On Tue, 31 Jul 2001, Kathryn Blackmond Laskey wrote:
> >The Jaynesian
> >viewpoint is that there are no "random" variables -- there are only
> >variables whose values may not be known with certainty, and there is no
> >logical distinction between these and any other variable.
>
> [...] But things get *really* squishy when we try
> quantifying over models.
I don't know what you mean by "quantifying over models". Could you elaborate?
> As long as we allow quantification only
> over what some people call "objective" [...]
> properties of the world, then all our probability models
> will satisfy what Ron Howard calls the "clarity test." [...]
>
> But just try applying the clarity test to questions such as:
>
> What did John really mean when he threatened to quit?
> Do you think Emily is in love with Joe?
> Do you agree that Fred is in denial over his anger toward Julio?
Let's back up a bit. Probability theory as a logic is about assigning or
inferring certainty levels to propositions. By definition, a proposition must
be either true or false, although we may not know, and may never know for
sure, which it is. These propositions are commonly formed by applying
predicates to variables representing real-word values. So if you want to
reason about statements that fail the clarity test ("John is a good boy"), or
entities for which one inherently cannot, *even in principle*, assign definite
values, then you've stepped outside of the realm of probability theory.
(However, see my remarks below as to why this is not a problem.)
> What is the ontological status of "that which John really meant," or
> "Emily's true feelings toward Joe," or "Fred's true disposition
> toward Julio?" It's pretty clear their values cannot be known with
> certainty. It's not at all clear whether they have "real values" at all.
Whether they can be known with certainty is irrelevant. For example, we can
reason about the prevalence of life in the universe even if this is impossible
to know with certainty (due to speed-of-light limitations and expansion of the
universe, we can never visit all of it.) But why do you say the above cannot
be known with certainty? Suppose that we had a complete understanding
of psychology and the operation of the human brain. It seems likely to me
that we would then be able to give precise definitions for the currently-vague
notions of intention and emotional state, such that complete knowledge of the
physical state of a person's brain would allow us to determine their mental
state. Whether this will ever be practical is beside the point, as long as it
is possible *in principle*.
> There are some who argue we must use some
> other formalism such as fuzzy logic for variables that don't satisfy
> the clarity test.
I don't consider the inability of probability theory to reason about
propositions or values that fail the clarity test to be a problem, for the
simple reason that I think it is a mistake to focus on such "values" as
objects of inference. Ultimately, what matters are the objective properties
of the real world. The only reason we care about "Emily's true feelings" or
"Fred's true disposition" is because we want to better predict what these
people may do in the future (or infer what they have already done). The only
reason we care about "that which John really meant" is because we wish to
extract useful information from John's statements.
I think that in many cases it is a mistake to even try to assign a "meaning"
to human utterances; rather, we can uses these utterances to make inference
about the objective state of the world through Bayes' rule:
P(state S | utterance U, X) =
P(state S | X) * P(utterance U | state S, X) / P(utterance U | X)
Rather than assigning a meaning to an utterance, we assign it a likelihood
function L(state S) = P(utterance U | state S, X). Note that this likelihood
function may depend on the speaker and circumstances in which the utterance is
made.
> But some people happily include such hypotheses in Bayesian networks and
> give them conditional probabilities just like other random variables.
> [...]
> The above does not stop me from defining random variables in a
> Bayesian network that refer to what an agent means or what an agent's
> feelings are toward another agent, or some other non-clarity-test
> phenomenon.
A model may have hidden (latent) variables that have no physical meaning, yet
are still useful for defining the model. For example, in speech recognition
we generally use a mixture of gaussians to approximate the observation
distribution for a given HMM state. The index of the mixture component then
is a hidden variable of our model, and we may reason with it in the same way
as any other variable, yet it has no physical meaning; it is an artifact of
how we chose to decompose our model. Similarly, I doubt that anyone can give
you a precise definition of "general intelligence", yet psychologists have
found that regression models incorporating a hidden variable g (for "general
intelligence") have substantial predictive power. As a third example, in
quantum mechanics we have a complex-valued wave function that can never be
observed itself, and which most physicists do not consider to have any
physical existence, but which is essential for making inferences about those
quantities that we can observe. So I do not see any philosophical problem
with including in our models, as intermediate, hidden variables, quantities
whose existence is questionable. We may view such variables as merely an
artifice to simplify the definition of a joint probability distribution on the
observable variables.
> For myself, I regard the value of the "Emily's true feelings" random
> variable as (approximately, where the approximation is good enough
> for my modeling purposes) a sufficient statistic for a highly complex
> brain state that it's simply not worth modeling in detail.
Sounds like we are in agreement, then.
> >Thus, measure theory helps us build consistent probabilistic models
> >involving continuous variables, [...]
>
> OK, but again, one may not need measure theory.
Sure. By analogy, it's well known in mathematical logic that a set of axioms
may have many different models.
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