Sampling Assumptions and the Size Principle in Property Induction

1
Sampling Assumptions and the Size Principle in
Property Induction
Philip M. Fernbach (philip_fernbach_at_brown.edu) Bro
wn University Department of Cognitive and
Linguistic Sciences, Box 1978 Providence, RI
02912 USA
Stimuli and Design
Results
Example Stimulus The Size Principle predicts a
non-monotonic effect of adding similar examples.
This is tested directly using an argument
preference task.
As predicted by the Bayesian Model, variation of
sampling assumptions yielded a statistically
significant treatment effect The direction of the
effect was also in line with the Bayesian model
as the strong sampling group displayed a greater
preference for one-premise arguments than did the
weak sampling or ambiguous groups.

• Three Conditions
• Strong Sampling Cover story implied that the
  categories were generated using a strong sampling
  procedure - students learning facts about animal
  categories by sampling at random from the set of
  animals to which the predicate applies.
• Weak Sampling Cover story implied that the
  categories were generated using a weak sampling
  procedure - students learning facts about animals
  categories by sampling from the set of all
  animals.
• Ambiguous No cover story. Participants asked to
  choose the stronger argument

Average preference for three-premise arguments on
a 1-7 scale. A score of 7 implies that the
three-premise argument was strongly preferred a
score of 1 implies that the one-premise argument
was strongly preferred and a score of 4 implies
that the one-premise and three-premise arguments
were judged equally strong.
However, inconsistent with the predictions of the
Bayesian framework, all groups showed an overall
preference for three-premise arguments over
one-premise arguments. In other words, most
participants failed to display the size principle
regardless of group .
Property Induction Argument
Categories
Predicate
Computational Model Predictions
Premises
Conclusion
A host of phenomena have been observed concerning
how people evaluate the strength of arguments
such as the one above, and several models have
been proposed to account for the phenomena.
Unlike previous models based on similarity
(Osherson et al, 1990) and feature matching
(Sloman, 1993), the Bayesian framework considers
how premise categories are sampled. Weak
sampling implies that the premise categories are
sampled randomly from the set of all relevant
categories and then given a label specifying
whether or not they belong to the set to which
the predicate applies. Strong sampling implies
that the premise categories are sampled
explicitly from the set of categories to which
the predicate applies. This makes the data more
informative about the nature of that set since
the probability of observing that category is
inversely proportional to the size of the
hypothesis.
Prediction whether three-premise arguments or
one-premise arguments should be judged stronger
for each model across the three conditions.
Percentage of scenarios in which three-premise
arguments were judged stronger, one-premise
arguments were judged stronger and arguments were
judged equally strong across all three
conditions.
Similarity Coverage Model (Osherson et al, 1990)
Similarity-based heuristic Feature-Based Model
(Sloman, 1993) Feature matching heuristic
Bayesian Models (Kemp Tenenbaum, 2003 Sanjana
Tenenbaum, 2003 Tenenbaum Griffiths, 2001)
Bayesian Inference over candidate hypotheses with
a sampling-sensitive likelihood calculation.
Generally assumes strong sampling.
Likelihood Calculation
Under Strong Sampling
Under Weak Sampling
Under strong sampling, the likelihood of a
hypothesis decreases exponentially in proportion
to its size as new categories are encountered.
Size Principle
Under strong sampling, smaller hypotheses are
favored over larger ones given evidence that is
consistent with both. In property induction tasks
the Size Principle implies that adding similar
premise categories should decrease the strength
of an argument whose conclusion is dissimilar.