Answer Sheet: Bayesian Data Analysis for Cognitive Science

Answer Sheet

This answer sheet is my solutions to exercises from “An Introduction to Bayesian Data Analysis for Cognitive Science” by Bruno Nicenboim, Daniel J. Schad, and Shravan Vasishth, demonstrating my proficiency in R programming. Chapters are separated into different files.

I welcome corrections for any mistakes and am open to discussions to further enhance understanding and application of these concepts.

Click here to access the book: An Introduction to Bayesian Data Analysis

---

Chapter 5: Bayesian Hierarchical Models

This page presents R codes for exercise 5.8 in chapter 5 Bayesian Hierarchical Models.

Exercise 5.8 The grammaticality illusion

Load the following two data sets:

library(brms)
library(dplyr)
library(bcogsci)

data("df_english")
df_en_5.8 <- df_english #renamed to follow my documentation structure
data("df_dutch")
df_nl_5.8 <- df_dutch

In an offline accuracy rating study on English double center-embedding constructions, Gibson and Thomas (1999) found that grammatical constructions (e.g., example 4a below) were no less acceptable than ungrammatical constructions (e.g., example 4b) where a middle verb phrase (e.g., was cleaning every week) was missing.

(4a) The apartment that the maid who the service had sent over was cleaning every week was well decorated. (4b) *The apartment that the maid who the service had sent over — was well decorated

Based on these results from English, Gibson and Thomas (1999) proposed that working- memory overload leads the comprehender to forget the prediction of the upcoming verb phrase (VP), which reduces working-memory load. This came to be known as the VP- forgetting hypothesis. The prediction is that in the word immediately following the final verb, the grammatical condition (which is coded as +1 in the data frames) should be harder to read than the ungrammatical condition (which is coded as -1).

The design shown above is set up to test this hypothesis using self-paced reading for English (Vasishth et al. 2011), and for Dutch (Frank, Trompenaars, and Vasishth 2015). The data provided are for the critical region (the noun phrase, labeled NP1, following the final verb); this is the region for which the theory predicts differences between the two conditions. We have reading times in log milliseconds.

1. First, fit a linear model with a full hierarchical structure by subjects and by items for the English data. Because we have log milliseconds data, we can simply use the normal likelihood (not the log-normal). What scale will be the parameters be in, milliseconds or log milliseconds?

2. Second, using the posterior for the effect of interest from the English data, derive a prior distribution for the effect in the Dutch data. Then fit two linear mixed models: (i) one model with relatively uninformative priors for (for example, ), and (ii) one model with the prior for you derived from the English data. Do the posterior distributions of the Dutch data’s effect show any important differences given the two priors? If yes, why; if not, why not?

3. Finally, just by looking at the English and Dutch posteriors, what can we say about the VP-forgetting hypothesis? Are the posteriors of the effect from these two languages consistent with the hypothesis?

#Define prior distribution for the model parameters
prior_en <- c(prior(normal(0,1), class = intercept),
              prior(normal(0,1), class = b, coef = grammatical),
              prior(normal(0,1), class = sd),
              prior(lkj(2), class = cor)
)

names(df_en_5.8) #checking column names

## [1] "subject"   "item"      "condition" "NP1"

#Task A
#Fitting the Bayesian hierarchical linear model
fit_en <- brm(
  formula = NP1 ~ condition + (1 + condition|subject) + ( 1 + condition|item),
  data = df_en_5.8,
  family = gaussian(),
  prior = c(prior(normal(0,1), class = "Intercept"),
            prior(normal(0,1), class = "b"),
            prior(normal(0,1), class = "sd")),
  iter = 4000,
  warmup = 1000,
  chains = 4,
  control = list(adapt_delta = 0.95))

## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 15.0.0 (clang-1500.0.40.1)’
## using SDK: ‘MacOSX14.0.sdk’
## clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DUSE_STANC3 -DSTRICT_R_HEADERS  -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION  -D_HAS_AUTO_PTR_ETC=0  -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/opt/R/arm64/include    -fPIC  -falign-functions=64 -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## #include <cmath>
##          ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1

## 
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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summary(fit_en)

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: NP1 ~ condition + (1 + condition | subject) + (1 + condition | item) 
##    Data: df_en_5.8 (Number of observations: 768) 
##   Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
##          total post-warmup draws = 12000
## 
## Multilevel Hyperparameters:
## ~item (Number of levels: 16) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept)                0.03      0.02     0.00     0.09 1.00     5394
## sd(condition)                0.03      0.02     0.00     0.09 1.00     5679
## cor(Intercept,condition)     0.14      0.57    -0.92     0.97 1.00    10280
##                          Tail_ESS
## sd(Intercept)                7106
## sd(condition)                6752
## cor(Intercept,condition)     8173
## 
## ~subject (Number of levels: 48) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept)                0.25      0.03     0.19     0.32 1.00     4628
## sd(condition)                0.12      0.02     0.08     0.17 1.00     6872
## cor(Intercept,condition)     0.81      0.13     0.51     0.99 1.00     4893
##                          Tail_ESS
## sd(Intercept)                7211
## sd(condition)                8901
## cor(Intercept,condition)     6427
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept     6.25      0.04     6.17     6.33 1.00     4313     6432
## condition     0.17      0.03     0.12     0.22 1.00     6727     8264
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.46      0.01     0.43     0.48 1.00    18805     9153
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Task A

• The fixed effect for condition shown as \(0.17\), as the the difference between two conditions is 2 unit (-1 to +1), then the predicted difference between two conditions is \(0.34\). As the data is in log-scale, when \(0.34\) exponentiated, it indicates that the NP1 value in grammatical condition (coded as +1) is 40% higher than the NP1 value in ungrammatical condition (coded as -1)

• \(cor(Intercept,condition) 0.81\) shows that there is variability in how strong the subject influenced by condition. The higher the baseline (intercept), the higher the effect (slope). Meanwhile item shows little variability.

#Extracting the posterior
posterior_summary(fit_en, pars = "b_condition")

##              Estimate  Est.Error      Q2.5     Q97.5
## b_condition 0.1737668 0.02600435 0.1215248 0.2245619

#Fitting model with uninformative prior
fit_nl_uninf <- brm(
  formula = NP1 ~ condition + (1 + condition|subject) + (1 + condition|item),
  data = df_nl_5.8,
  family = gaussian(),
  prior = c(prior(normal(0,1), class = "Intercept"),
            prior(normal(0,1), class = "b", coef = "condition"),
            prior(normal(0,1), class = "sd")),
  iter = 4000,
  warmup = 1000,
  control = list(adapt_delta = 0.95))

## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 15.0.0 (clang-1500.0.40.1)’
## using SDK: ‘MacOSX14.0.sdk’
## clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DUSE_STANC3 -DSTRICT_R_HEADERS  -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION  -D_HAS_AUTO_PTR_ETC=0  -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/opt/R/arm64/include    -fPIC  -falign-functions=64 -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## #include <cmath>
##          ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1

## 
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
## Chain 1: 
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## Chain 4:

#Fitting model with informative prior from English data
fit_nl_info <- brm(
  formula = NP1 ~ condition + (1 + condition|subject) + (1 + condition|item),
  data = df_nl_5.8,
  family = gaussian(),
  prior = c(prior(normal(0,1), class = "Intercept"),
            prior(normal(0.174,0.10), class = "b", coef = "condition"),
            prior(normal(0,1), class = "sd")),
  iter = 4000,
  warmup =1000,
  control = list(adapt_delta = 0.95))

## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 15.0.0 (clang-1500.0.40.1)’
## using SDK: ‘MacOSX14.0.sdk’
## clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DUSE_STANC3 -DSTRICT_R_HEADERS  -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION  -D_HAS_AUTO_PTR_ETC=0  -include '/Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/opt/R/arm64/include    -fPIC  -falign-functions=64 -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## #include <cmath>
##          ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1

## 
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
## Chain 1: 
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#Comparing the posteriors from uninformative and informative model
post_nl_uninf <- posterior_summary(fit_nl_uninf, pars = "b_condition")
post_nl_info <- posterior_summary(fit_nl_info, pars = "b_condition")
print(post_nl_uninf)

##                Estimate Est.Error       Q2.5      Q97.5
## b_condition -0.04536502 0.0320253 -0.1080663 0.01915261

print(post_nl_info)

##                Estimate  Est.Error        Q2.5      Q97.5
## b_condition -0.02446335 0.03146069 -0.08425477 0.04081138

#Plotting to inspect result graphically
library(bayesplot)

draws_nl_uninf <- as.matrix(fit_nl_uninf, pars = "b_condition")
draws_nl_info <- as.matrix(fit_nl_info, pars = "b_condition")

mcmc_dens_overlay(list(uninformative = draws_nl_uninf,
                       informative = draws_nl_info),
                  xlab = "Effect of Condition Comparison")

Task B

• Both the summaries and plot show that uninformative and informative model are almost overlap. This indicates that the data has stronger impact than the priors.

Task C

• The posterior estimate of \(b_condition\) in English data is \(0.174\), which indicates grammatical condition is linked to longer reading times. On the other hand, in Dutch data both uninformative and informative model gave estimates very close to \(0\). This means, the posterior of the effect from English data is consistent with VP-forgetting hypothesis, while the another from Dutch data is not.
• In addition, credible interval in English data (\(0.124, 0.227\)) does not include \(0\), which indicates positive effect. Meanwhile in Dutch models, \(0\) is included, which means VP-forgetting hypothesis is not supported.