Setting the stage
1
Introduction
1.1
What does your theory say about your data?
1.2
What do your data say about your theory?
1.3
What do your parameters say about other things?
1.4
What does your expertise say about your parameters?
2
Getting started in
Stan
2.1
Installation in
R
2.2
The anatomy of a
Stan
program
2.2.1
Data block
2.2.2
Transformed data block
2.2.3
Parameters block
2.2.4
Model block
2.2.5
Generated quantities block
2.2.6
The final product
2.3
Estimating a model
2.4
Looking at the results
3
Probabilistic models of behavior
3.1
The problem with deterministic models
3.2
What
is
a probabilistic model?
3.3
Example dataset and model
3.4
Optimal choice plus an error
3.4.1
Estimating a model
3.5
Utility-based models
3.5.1
Estimating a model
3.5.2
Doing something with the estimates
4
Considerations for choosing a prior
4.1
Example model and experiment
4.2
Getting the support right
4.3
Eliciting reasonable priors
4.3.1
Parameter values and the prior pushforward check
4.3.2
Predictions and other derived quantities: The prior predictive check
4.4
Assessing the sampling performance of a prior
4.4.1
Does our model recover its parameters well?
4.4.2
Do we see any pathologies in the estimation process?
4.5
R
code used for this chapter
Building blocks
5
Representative agent and participant-specific models
5.1
Participant-specific models
5.1.1
Example data and economic model
5.1.2
Going to the probabilistic model
5.1.3
A short side quest into canned estimation techniques
5.1.4
Assigning priors
5.1.5
Estimating the model for one participant
5.1.6
Estimating the model for all participants
5.1.7
But we could be learning more!
5.2
Actual representative agent models (pooled models)
6
Hierarchical models
6.1
A random sample of participants walks into your lab
6.2
The anatomy of a basic hierarchical model
6.3
Accounting for unobserved heterogeneity
6.3.1
The last time you will integrate the likelihood, probably
6.3.2
Data augmentation
6.4
A multivariate normal hierarchical model
6.4.1
Decomposing the variance-covariance matrix
6.4.2
Transformed parameters and normal distributions
6.5
Example: again with
Bruhin, Fehr, and Schunk (2019)
6.5.1
No correlation between individual-level parameters
6.5.2
Correlation between individual-level parameters
7
Mixture models
7.1
A menu of models
7.2
Dichotomous and toolbox mixture models
7.3
Coding peculearities
7.4
Example experiment:
Andreoni and Vesterlund (2001)
7.4.1
As basic as it gets
7.4.2
Adding some heterogeneity
7.5
Some code used to estimate the models
8
Model evaluation
8.1
Example dataset and models
8.2
Model posterior probabilities
8.2.1
Implementation using bridge sampling and the
bridgesampling
library
8.3
Cross-validation
8.3.1
Expected Log Predicted Density (ELPD) and other measures of goodness of fit
8.3.2
1-round cross-validation
8.3.3
Leave-one-out cross-validation (LOO)
8.3.4
Approximate LOO
8.3.5
\(k\)
-fold cross-validation
9
Speeding up your
Stan
code
9.1
Example dataset and model
9.2
A really slow way to estimate the model
9.3
Pre-computing things
9.4
Vectorization
9.5
Within-chain parallelization with
reduce_sum()
9.6
Evaluating the implementations
9.6.1
Pre-computing and vectorization
9.6.2
Within-chain parallelization
9.7
R
code to estimate models
9.7.1
Slow, pre-computed, and vectorized models
9.7.2
Parallelized model
Applications
10
Application: Experience-Weighted Attraction
10.1
The model at the individual level
10.2
Some computational and coding issues
10.3
Representative agent models
10.3.1
Prior calibration
10.3.2
The
Stan
model
10.3.3
Results
10.4
Hierarchical model
10.4.1
Prior calibration
10.4.2
The
Stan
model
10.4.3
Results
10.5
Some code used to estimate the models
10.5.1
Loading the data
10.5.2
Estimating the representative agent models
10.5.3
Estimating the hierarchical model
11
Application: Strategy Frequency Estimation
11.1
Simplifying the individual likelihood functions
11.2
Example experiment:
Dal Bó and Fréchette (2011)
11.2.1
The SFEM with homogeneous trembles
11.2.2
Adding heterogeneous trembles and integrating the likelihood
11.3
R
code to do these estimations
12
Application: Strategy frequency estimation with a mixed strategy
12.1
Example dataset and strategies
12.2
The likelihood function
12.3
Implementation in
Stan
12.4
Results
12.5
R
code used to estimate the models
13
Computing Quantal Response Equilibrium
13.1
Overview of quantal response equilibrium
13.2
Computing Quantal Response Equilibrium
13.2.1
Setting up the problem
13.2.2
A predictor-corrector algorithm
13.2.3
Initial conditions
13.2.4
Algorithm tuning
13.3
The predictor-corrector algorithm in
R
13.4
Some example games
13.4.1
Generalized matching pennies
(Ochs 1995)
13.4.2
Stag hunt
13.4.3
\(n\)
-player Volunteer’s Dilemma imposing symmetric strategies
14
Application: Quantal Response Equilibrium and the Volunteer’s Dilemma
(Goeree, Holt, and Smith 2017)
14.1
Solving logit QRE and estimating the model
14.2
Adding some heterogeneity
14.2.1
Computing quantal response equilibrium with heterogeneous parameters
14.2.2
Warm glow volunteering
14.2.3
Duplicate aversion
14.2.4
Results
14.3
R
code to run estimations
15
Application: A Quantal Response Equilibrium with discrete types
15.1
Example dataset and models
15.2
A note on replication
15.3
Three models that make different assumptions about bracketing
15.3.1
Broad bracketing only
15.3.2
Narrow bracketing only
15.3.3
A mixture of broad and narrow bracketing
15.4
Results
15.5
R
code used to estimate these models
16
Application: Level-
\(k\)
models
16.1
Data and game
16.2
The level-
\(k\)
model
16.2.1
The deterministic component of the model
16.2.2
Exact and probabilistic play
16.3
Assigning probabilities to types for each participant separately
16.3.1
The
Stan
program
16.3.2
Prior calibration
16.3.3
Results
16.4
Doing the averaging within one program
16.4.1
The
Stan
program
16.4.2
Results
16.5
A mixture model
16.5.1
Stan
program
16.5.2
A prior for
\(\psi\)
16.5.3
Results
16.6
A mixture over levels and hierarchical nuisance parameters
16.6.1
Prior calibration
16.6.2
Stan
program
16.6.3
Results
16.7
A different assumption about mixing
16.7.1
Stan
program
16.7.2
Results
16.8
R
code to estimate the models
16.8.1
Participant-specific estimation conditional on
\(k\)
with Bayesian model averaging
16.8.2
Participant-specific estimation with a prior over
\(k\)
16.8.3
Mixture model
16.8.4
Hierarchical model
16.8.5
Mixture model with beliefs consistent with truncated type distribution
17
Application: Estimating risk preferences
17.1
Example dataset
17.2
We might not just be interested in the parameters
17.3
Introducing some important models
17.3.1
Expected utility theory
17.3.2
Rank-dependent utility
(Quiggin 1982)
17.3.3
Comparing the certainty equivalents estimated using EUT and RDU
17.4
A hierarchical specification
17.4.1
Population-level estimates
17.4.2
Participant-level estimates
17.5
R
code used to estimate these models
18
Application: Meta-analysis using (some of) the METARET data
18.1
Data
18.2
A basic model
18.3
But the data are really interval-valued!
18.4
Heterogeneous standard deviations
18.5
Student-
\(t\)
distributions, because why not?
18.6
R
code to estimate the models
19
Application: Ranked choices and the Thurstonian model
19.1
The Thurstonian model
19.2
Computational issues
19.3
Example dataset and model
19.4
A representative agent model
19.5
A hierarchical model
19.6
R
code used to run this
Links to data
References
Structural Bayesian Techniques for Experimental and Behavioral Economics
Building blocks