Induction and Learning
PSY 322/ORF 322: Human Machine Interactions
Gilbert Harman
Department of Philosophy
Princeton University
Monday, February 28, 2005
Cognitive Science Talks
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Note this talk coming up: 4 PM, Wednesday, March 30.
Ben Schneiderman, author of Leonardo’s Laptop. Small
auditorium, Computer Science.
Report on Homework 2: Rosenthal’s HOT Theory
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Abbreviated sample good answers
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Advantage: intuitively, conscious states are states you can
report being in.
Problem: you can have such a higher order thought without
being in S, but intuitively you cannot be in a conscious
state S if you are not in that state S.
Sample less good answers
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An advantage of R’s theory is that it identifies a conscious
experience with an experience of which one is conscious.
(True, but that’s just the theory, not an advantage of the
theory).
A disadvantage of the theory is that there is no way to tell
whether there is a higher-order thought if one is not
conscious of that thought. (False, there are many ways to
find out one what states one is in without the states being
conscious. Blindsight example.)
Review: Where Are We?
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Human machine interaction. Body as machine we interact
with. Maybe we are our bodies?
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Can a machine be conscious? Can a machine think?
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Programs that think. Following deductive rules.
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Human thinking. Reasoning about what follows from what.
Using rules? Using mental models? Probabilistic reasoning.
Creative reasoning.
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Do people ever use rules? Calculation.
Induction and Deduction
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Deduction: conclusion guaranteed by premises.
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Induction: conclusion warranted but not guaranteed.
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Complication: inductive arguments in math
Induction and Deduction as Two Kinds of Reasoning?
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Deduction has to do with what follows from what.
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Induction has to do with what can be inferred from what.
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Deductive rules are rules that proofs or arguments must
satisfy, not directly rules that people should follow in
reasoning.
Modus ponens
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P and if P, then Q imply Q.
NOT: if your believe P and if P, then Q your should or
may infer Q. (What if you also believe not Q?
Reasoning as Change in View
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Reasoning often involves giving up something previously
accepted.
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Not just inferring new things.
What Follows from a Contraction?
1. P&not-P
2. P (from 1)
3. not-P (from 1)
4. P or Q (from 2)
5. Q (from 3 and 4)
What Would You Infer from Contradictory Beliefs?
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You would not infer every proposition you can think of.
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You would try to modify your beliefs in order to eliminate
the contradiction.
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What follows from your beliefs is not always something you
can infer from your beliefs.
The Problem of Induction
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Deduction is reliable in that it guarantees true results given
true premises
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Can we show something similar for induction? That it
tends to yield true results?
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But what about the fact that good inductive reasoning can
abandon “premises”?
Can We Formulate Inductive Principles?
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Enumerative induction: given that all observed As are F ,
infer that all As are F (or most are, or the next will be).
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Inference to the best explanation: given various data and
background assumptions, if E best explains the data in the
light of the background assumptions, infer E.
New Riddle of Induction
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The fact that all observed As are F is not in general a
reliable indicator of whether all, most, or even the next A
will be F . It depends at least on what F is. Suppose F
means “observed”!
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There has to be some sort of “inductive bias” favoring
some hypotheses that are compatible with the evidence
over others.
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Consider the general problem of curve-fitting. We do not
just look for the hypothesis that gives the best fit but look
for something like the simplest hypothesis that gives the
best fit, where simpler hypotheses get treated differently
from less simple hypotheses.
Justifying Induction
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After we figure out what rule seems right, can we justify
our using it?
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The philosopher Nelson Goodman says: we justify a rule
by seeing how it fits with various examples of what we
think are good inferences, and we justify particular
inferences in terms of rules
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In other words: justification consists in getting rules and
particular judgments in sync with each other so that we are
satisfied with them and they do not conflict with each
other.
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The political philosopher John Rawls applies this idea to
justifying principles of justice: we try to get our ideas into
“reflective equilibrium”
Does Reflective Equilibrium Justify?
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We do seem to reason in the way Goodman and Rawls
suggest.
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But why think it is reasonable to reason in that way.
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Answer: that is what we mean by “reasonable”?
But many ordinary reasoning practices are clearly no good
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Jumping to conclusions
Gambler’s fallacy
Bias toward looking for positive evidence, ignoring negative
evidence
Effects of framing
Connectionist Constraint Satisfaction Model
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Positive (excitatory) links between evidence, evidential
principles, and conclusions supported by the evidence on
those principles
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Negative (inhibiting) links between competing conclusions
and between competing evidential principles
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Excitation spreads throughout the network
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Reach a steady state
Necker Cube Analogy
Empirical Studies of Reasoning of Jurors
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Test subjects’ confidence in various principles
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Reliability of certain type of eye-witness testimony
Whether posting on a computer bulletin board is more like
talking on the telephone or writing an article for a
newspaper . . .
Give subjects details of complex case
Results
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Subjects divide on their verdicts
Subjects are very confident of their verdicts
Subjects confidence or lack of confidence in various
principles changes to fit their verdicts
Reliability
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We want reliable methods: methods that do as well as
possible in getting us true beliefs
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Deductive methods are reliable in the sense that they
guarantee the truth of conclusions when the premises are
true.
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But what if the premises are not true?
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Reasoning sometimes leads us to give up things previously
believed.
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How then can we define a useful notion of reliability?
Suggestion from Statistical Learning Theory
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Suppose an unknown background statistical probability
distribution that produces items with certain observable
features and classifications in which we are interested
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How can we use data to come up with a rule for classifying
new cases?
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We would like to find a rule that minimizes expected error
on the new cases (perhaps weighted by expected costs of
different sorts of errors)
What can we know about this?
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A method will not get anywhere unless it has some
inductive bias
Perhaps: we only consider certain sorts of rules
Then task is to find the best of the rules we are considering
Statistical learning theory has results about when this is
possible.
Pattern Recognition or Classification
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Given some information about a case, we want to classify it
in some way
Examples
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Automatic mail sorting based on zip codes
Computer speech recognition of commands
Email spam detection
Automatic medical diagnosis based on X-rays or blood
samples
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Quality of the system: the accuracy of the classifications as
measured for example by the percentage of errors
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We might want to distinguish different sorts of errors.