Download Day 3 - University of California San Diego

Day 3:
Competition models
Roger Levy
University of Edinburgh
&
University of California – San Diego
Today
• Probability/Statistics concept: linear models
• Finish competition models
Linear models
• y = a + bx
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y
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Linear fit
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x
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Linear models (2)
• Linear regression is often formulated in terms of
“least squares” or minimizing “sum of squared
error”
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y
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error
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x
• For us, an alternative interpretation is more
important
• Assume that the datapoints were generated
stochastically with normally distributed error
• Least-squares fit is maximum-likelihood estimate
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y
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true
params
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estimated
parameters
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Generalized
Linear models (3)
outcome
• The task of fitting a (linear) model to a
continuous real-valued output variable is called
(linear) regression
• But if our output variable is discrete and
+
unordered, then
linear regression doesn’t make
sense
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x
• We can generalize linear regression by allowing
+
1
Logistic fit
outcome
new scale
interpretable
as probability
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x
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1.0
Linear models (4)
• We can also generalize linear models beyond one
“input” variable (also called independent variable,
covariate, feature,…)
U
• We can generalize to >2 classes by introducing
multiple predictors Ui
Today
• Probability/Statistics concept: linear models
• Finish competition models
Case study: McRae et al. 1998
• Variant of the famous garden-path sentences
• The {crook/cop} arrested by the detective was
guilty
• Ambiguity at the first verb is (almost) completely
resolved by the end of the PP
• But the viability of RR versus MC interpretations
at the temporary ambiguity is affected by a
number of non-categorical factors
• McRae et al. constructed a model incorporating
the use of these factors for incremental
constraint-based disambiguation
• linking hypothesis: competition among alternatives
drives reading times
Modeling procedure
• First, define a model of incremental online
disambiguation
• Second, fit the model parameters based on
“naturally-occurring data”
• Third, test the model predictions against
experimentally derived behavioral data, using the
linking hypothesis between model structure and
behavioral measures
Constraint types
• Configurational bias: MV vs. RR
• Thematic fit (initial NP to verb’s roles)
• i.e., Plaus(verb,noun), ranging from 0 through 6
• Bias of verb: simple past vs. past participle
• i.e., P(past | verb)*
• Support of by
• i.e., P(MV | <verb,by>) [not conditioned on specific
verb]
• That these factors can affect processing in the
MV/RR ambiguity is motivated by a variety of
previous studies (MacDonald et al. 1993, Burgess et al.
1993, Trueswell et al. 1994 (c.f. Ferreira & Clifton 1986),
cally not
calculated
this way, but this would be the rational reconstruction
Trueswell
1996)
The competition model
• Constraint
strength
determines
degree of bias
• Constraint weight
determines its
importance in the
RR/MC
decision
Interpretation
nodes
strength
Constraint nodes
weight
Evaluating the model
• The support ci,j at each constraint is normalized
• Each interpretation Ai receives support from each
constraint ci,j proportionate to constraint weight
wj
• The interpretation nodes feed additional support
back into each constraint node, at a growth rate
of wjAi
[ci step
modelt demo]
• …at each time
i
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Probability of preferred candidate
1.0
CI simulation
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5
10
15
Cycle
20
25
30
The feedback process
• Generally,* the positive feedback process means
that the interpretation Ij that has greatest
activation after step 1 will have its activation
increased more and more with each iteration of
the model
when there are ≥3 interpretation nodes, leader is not guaranteed to win
Estimating constraint strength
• RR/MC bias: corpus study
• conditional probability of RR or MC given “NP V”
sequence
• Verb tense bias: corpus study
• conditional probability (well, almost*) of simple
past/past participle given the verb
• by bias: corpus study
• conditional probability of RR or MC given “V-ed by”
• thematic fit: offline judgment study
• mean typicality rating for “cop” + “arrested” (not a
probability, though normalized)
McRae et al. 1998
Estimating constraint weight
• The idea: constraint weights that best fit offline
sentence continuations should also fit online
reading data
• Empirical data collection: gated sentence
completions
The cop arrested…
arrested by…
by the…
the detective…
• The learning procedure: minimize root mean
square error of model predictions
• …for a variety of # of time steps k
• optimal constraint weights determined by grid
search between [0,1]
Fit to offline gated completion
data
• initial MV
bias
• more
words→
increasing
RR bias
• 100% RR
after seeing
agent
• Before then,
good patient
biases
toward RR
Fit against self-paced reading data
• Competition hypothesis: associate processing
time with the number of steps required for a
model to run to a certain threshold
• Intuition: at every word, readers hesitate until one
interpretation is salient enough
• Dynamic threshold at time step i: 1 – Δcrit × i
• Intuition: the more time spent, the less fussy
readers become about requiring a salient
interpretation
• Usually,* the initially-best interpretation will reach
the threshold first
0.8
0.4
0.6
decreasing
threshold
0.2
point of intersection
determines predicted RT
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Probability of preferred candidate
1.0
CI simulation with decreasing threshold (slope = 0.007)
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5
10
15
Cycle
20
25
30
Self-paced reading expt
• Contrasted good-patient versus good-agent
subject NPs
• Compared reading time in unreduced versus
reduced relatives
The {cop/crook} (who was) arrested by the detective was guilty…
• Measurement of interest: slowdown per region in
reduced with respect to unreduced condition
• Linking hypothesis: cycles to threshold ~ reading
time per region
McRae et al. results
model still prefers MV
in good-agent case
mean activation
of RR node
Results
• Good match between reading time patterns and #
cycles in model
• Model-predicted RTs are basically monotonic in
initial equibias of candidates
• At end of agent PP, the model prefers main-clause
interpretation in good-agent (the cop arrested)
condition!
• is this consistent with gated completion results?
• Critical analysis: intuitively, do the data admit to
other interpretations besides competition?
What kind of probabilistic model is
this?
• Without feedback, this is a kind of linear model
• In general, a linear model has form (e.g., NLP
MaxEnt)

Ai  cij wij
F ( x)  ( x)  
or

j
• McRae et al. have added the requirement that
values of {wij} are independent of i
Ai   cij w j
j
• This is a discriminative model -- fits
P({MV,RR}|string)
Ai  c j w
• A more commonly
seen
assumption (e.g., in much
ij
j
of statistics) is that
values of {c } are independent

Conclusions for Competition
• General picture: there is support for deployment
of a variety of probabilistic information in online
reading
• More detailed idea: when multiple possible
analyses are salient, processing gets
{slower/more difficult}
• This is uncertainty only about what has been said
• Specific formalization: a type of linear model is
coupled with feedback to simulate online
competition
• cycles-to-threshold of competition process
*CI model predictions
implementation
downloadable
from course we
determines
about
reading time
http://homepages.inf.ed.ac.uk/rlevy/esslli2006
• [one more brainstorming session…]
A recent challenge to competition
• Sometimes, ambiguity seems to facilitate
processing:
The daughteri of the colonelj who shot himself*i/j
The daughteri of the colonelj who shot herselfi/*j
harder
easier
The soni
of the colonelj who shot himselfi/j
• [colonel has stereotypical male gender]
• Argued to be problematic for parallel constraintbased competition models
(Traxler et al. 1998; Van Gompel et al. 2001,
A recent challenge to competition
(2)
The soni of the colonelj who shot himselfi/j
• The reasoning here is that when there are two
valid attachments for the RC, there is a syntactic
ambiguity that doesn’t exist when there is only
one valid attachment
• This has also been demonstrated for other
disambiguations, e.g., animacy-based:
The bodyguardi of the governorj
The governori of the provincei
easier
retiringi/j
retiringi/*j
harder
Where is CI on the serial↔parallel
gradient?
• CI is widely recognized as a parallel model
• But because of the positive feedback cycle, it can
also behave like a serial model!
• [explain on the board]
• In some ways it is intermediate serial/parallel:
• After reading of wi is complete, the top-ranked
interpretation I1 will usually* have activation a1≥p1
• This can cause pseudo-serial behavior
• We saw this at “the detective” in good-agent
condition
High-level issues
• Granularity level of competing candidates
• the old question of granularity for estimating
probs
• also: more candidates → often more cycles to
threshold
• Window size for threshold requirement
• self-paced reading: the region displayed
• eye-tracking: fixation? word? (Spivey & Tanenhaus
1998)
Further reading
• Origin of normalized recurrence algorithm:
Spivey-Knowlton’s 1996 dissertation
• Spivey and Tanenhaus 1998
• Ferretti & McRae 1999
• Green & Mitchell 2006