) are imprecise, uncertain and not totally
reliable. A further complicating factor is that an event may have a
positive or negative polarity. To illustrate, in the raincoats example,
rainy weather has positive polarity, while dry weather has negative
polarity. The issue of polarities makes it much more difficult to
aggregate contributions of collateral events to the consequent event,
and expressing the aggregate as a weighted combination in the manner
suggested by Marianne Belis.
A conclusion which emerges is that in realistic settings it
would be unrealistic to aim at expressing the degree of causality or
materiality as a sharply defined number. What is necessary is a recourse
to granulation of variables and their probabilities, resulting in what
may be called a bimodal distribution, with �bimodal� signifying that
granulation is applied to both variables and their probability
distributions. More specifically, if granulation is coarse, the granular
values of degree may be zero, low, medium and high, and likewise for the
values of probabilities. As an example, a bimodal distribution of degree
may be of the form ((low, low), (high, medium), (low, high)), meaning
that the granular probabilities of low, medium and high are low, high
and low, respectively. What this suggests is that, in realistic
settings, causal dependence is certainly not a matter of yes or no;
rather, it is a matter of degree. However, in most cases, the degree
cannot be represented as a number, an interval or a probability
distribution. It may be representable as a fuzzy interval or, more
generally, as a bimodal distribution.
Obviously, more can be said when the experiment can be
repeated�as in the realm of medical experimentation, and/or when at
least a partial model of the collateral network is known. But what could
be said would not be in the spirit of theories based on bivalent logic
and bivalent-logic-based probability theory. I hope to have an
opportunity to say more about this important issue at a later time.
Regards to all
Lotfi
--
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
