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Cognitive modeling
Bram Zandbelt
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Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
1.1 Why ask the question in the first place?
Source: Google
Bram Zandbelt
Source: http://www.scientificamerican.com/article/just-a-theory-7-misused-science-words/
1.1 Why ask the question in the first place?
Bram Zandbelt
1.2 Examples of models
Models
Schematic PhysicalSymbolic
Verbal Formal
Bram Zandbelt
1.2 Examples of models
Models
Schematic PhysicalSymbolic
Verbal Formal
Bram Zandbelt
1.3 Definition of a model
Fum et al. (2007) Cog Sys Res 8:135
“A model is a simpler and more abstract version of a
system that keeps its essential features
while omitting unnecessary details”
–Howard Skipper
“A model is a lie that helps you see the truth”
Bram Zandbelt
Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
2.1 Why use models?
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions

Models are tools serving various purposes, including
description, prediction, and explanation
… lead to new experiments

[…] leading to new hypotheses, guiding experiments,
and findings
… and are often complex

Abstractions help to see the big picture
… promote a scientific habit

Formulating models forces you to think logically and
clearly about what you know and don’t know
Bram Zandbelt
Same data,
different conclusions
Optimist:
glass
half
full
Pessimist:
glass
half
empty
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
2.1 Why use models?
Bram Zandbelt
Sources: fieltriptoolbox.org
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
… and are often complex

Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
Sources: Van Belle et al. (2014) Neuroimage; fieltriptoolbox.org
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
… and are often complex

Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions

Models are tools serving various purposes, including
description, prediction, and explanation
… and are often complex

Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions

Models are tools serving various purposes, including
description, prediction, and explanation
… and are often complex

Abstractions help to see the big picture
… promote a scientific habit

Formulating models forces you to think logically and
clearly about what you know and don’t know
2.1 Why use models?
Bram Zandbelt
Data never speak for themselves

A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions

Models are tools serving various purposes, including
description, prediction, and explanation
… lead to new experiments

They suggest novel hypotheses, predicting future
findings, and guide experimental design
… and are often complex

Abstractions help to see the big picture
… promote a scientific habit

Formulating models forces you to think logically and
clearly about what you know and don’t know
2.1 Why use models?
Bram Zandbelt
2.2 Why use (formal) models in cognitive neuroscience?
The brain is a complex system
Complex systems need to be understood at
multiple levels
Bram Zandbelt
Marr’s level Question
1
Computational/
Functional
What is the function? 

Why is it performed?
input i output o
Sources: Marr (1982)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Marr’s level Question
1
Computational/
Functional
What is the function? 

Why is it performed?
2 Algorithm
What algorithm achieves this function?
How are inputs and outputs represented?
f(i)input i output o
Sources: Marr (1982)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Marr’s level Question
1
Computational/
Functional
What is the function? 

Why is it performed?
2 Algorithm
What algorithm achieves this function?
How are inputs and outputs represented?
3 Implementation
How are algorithm and representation
realized physically?
Sources: Marr (1982)
input i output o
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
2.2 Why use (formal) models in cognitive neuroscience?
The brain is a complex system
Complex systems need to be understood at
multiple levels
Models can help to bridge the gap
between brain and behavior
Understanding computation guides research in
the underlying circuits and provides a language
for theories of behavior
Bram Zandbelt
Implementation Algorithm Function
Sources: Carandini (2012) Nat Neurosci
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
The brain is a complex system
Complex systems need to be understood at
multiple levels
Models can take different forms
Verbal: words or box-and-arrow diagrams
Formal: axioms, equations, computer code
2.2 Why use (formal) models in cognitive neuroscience?
Models can help to bridge the gap
between brain and behavior
Understanding computation guides research in
the underlying circuits and provides a language
for theories of behavior
Bram Zandbelt
Cognitive process
input i output o = f(i)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Cognitive processBrain… …
Model inputs
Predicted
behavior
Unobserved
neural/mental
process
N
Brain
signal
detection … …
response
preparation
response
execution
qualitative
fit
RT
Data/Reality
2.2 Why use (formal) models in cognitive neuroscience?
Verbal model
Observed
neural
process
1
Unobserved
neural/mental
process
i
qualitative
constraints
Bram Zandbelt
Brain
Observed
neural
process
1
… …
Model inputs
Predicted
behavior
Unobserved
neural/mental
process
i
Unobserved
neural/mental
process
N
Brain… …
∫dt
qualitative
and
quantitative
fit
Model parameters
RT
qualitative
and
quantitative
constraints
RT
signal
detection
response
preparation
response
execution
2.2 Why use (formal) models in cognitive neuroscience?
Formal model
Data/Reality
Bram Zandbelt
Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Does the conclusion logically follow from the premises?
Premise 1
No police dogs are
vicious
No nutritional things
are inexpensive
No addictive things are
inexpensive
No millionaires are
hard workers
Premise 2
Some highly trained
dogs are vicious
Some vitamin tablets
are inexpensive
Some cigarettes are
inexpensive
Some rich people are
hard works
Conclusion
Therefore, some highly
trained dogs are not
police dogs
Therefore, some
vitamin tablets are not
nutritional
Therefore, some
addictive things are not
cigarettes
Therefore, some
millionaires are not rich
people
Valid
Believable
Valid
Unbelievable
Invalid
Believable
Invalid
Unbelievable
Sources: Evans et al. (1983) Mem Cogn
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Limits of human thinking
(imagination, working memory)
e.g. reasoning about complex systems
Computational power
(in-depth exploration, no memory issues)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Sources: Marder (2014) Ann Rev Neurosci
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Limits of human thinking
(imagination, working memory)
e.g. reasoning about complex systems
Computational power
(in-depth exploration, no memory issues)
Misunderstanding
(hidden assumptions, vague definitions)
e.g. concept of inhibition
Computer code and equations
(explicit assumptions, precise definitions)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Sources: Aron (2007) Neuroscientist; see also MacLeod et al. (2003) in Psychology of learning and motivation, B. Ross, Ed., vol. 43, pp. 163–214.
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
Sources: Lewandowsky, S. (1993) Psych Sci; Jacobs, A. M., & Grainger, J. (1994). J Exp Psychol um Percept Perform; Ulrich (2009) in: Rösler, Ranganath, Röder,
Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford University Press
… overspecification of irrelevant details

Obscures the discovery of general principles
… less suitable for new research fields

Sometimes we only have vague ideas
… overparameterization

Good fits can be bad; simpler models may exist
… realism comes at a cost

Bonini’s paradox: as a model becomes more realistic, it
becomes increasingly difficult to understand
2.2 Why use (formal) models in cognitive neuroscience?
Formal models have limitations, too:
Bram Zandbelt
Psychophysical models

Relate physical stimuli to sensation/perception
2.3 Formal models come in different flavors
Bram Zandbelt
Expected Utility
Axiomatic models

Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models

Relate physical stimuli to sensation/perception
2.3 Formal models come in different flavors
Source: von Neumann & Morgenstern (1944)
Bram Zandbelt
Axiomatic models

Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models

Relate physical stimuli to sensation/perception
Algebraic models

Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
2.3 Formal models come in different flavors
Source: Logan (1988) Psych Rev; Logan (2002) Psych Rev
Bram Zandbelt
Algorithmic models

Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models

Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models

Relate physical stimuli to sensation/perception
Algebraic models

Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
2.3 Formal models come in different flavors
Bram Zandbelt
Algorithmic models

Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models

Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models

Relate physical stimuli to sensation/perception
Algebraic models

Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
Connectionist models

Describe behavior with multilayer networks of
interconnected units
2.3 Formal models come in different flavors
Bram Zandbelt
Schizophrenia is not a rare disorder. It has a lifetime
risk of ~0.7%1
(similar to that of rheumatoid arthritis).
It has a genetic basis, but the importance of social fac-
tors in its emergence is also recognized. Schizophrenia
is devastating for both sufferers and their carers.
Patients are likely to be unemployed or fail to fulfil
their original potential. Contact with the police result-
ing from socially unacceptable behaviour is common,
and the risk of suicide is high. The first episode typi-
cally occurs when patients are in their mid 20s, and
most sufferers never fully recover. Although drug treat-
ment and, more recently, cognitive behavioural therapy
can reduce suffering, there is as yet no cure for this
disorder. Furthermore, although schizophrenia clearly
has a strong biological component (BOX 1), no diagnos-
tic physiological markers have been found. Diagnosis,
therefore, is made on the basis of symptoms described
by the patient, signs observed by the clinician and the
history of the disorder (BOX 2).
The most striking and characteristic features
of the disorder are hallucinations and delusions.
Hallucinations are false perceptions, such as patients
hearing people talking about them or hearing their
thoughts spoken aloud (TABLE 1). Delusions are per-
sistent bizarre or irrational beliefs that are not easily
understood in terms of an individual’s social or cul-
tural background. For example, patients may believe
that other people can hear their thoughts or that
the government is monitoring their every action.
Hallucinations and delusions are examples of positive
symptoms, which are so called because the abnormal-
ity lies in their presence. Positive symptoms contrast
with negative symptoms (also known as signs), which
are defined by the absence of normal functions, as is
the case with reduced speech output (alogia) or loss
of motivation (avolition). There is evidence that posi-
tive and negative symptoms reflect different underlying
physiological disorders2,3
. Although an important chal-
lenge for future work will be to find an explanation
for both positive and negative symptoms, we believe
that the current state of the field and the fact that these
symptoms seem to dissociate across groups of patients
make it sensible to confine our ideas in this Review
to the positive symptoms. Our aim is to consider how
abnormal physiological responses in the brains of peo-
ple with schizophrenia might be linked to the positive
symptoms that they experience. We show that a com-
mon mechanism, involving minimization of predic-
tion error, may underlie perception and inference, and
that a disruption in this mechanism may cause both
abnormal perceptions (hallucinations) and abnormal
beliefs (delusions). We are not concerned with the ulti-
mate causes of the disorder, in which both genetic and
environmental factors play a part.
*University of Cambridge,
Department of Psychiatry,
Addenbrooke’s Hospital,
Hills Road, Cambridge,
CB2 2QQ, UK.
‡
Centre for Functionally
Integrative Neuroscience,
Aarhus University Hospital,
8000 Aarhus C, Denmark.
§
Wellcome Trust Centre for
Neuroimaging, Functional
Imaging Laboratory,
University College London,
London, WC1N 3BG, UK.
Correspondence to C.D.F.
e-mail: c.frith@ucl.ac.uk
doi:10.1038/nrn2536
Published online
3 December 2008
Cognitive behavioural
therapy
A form of psychotherapy in
which the patient is
encouraged to examine the
cognitive processes by which
they arrive at a particular state
of mind, and to change these
processes together with the
accompanying behaviours that
may reinforce them.
Perceiving is believing: a Bayesian
approach to explaining the positive
symptoms of schizophrenia
Paul C. Fletcher* and Chris D. Frith‡§
Abstract | Advances in cognitive neuroscience offer us new ways to understand the
symptoms of mental illness by uniting basic neurochemical and neurophysiological
observations with the conscious experiences that characterize these symptoms. Cognitive
theories about the positive symptoms of schizophrenia — hallucinations and delusions —
have tended to treat perception and belief formation as distinct processes. However, recent
advances in computational neuroscience have led us to consider the unusual perceptual
experiences of patients and their sometimes bizarre beliefs as part of the same core
abnormality — a disturbance in error-dependent updating of inferences and beliefs about
the world. We suggest that it is possible to understand these symptoms in terms of a
disturbed hierarchical Bayesian framework, without recourse to separate considerations of
experience and belief.
REVIEWS
48 | JANUARY 2009 | VOLUME 10 www.nature.com/reviews/neuro
Source: Fletcher & Frith (2009) Nat Rev Neurosci
Algorithmic models

Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models

Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models

Relate physical stimuli to sensation/perception
Algebraic models

Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
Connectionist models

Describe behavior with multilayer networks of
interconnected units
Bayesian models

Assume that we make inferences using Bayesian
statistics
2.3 Formal models come in different flavors
Bram Zandbelt
Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
Steps in cognitive modeling
Formulation
Estimation
Evaluation
Bram Zandbelt
3.1 How to formulate a model?
Core assumptions (A)

Based on conceptual theory of
underlying mechanism
Auxilliary assumptions

Conceptual theories often lack
important details
Definitions

Of dependent variables, such as RT
Theorems (T)

Combine assumptions & definitions to
derive abstract predictions
Predictions (P)

Add parameters for concrete predictions
that can be compared with data
Parameters

Tuning knobs of the model
Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford
University Press
Bram Zandbelt
Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford
University Press
Model of cross modal temporal discrimination
3.1 How to formulate a model?
Bram Zandbelt
3.1 How to formulate a model?
Bram Zandbelt
3.2 How to estimate a model?
Main estimation methods: LSE & MLE

LSE: finds parameters that most accurately
describe the data
MLE: finds parameters that most likely have
generated the data
Least-squares estimation (LSE)
Maximum likelihood estimation (LSE)
Bram Zandbelt
3.2 How to estimate a model?
Source: Lewandowsky, S., & Farrell, S. (2010). Computational modeling in cognition: Principles and practice. Sage.
Main estimation methods: LSE & MLE

LSE: finds parameters that most accurately
describe the data
MLE: finds parameters that most likely have
generated the data
Various approaches to find best fit

Grid search - easy but laborious
Simplex - efficient but risk ending in local
minimum
Simulated annealing, genetic algorithm - likely
to end in global minimum but time-consumingparam X
param Y
cost
fun
Bram Zandbelt
3.3 How to evaluate a model?
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little; see also Jacobs & Grainger (1994) J Exp
Psychol Hum Percept Perform
Bram Zandbelt
Goodness of fit can be quantified with
likelihood or root mean squared error
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
3.3 How to evaluate a model?
Bram Zandbelt
Complexity can be quantified with
Akaike and Bayesian Information Criterion (AIC,BIC)
Goodness of fit
Penalty for
free parameters
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
3.3 How to evaluate a model?
Bram Zandbelt
Generalizability can be quantified with
cross validation
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
Same model,
new data
3.3 How to evaluate a model?
Bram Zandbelt
Further reading
Lewandowsky, S., & Farrell, S. (2010). Computational
modeling in cognition: Principles and practice. Sage.
Cavagnaro, D. R., Myung, J. I., & Pitt, M. A. (2010).
Mathematical modeling. In T. D. Little (Ed.), The Oxford
Handbook of Quantitative Methods (Vol. 1, pp. 438–
453). New York, NY: Oxford University Press.
C H A P T E R
21
Mathematical Modeling
Daniel R. Cavagnaro, Jay I. Myung, and Mark A. Pitt
Abstract
Explanations of human behavior are most often presented in a verbal form as theories. Psychologists
can also harness the power and precision of mathematics by explaining behavior quantitatively. This
chapter introduces the reader to how this is done and the advantages of doing so. It begins by
contrasting mathematical modeling with hypothesis testing to highlight how the two methods of
knowledge acquisition differ. The many styles of modeling are then surveyed, along with their
advantages and disadvantages. This is followed by an in-depth example of how to create a
mathematical model and fit it to experimental data. Issues in evaluating models are discussed, including
a survey of quantitative methods of model selection. Particular attention is paid to the concept of
generalizability and the trade-off of model fit with model complexity. The chapter closes by describing
some of the challenges for the discipline in the years ahead.
Key Words: Cognitive modeling, model testing, model evaluation, model comparison
Introduction
Psychologists study behavior. Data, acquired
through experimentation, are used to build theo-
ries that explain behavior, which in turn provide
meaning and understanding. Because behavior is
complex, a complete theory of any behavior (e.g.,
depression, reasoning, motivation) is likely to be
complex as well, having many variables and condi-
tions that influence it.
Mathematical models are tools that assist in the-
ory development and testing. Models are theories, or
parts of theories, formalized mathematically. They
complement theorizing in many ways, as discussed
in the following pages, but their ultimate goal
is to promote understanding of the theory, and
thus behavior, by taking advantage of the precision
offered by mathematics. Although they have been
part of psychology since its inception, their popu-
larity began to rise in the 1950s and has increased
substantially since the 1980s, in part because of the
introduction of personal computers. This interest is
not an accident or fad. Every style of model that
has been introduced has had a significant impact
in its discipline, and sometimes far beyond that.
After reading this chapter, the reader should begin
to understand why.
This chapter is written as a first introduction to
mathematical modeling in psychology for those with
little or no prior experience with the topic. Our aim
is to provide a good conceptual understanding of
the topic and make the reader aware of some of
the fundamental issues in mathematical modeling
but not necessarily to provide an in-depth step-by-
step tutorial on how to actually build and evaluate a
mathematical model from scratch. In doing so, we
assume no more of the reader than a year-long course
in graduate-level statistics. For related publications
on the topic, the reader is directed to Busemeyer and
Diederich (2010), Fum, Del Missier, and Stocco
(2007), and Myung and Pitt (2002). In particular,
437
Bram Zandbelt
Feel free to distribute, remix, tweak, and build upon these slides.
Please attribute Bram Zandbelt with a link to
http://www.slideshare.net/bramzandbelt/cognitive-modeling
Except where otherwise noted, this work is licensed under
http://creativecommons.org/licenses/by/4.0/
Bram Zandbelt

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Cognitive modeling

  • 2. Preview Cognitive modeling 1. What is a model? 1.1. Why ask this question in the first place? 1.2. Examples of models 1.3. Definition of a model 2. Why use models? 2.1. Why use models in general? 2.2. Why use models in cognitive neuroscience? 2.3. Formal models come in di erent flavors 3. How to use models? 3.1. How to formulate a model? 3.2. How to estimate a model? 3.3. How to evaluate a model? Bram Zandbelt
  • 3. Preview Cognitive modeling 1. What is a model? 1.1. Why ask this question in the first place? 1.2. Examples of models 1.3. Definition of a model 2. Why use models? 2.1. Why use models in general? 2.2. Why use models in cognitive neuroscience? 2.3. Formal models come in di erent flavors 3. How to use models? 3.1. How to formulate a model? 3.2. How to estimate a model? 3.3. How to evaluate a model? Bram Zandbelt
  • 4. 1.1 Why ask the question in the first place? Source: Google Bram Zandbelt
  • 6. 1.2 Examples of models Models Schematic PhysicalSymbolic Verbal Formal Bram Zandbelt
  • 7. 1.2 Examples of models Models Schematic PhysicalSymbolic Verbal Formal Bram Zandbelt
  • 8. 1.3 Definition of a model Fum et al. (2007) Cog Sys Res 8:135 “A model is a simpler and more abstract version of a system that keeps its essential features while omitting unnecessary details” –Howard Skipper “A model is a lie that helps you see the truth” Bram Zandbelt
  • 9. Preview Cognitive modeling 1. What is a model? 1.1. Why ask this question in the first place? 1.2. Examples of models 1.3. Definition of a model 2. Why use models? 2.1. Why use models in general? 2.2. Why use models in cognitive neuroscience? 2.3. Formal models come in di erent flavors 3. How to use models? 3.1. How to formulate a model? 3.2. How to estimate a model? 3.3. How to evaluate a model? Bram Zandbelt
  • 10. 2.1 Why use models? Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data Models address scientific questions
 Models are tools serving various purposes, including description, prediction, and explanation … lead to new experiments
 […] leading to new hypotheses, guiding experiments, and findings … and are often complex
 Abstractions help to see the big picture … promote a scientific habit
 Formulating models forces you to think logically and clearly about what you know and don’t know Bram Zandbelt
  • 11. Same data, different conclusions Optimist: glass half full Pessimist: glass half empty Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data 2.1 Why use models? Bram Zandbelt
  • 12. Sources: fieltriptoolbox.org Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data … and are often complex
 Abstractions help to see the big picture 2.1 Why use models? Bram Zandbelt
  • 13. Sources: Van Belle et al. (2014) Neuroimage; fieltriptoolbox.org Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data … and are often complex
 Abstractions help to see the big picture 2.1 Why use models? Bram Zandbelt
  • 14. Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data Models address scientific questions
 Models are tools serving various purposes, including description, prediction, and explanation … and are often complex
 Abstractions help to see the big picture 2.1 Why use models? Bram Zandbelt
  • 15. Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data Models address scientific questions
 Models are tools serving various purposes, including description, prediction, and explanation … and are often complex
 Abstractions help to see the big picture … promote a scientific habit
 Formulating models forces you to think logically and clearly about what you know and don’t know 2.1 Why use models? Bram Zandbelt
  • 16. Data never speak for themselves
 A framework, theory, causal model, logical construct, perception of the world, etc. is necessary to make sense of data Models address scientific questions
 Models are tools serving various purposes, including description, prediction, and explanation … lead to new experiments
 They suggest novel hypotheses, predicting future findings, and guide experimental design … and are often complex
 Abstractions help to see the big picture … promote a scientific habit
 Formulating models forces you to think logically and clearly about what you know and don’t know 2.1 Why use models? Bram Zandbelt
  • 17. 2.2 Why use (formal) models in cognitive neuroscience? The brain is a complex system Complex systems need to be understood at multiple levels Bram Zandbelt
  • 18. Marr’s level Question 1 Computational/ Functional What is the function? 
 Why is it performed? input i output o Sources: Marr (1982) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 19. Marr’s level Question 1 Computational/ Functional What is the function? 
 Why is it performed? 2 Algorithm What algorithm achieves this function? How are inputs and outputs represented? f(i)input i output o Sources: Marr (1982) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 20. Marr’s level Question 1 Computational/ Functional What is the function? 
 Why is it performed? 2 Algorithm What algorithm achieves this function? How are inputs and outputs represented? 3 Implementation How are algorithm and representation realized physically? Sources: Marr (1982) input i output o 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 21. 2.2 Why use (formal) models in cognitive neuroscience? The brain is a complex system Complex systems need to be understood at multiple levels Models can help to bridge the gap between brain and behavior Understanding computation guides research in the underlying circuits and provides a language for theories of behavior Bram Zandbelt
  • 22. Implementation Algorithm Function Sources: Carandini (2012) Nat Neurosci 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 23. The brain is a complex system Complex systems need to be understood at multiple levels Models can take different forms Verbal: words or box-and-arrow diagrams Formal: axioms, equations, computer code 2.2 Why use (formal) models in cognitive neuroscience? Models can help to bridge the gap between brain and behavior Understanding computation guides research in the underlying circuits and provides a language for theories of behavior Bram Zandbelt
  • 24. Cognitive process input i output o = f(i) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 25. Cognitive processBrain… … Model inputs Predicted behavior Unobserved neural/mental process N Brain signal detection … … response preparation response execution qualitative fit RT Data/Reality 2.2 Why use (formal) models in cognitive neuroscience? Verbal model Observed neural process 1 Unobserved neural/mental process i qualitative constraints Bram Zandbelt
  • 26. Brain Observed neural process 1 … … Model inputs Predicted behavior Unobserved neural/mental process i Unobserved neural/mental process N Brain… … ∫dt qualitative and quantitative fit Model parameters RT qualitative and quantitative constraints RT signal detection response preparation response execution 2.2 Why use (formal) models in cognitive neuroscience? Formal model Data/Reality Bram Zandbelt
  • 27. Potential problems of verbal models Solutions from formal models Flawed reasoning (inconsistencies, contradictions, gaps) e.g. belief bias Formal system (clarity, coherence, completeness) Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 28. Does the conclusion logically follow from the premises? Premise 1 No police dogs are vicious No nutritional things are inexpensive No addictive things are inexpensive No millionaires are hard workers Premise 2 Some highly trained dogs are vicious Some vitamin tablets are inexpensive Some cigarettes are inexpensive Some rich people are hard works Conclusion Therefore, some highly trained dogs are not police dogs Therefore, some vitamin tablets are not nutritional Therefore, some addictive things are not cigarettes Therefore, some millionaires are not rich people Valid Believable Valid Unbelievable Invalid Believable Invalid Unbelievable Sources: Evans et al. (1983) Mem Cogn 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 29. Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991) Potential problems of verbal models Solutions from formal models Flawed reasoning (inconsistencies, contradictions, gaps) e.g. belief bias Formal system (clarity, coherence, completeness) Limits of human thinking (imagination, working memory) e.g. reasoning about complex systems Computational power (in-depth exploration, no memory issues) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 30. Sources: Marder (2014) Ann Rev Neurosci 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 31. Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991) Potential problems of verbal models Solutions from formal models Flawed reasoning (inconsistencies, contradictions, gaps) e.g. belief bias Formal system (clarity, coherence, completeness) Limits of human thinking (imagination, working memory) e.g. reasoning about complex systems Computational power (in-depth exploration, no memory issues) Misunderstanding (hidden assumptions, vague definitions) e.g. concept of inhibition Computer code and equations (explicit assumptions, precise definitions) 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 32. Sources: Aron (2007) Neuroscientist; see also MacLeod et al. (2003) in Psychology of learning and motivation, B. Ross, Ed., vol. 43, pp. 163–214. 2.2 Why use (formal) models in cognitive neuroscience? Bram Zandbelt
  • 33. Sources: Lewandowsky, S. (1993) Psych Sci; Jacobs, A. M., & Grainger, J. (1994). J Exp Psychol um Percept Perform; Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford University Press … overspecification of irrelevant details
 Obscures the discovery of general principles … less suitable for new research fields
 Sometimes we only have vague ideas … overparameterization
 Good fits can be bad; simpler models may exist … realism comes at a cost
 Bonini’s paradox: as a model becomes more realistic, it becomes increasingly difficult to understand 2.2 Why use (formal) models in cognitive neuroscience? Formal models have limitations, too: Bram Zandbelt
  • 34. Psychophysical models
 Relate physical stimuli to sensation/perception 2.3 Formal models come in different flavors Bram Zandbelt
  • 35. Expected Utility Axiomatic models
 Replace the phenomenon to be modeled with logical propositions from which behavior can be derived Psychophysical models
 Relate physical stimuli to sensation/perception 2.3 Formal models come in different flavors Source: von Neumann & Morgenstern (1944) Bram Zandbelt
  • 36. Axiomatic models
 Replace the phenomenon to be modeled with logical propositions from which behavior can be derived Psychophysical models
 Relate physical stimuli to sensation/perception Algebraic models
 Simple equations that describe how input stimuli and model parameters are combined to produce behavior 2.3 Formal models come in different flavors Source: Logan (1988) Psych Rev; Logan (2002) Psych Rev Bram Zandbelt
  • 37. Algorithmic models
 Defined in terms of a computer simulation that describes how processes interact to produce behavior Axiomatic models
 Replace the phenomenon to be modeled with logical propositions from which behavior can be derived Psychophysical models
 Relate physical stimuli to sensation/perception Algebraic models
 Simple equations that describe how input stimuli and model parameters are combined to produce behavior 2.3 Formal models come in different flavors Bram Zandbelt
  • 38. Algorithmic models
 Defined in terms of a computer simulation that describes how processes interact to produce behavior Axiomatic models
 Replace the phenomenon to be modeled with logical propositions from which behavior can be derived Psychophysical models
 Relate physical stimuli to sensation/perception Algebraic models
 Simple equations that describe how input stimuli and model parameters are combined to produce behavior Connectionist models
 Describe behavior with multilayer networks of interconnected units 2.3 Formal models come in different flavors Bram Zandbelt
  • 39. Schizophrenia is not a rare disorder. It has a lifetime risk of ~0.7%1 (similar to that of rheumatoid arthritis). It has a genetic basis, but the importance of social fac- tors in its emergence is also recognized. Schizophrenia is devastating for both sufferers and their carers. Patients are likely to be unemployed or fail to fulfil their original potential. Contact with the police result- ing from socially unacceptable behaviour is common, and the risk of suicide is high. The first episode typi- cally occurs when patients are in their mid 20s, and most sufferers never fully recover. Although drug treat- ment and, more recently, cognitive behavioural therapy can reduce suffering, there is as yet no cure for this disorder. Furthermore, although schizophrenia clearly has a strong biological component (BOX 1), no diagnos- tic physiological markers have been found. Diagnosis, therefore, is made on the basis of symptoms described by the patient, signs observed by the clinician and the history of the disorder (BOX 2). The most striking and characteristic features of the disorder are hallucinations and delusions. Hallucinations are false perceptions, such as patients hearing people talking about them or hearing their thoughts spoken aloud (TABLE 1). Delusions are per- sistent bizarre or irrational beliefs that are not easily understood in terms of an individual’s social or cul- tural background. For example, patients may believe that other people can hear their thoughts or that the government is monitoring their every action. Hallucinations and delusions are examples of positive symptoms, which are so called because the abnormal- ity lies in their presence. Positive symptoms contrast with negative symptoms (also known as signs), which are defined by the absence of normal functions, as is the case with reduced speech output (alogia) or loss of motivation (avolition). There is evidence that posi- tive and negative symptoms reflect different underlying physiological disorders2,3 . Although an important chal- lenge for future work will be to find an explanation for both positive and negative symptoms, we believe that the current state of the field and the fact that these symptoms seem to dissociate across groups of patients make it sensible to confine our ideas in this Review to the positive symptoms. Our aim is to consider how abnormal physiological responses in the brains of peo- ple with schizophrenia might be linked to the positive symptoms that they experience. We show that a com- mon mechanism, involving minimization of predic- tion error, may underlie perception and inference, and that a disruption in this mechanism may cause both abnormal perceptions (hallucinations) and abnormal beliefs (delusions). We are not concerned with the ulti- mate causes of the disorder, in which both genetic and environmental factors play a part. *University of Cambridge, Department of Psychiatry, Addenbrooke’s Hospital, Hills Road, Cambridge, CB2 2QQ, UK. ‡ Centre for Functionally Integrative Neuroscience, Aarhus University Hospital, 8000 Aarhus C, Denmark. § Wellcome Trust Centre for Neuroimaging, Functional Imaging Laboratory, University College London, London, WC1N 3BG, UK. Correspondence to C.D.F. e-mail: c.frith@ucl.ac.uk doi:10.1038/nrn2536 Published online 3 December 2008 Cognitive behavioural therapy A form of psychotherapy in which the patient is encouraged to examine the cognitive processes by which they arrive at a particular state of mind, and to change these processes together with the accompanying behaviours that may reinforce them. Perceiving is believing: a Bayesian approach to explaining the positive symptoms of schizophrenia Paul C. Fletcher* and Chris D. Frith‡§ Abstract | Advances in cognitive neuroscience offer us new ways to understand the symptoms of mental illness by uniting basic neurochemical and neurophysiological observations with the conscious experiences that characterize these symptoms. Cognitive theories about the positive symptoms of schizophrenia — hallucinations and delusions — have tended to treat perception and belief formation as distinct processes. However, recent advances in computational neuroscience have led us to consider the unusual perceptual experiences of patients and their sometimes bizarre beliefs as part of the same core abnormality — a disturbance in error-dependent updating of inferences and beliefs about the world. We suggest that it is possible to understand these symptoms in terms of a disturbed hierarchical Bayesian framework, without recourse to separate considerations of experience and belief. REVIEWS 48 | JANUARY 2009 | VOLUME 10 www.nature.com/reviews/neuro Source: Fletcher & Frith (2009) Nat Rev Neurosci Algorithmic models
 Defined in terms of a computer simulation that describes how processes interact to produce behavior Axiomatic models
 Replace the phenomenon to be modeled with logical propositions from which behavior can be derived Psychophysical models
 Relate physical stimuli to sensation/perception Algebraic models
 Simple equations that describe how input stimuli and model parameters are combined to produce behavior Connectionist models
 Describe behavior with multilayer networks of interconnected units Bayesian models
 Assume that we make inferences using Bayesian statistics 2.3 Formal models come in different flavors Bram Zandbelt
  • 40. Preview Cognitive modeling 1. What is a model? 1.1. Why ask this question in the first place? 1.2. Examples of models 1.3. Definition of a model 2. Why use models? 2.1. Why use models in general? 2.2. Why use models in cognitive neuroscience? 2.3. Formal models come in di erent flavors 3. How to use models? 3.1. How to formulate a model? 3.2. How to estimate a model? 3.3. How to evaluate a model? Bram Zandbelt
  • 41. Steps in cognitive modeling Formulation Estimation Evaluation Bram Zandbelt
  • 42. 3.1 How to formulate a model? Core assumptions (A)
 Based on conceptual theory of underlying mechanism Auxilliary assumptions
 Conceptual theories often lack important details Definitions
 Of dependent variables, such as RT Theorems (T)
 Combine assumptions & definitions to derive abstract predictions Predictions (P)
 Add parameters for concrete predictions that can be compared with data Parameters
 Tuning knobs of the model Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford University Press Bram Zandbelt
  • 43. Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford University Press Model of cross modal temporal discrimination 3.1 How to formulate a model? Bram Zandbelt
  • 44. 3.1 How to formulate a model? Bram Zandbelt
  • 45. 3.2 How to estimate a model? Main estimation methods: LSE & MLE
 LSE: finds parameters that most accurately describe the data MLE: finds parameters that most likely have generated the data Least-squares estimation (LSE) Maximum likelihood estimation (LSE) Bram Zandbelt
  • 46. 3.2 How to estimate a model? Source: Lewandowsky, S., & Farrell, S. (2010). Computational modeling in cognition: Principles and practice. Sage. Main estimation methods: LSE & MLE
 LSE: finds parameters that most accurately describe the data MLE: finds parameters that most likely have generated the data Various approaches to find best fit
 Grid search - easy but laborious Simplex - efficient but risk ending in local minimum Simulated annealing, genetic algorithm - likely to end in global minimum but time-consumingparam X param Y cost fun Bram Zandbelt
  • 47. 3.3 How to evaluate a model? Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little; see also Jacobs & Grainger (1994) J Exp Psychol Hum Percept Perform Bram Zandbelt
  • 48. Goodness of fit can be quantified with likelihood or root mean squared error Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little 3.3 How to evaluate a model? Bram Zandbelt
  • 49. Complexity can be quantified with Akaike and Bayesian Information Criterion (AIC,BIC) Goodness of fit Penalty for free parameters Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little 3.3 How to evaluate a model? Bram Zandbelt
  • 50. Generalizability can be quantified with cross validation Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little Same model, new data 3.3 How to evaluate a model? Bram Zandbelt
  • 51. Further reading Lewandowsky, S., & Farrell, S. (2010). Computational modeling in cognition: Principles and practice. Sage. Cavagnaro, D. R., Myung, J. I., & Pitt, M. A. (2010). Mathematical modeling. In T. D. Little (Ed.), The Oxford Handbook of Quantitative Methods (Vol. 1, pp. 438– 453). New York, NY: Oxford University Press. C H A P T E R 21 Mathematical Modeling Daniel R. Cavagnaro, Jay I. Myung, and Mark A. Pitt Abstract Explanations of human behavior are most often presented in a verbal form as theories. Psychologists can also harness the power and precision of mathematics by explaining behavior quantitatively. This chapter introduces the reader to how this is done and the advantages of doing so. It begins by contrasting mathematical modeling with hypothesis testing to highlight how the two methods of knowledge acquisition differ. The many styles of modeling are then surveyed, along with their advantages and disadvantages. This is followed by an in-depth example of how to create a mathematical model and fit it to experimental data. Issues in evaluating models are discussed, including a survey of quantitative methods of model selection. Particular attention is paid to the concept of generalizability and the trade-off of model fit with model complexity. The chapter closes by describing some of the challenges for the discipline in the years ahead. Key Words: Cognitive modeling, model testing, model evaluation, model comparison Introduction Psychologists study behavior. Data, acquired through experimentation, are used to build theo- ries that explain behavior, which in turn provide meaning and understanding. Because behavior is complex, a complete theory of any behavior (e.g., depression, reasoning, motivation) is likely to be complex as well, having many variables and condi- tions that influence it. Mathematical models are tools that assist in the- ory development and testing. Models are theories, or parts of theories, formalized mathematically. They complement theorizing in many ways, as discussed in the following pages, but their ultimate goal is to promote understanding of the theory, and thus behavior, by taking advantage of the precision offered by mathematics. Although they have been part of psychology since its inception, their popu- larity began to rise in the 1950s and has increased substantially since the 1980s, in part because of the introduction of personal computers. This interest is not an accident or fad. Every style of model that has been introduced has had a significant impact in its discipline, and sometimes far beyond that. After reading this chapter, the reader should begin to understand why. This chapter is written as a first introduction to mathematical modeling in psychology for those with little or no prior experience with the topic. Our aim is to provide a good conceptual understanding of the topic and make the reader aware of some of the fundamental issues in mathematical modeling but not necessarily to provide an in-depth step-by- step tutorial on how to actually build and evaluate a mathematical model from scratch. In doing so, we assume no more of the reader than a year-long course in graduate-level statistics. For related publications on the topic, the reader is directed to Busemeyer and Diederich (2010), Fum, Del Missier, and Stocco (2007), and Myung and Pitt (2002). In particular, 437 Bram Zandbelt
  • 52. Feel free to distribute, remix, tweak, and build upon these slides. Please attribute Bram Zandbelt with a link to http://www.slideshare.net/bramzandbelt/cognitive-modeling Except where otherwise noted, this work is licensed under http://creativecommons.org/licenses/by/4.0/ Bram Zandbelt