## Bayesian linear regression with informed priors
## Define and fit the Bayesian linear regression model
## Print summary of the Bayesian linear regression model
summary(bayesian_model)
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: y ~ x
## Data: plot_data (Number of observations: 200)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.60 0.07 1.45 1.74 1.00 3876 3112
## x 0.81 0.07 0.68 0.94 1.00 3921 2915
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.05 0.05 0.95 1.15 1.00 4031 2761
##
## 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).
## Plot posterior distributions of parameters
plot(bayesian_model)
## Bayesian linear regression with uninformed priors
## Fit the Bayesian linear regression model with default priors
bayesian_model_uninformed <- brm(
formula = y ~ x, # Formula for the regression model
data = plot_data, # Data
family = gaussian(), # Gaussian likelihood for linear regression
iter = 2000, # Number of iterations for sampling
chains = 4, # Number of chains for sampling
cores = 2 # Number of CPU cores to use
)
## Compiling Stan program...
## Trying to compile a simple C file
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 17.0.0 (clang-1700.0.13.5)’
## using SDK: ‘MacOSX15.5.sdk’
## clang -arch x86_64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## 679 | #include <cmath>
## | ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1
## Start sampling
## Print summary of the Bayesian linear regression model with uninformed priors
summary(bayesian_model_uninformed)
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: y ~ x
## Data: plot_data (Number of observations: 200)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.60 0.08 1.45 1.75 1.00 4262 3084
## x 0.81 0.07 0.68 0.94 1.00 4292 2682
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.05 0.05 0.95 1.16 1.00 4537 2991
##
## 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).
## ng)
## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup)
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## Chain 3:
## Chain 3: Elapsed Time: 0.022 seconds (Warm-up)
## Chain 3: 0.016 seconds (Sampling)
## Chain 3: 0.038 seconds (Total)
## Chain 3:
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## Chain 4:
## Chain 4: Elapsed Time: 0.023 seconds (Warm-up)
## Chain 4: 0.019 seconds (Sampling)
## Chain 4: 0.042 seconds (Total)
## Chain 4:
## Moderate prior (centered near truth, reasonable SD)
prior_moderate <- c(
prior(normal(2, 1), class = Intercept),
prior(normal(3, 1), class = b)
)
fit_moderate <- brm(
y ~ x,
data = plot_data,
family = gaussian(),
prior = prior_moderate,
iter = 2000,
chains = 4,
cores = 2,
seed = 123
)
## Compiling Stan program...
## Trying to compile a simple C file
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 17.0.0 (clang-1700.0.13.5)’
## using SDK: ‘MacOSX15.5.sdk’
## clang -arch x86_64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## 679 | #include <cmath>
## | ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1
## Start sampling
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## Chain 1:
## Chain 1: Elapsed Time: 0.023 seconds (Warm-up)
## Chain 1: 0.02 seconds (Sampling)
## Chain 1: 0.043 seconds (Total)
## Chain 1:
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## Chain 2:
## Chain 2: Elapsed Time: 0.026 seconds (Warm-up)
## Chain 2: 0.02 seconds (Sampling)
## Chain 2: 0.046 seconds (Total)
## Chain 2:
summary(fit_moderate)
##
## 2: Iteration: 600 / 2000 [ 30%] (Warmup)
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## Chain 1:
## Chain 1: Elapsed Time: 0.023 seconds (Warm-up)
## Chain 1: 0.02 seconds (Sampling)
## Chain 1: 0.043 seconds (Total)
## Chain 1:
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## Chain 2:
## Chain 2: Elapsed Time: 0.026 seconds (Warm-up)
## Chain 2: 0.02 seconds (Sampling)
## Chain 2: 0.046 seconds (Total)
## Chain 2:
plot(fit_moderate)
## Weak prior (vague belief, large SD)
prior_weak <- c(
prior(normal(0, 10), class = Intercept),
prior(normal(0, 10), class = b)
)
fit_weak <- brm(
y ~ x,
data = plot_data,
family = gaussian(),
prior = prior_weak,
iter = 2000,
chains = 4,
cores = 2,
seed = 123
)
## Compiling Stan program...
## Trying to compile a simple C file
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 17.0.0 (clang-1700.0.13.5)’
## using SDK: ‘MacOSX15.5.sdk’
## clang -arch x86_64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## 679 | #include <cmath>
## | ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1
## Start sampling
summary(fit_weak)
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: y ~ x
## Data: plot_data (Number of observations: 200)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.60 0.08 1.45 1.75 1.00 4135 2600
## x 0.81 0.06 0.68 0.94 1.00 4278 2849
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.05 0.05 0.95 1.16 1.00 4317 2722
##
## 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).
## up)
## Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling)
## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup)
## Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling)
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## Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling)
## Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
## Chain 3:
## Chain 3: Elapsed Time: 0.027 seconds (Warm-up)
## Chain 3: 0.017 seconds (Sampling)
## Chain 3: 0.044 seconds (Total)
## Chain 3:
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## Chain 4:
## Chain 4: Elapsed Time: 0.025 seconds (Warm-up)
## Chain 4: 0.021 seconds (Sampling)
## Chain 4: 0.046 seconds (Total)
## Chain 4:
plot(fit_weak)
## Extreme prior (wrong belief, small SD — strong bias)
prior_extreme <- c(
prior(normal(0, 0.5), class = Intercept),
prior(normal(-2, 0.5), class = b)
)
fit_extreme <- brm(
y ~ x,
data = plot_data,
family = gaussian(),
prior = prior_extreme,
iter = 2000,
chains = 4,
cores = 2,
seed = 123
)
## Compiling Stan program...
## Trying to compile a simple C file
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 17.0.0 (clang-1700.0.13.5)’
## using SDK: ‘MacOSX15.5.sdk’
## clang -arch x86_64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.5-x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/x86_64/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/4.5-x86_64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/Core:19:
## /Library/Frameworks/R.framework/Versions/4.5-x86_64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
## 679 | #include <cmath>
## | ^~~~~~~
## 1 error generated.
## make: *** [foo.o] Error 1
## Start sampling
summary(fit_extreme)
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: y ~ x
## Data: plot_data (Number of observations: 200)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.57 0.07 1.42 1.71 1.00 3627 2663
## x 0.77 0.07 0.64 0.89 1.00 3331 3057
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.05 0.05 0.95 1.15 1.00 3333 2436
##
## 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).
## ng)
## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup)
## Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling)
## Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling)
## Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling)
## Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling)
## Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling)
## Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling)
## Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling)
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## Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling)
## Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
## Chain 3:
## Chain 3: Elapsed Time: 0.025 seconds (Warm-up)
## Chain 3: 0.021 seconds (Sampling)
## Chain 3: 0.046 seconds (Total)
## Chain 3:
## Chain 4: Iteration: 2000 / 2000 [100%] (Sampling)
## Chain 4:
## Chain 4: Elapsed Time: 0.024 seconds (Warm-up)
## Chain 4: 0.02 seconds (Sampling)
## Chain 4: 0.044 seconds (Total)
## Chain 4:
plot(fit_extreme)
# Load packages
library(tidybayes)
##
## Attaching package: 'tidybayes'
## The following objects are masked from 'package:brms':
##
## dstudent_t, pstudent_t, qstudent_t, rstudent_t
library(dplyr)
library(ggplot2)
# --- Reference line for the plot (pick ONE of the two lines below) ---
# (A) If you KNOW the true slope from your simulation (e.g., y = 1.5 + 0.8*x + noise):
# beta1_true <- 0.8
# (B) Otherwise, use the OLS slope from the same data:
m1 <- lm(y ~ x, data = plot_data)
beta1_true <- unname(coef(m1)["x"])
# --- Collect posterior draws for the slope from each model ---
posterior_draws <- bind_rows(
fit_extreme %>% spread_draws(b_x) %>% mutate(prior = "Extreme"),
fit_moderate %>% spread_draws(b_x) %>% mutate(prior = "Moderate"),
fit_weak %>% spread_draws(b_x) %>% mutate(prior = "Weak")
)
# --- Plot posterior densities with a vertical reference line ---
ggplot(posterior_draws, aes(x = b_x, fill = prior, color = prior)) +
geom_density(alpha = 0.4) +
geom_vline(xintercept = beta1_true, linetype = "dashed", color = "black") +
labs(
title = "Posterior Distributions of Slope (b_x)",
subtitle = "Comparison under different priors",
x = "Slope Estimate (b_x)",
y = "Density"
) +
theme_minimal(base_size = 14)
