Explore the impact of prior distributions in a simple single predictor Bayesian regression model
Andi Fugard (@andi@sciences.social)
Last knitted 25 August 2024
Includes required packages:
library(conflicted)
library(brms)Loading required package: Rcpp
Loading 'brms' package (version 2.21.0). Useful instructions
can be found by typing help('brms'). A more detailed introduction
to the package is available through vignette('brms_overview').library(tidyverse)── Attaching core tidyverse packages ─────────────────────────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 3.5.1 ✔ tibble 3.2.1
✔ lubridate 1.9.3 ✔ tidyr 1.3.1
✔ purrr 1.0.2 library(gtable)
library(tidybayes)
library(ggthemes)
library(tictoc)
library(beepr)
conflicts_prefer(
brms::ar,
stats::filter,
stats::lag,
brms::dstudent_t,
brms::pstudent_t,
brms::qstudent_t,
brms::rstudent_t
)[conflicted] Will prefer brms::ar over any other package.
[conflicted] Will prefer stats::filter over any other package.
[conflicted] Will prefer stats::lag over any other package.
[conflicted] Will prefer brms::dstudent_t over any other package.
[conflicted] Will prefer brms::pstudent_t over any other package.
[conflicted] Will prefer brms::qstudent_t over any other package.
[conflicted] Will prefer brms::rstudent_t over any other package.Simulate a tiny dataset with n = 15.
set.seed(1335)
the_n <- 15
dat <- tibble(x = rnorm(the_n, 0, 1),
y = 2 * x + rnorm(the_n, 0, 10))
ggplot(dat, aes(x,y)) +
geom_point()
ols_tiny <- lm(y ~ x, data = dat)
summary(ols_tiny)
Call:
lm(formula = y ~ x, data = dat)
Residuals:
Min 1Q Median 3Q Max
-13.2888 -4.9870 0.0014 4.2663 17.2071
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.365 2.060 -0.662 0.519
x 3.871 2.738 1.414 0.181
Residual standard error: 7.976 on 13 degrees of freedom
Multiple R-squared: 0.1333, Adjusted R-squared: 0.06662
F-statistic: 1.999 on 1 and 13 DF, p-value: 0.1809So the true slope for x is 2, the OLS estimate from this sample is 3.87.
Here are the prior distributions on x I want to try:
priors <- expand.grid(mean = c(0, 2), sd = c(0.1, 1, 10)) |>
mutate(prior = paste0("normal(", mean, ", ", sd, ")"))
priorsFit the models:
tic()
res <- map(
priors$prior,
\(p) brm(
y ~ x,
data = dat,
prior = set_prior(p, class = "b", coef = "x"),
silent = 0
)
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Chain 4: toc()412.34 sec elapsedbeep(5)Save draws from the posteriors of all these models into one data frame:
names(res) <- priors$prior
grab_posteriors <- function(prior) {
res[[prior]] |>
spread_draws(b_x) |>
mutate(prior = prior)
}
all_posteriors <- map(names(res), grab_posteriors) |> bind_rows()Go again with a bigger dataset:
set.seed(1335)
the_n <- 1500
dat_bigger <- tibble(x = rnorm(the_n, 0, 1),
y = 2 * x + rnorm(the_n, 0, 10))
ggplot(dat_bigger, aes(x, y)) +
geom_point()
lm(y ~ x, data = dat_bigger) |> summary()
Call:
lm(formula = y ~ x, data = dat_bigger)
Residuals:
Min 1Q Median 3Q Max
-28.603 -6.372 -0.289 6.817 35.387
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.1716 0.2580 -0.665 0.506
x 1.8528 0.2565 7.224 7.99e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.989 on 1498 degrees of freedom
Multiple R-squared: 0.03367, Adjusted R-squared: 0.03302
F-statistic: 52.19 on 1 and 1498 DF, p-value: 7.986e-13tic()
res_bigger <- map(
priors$prior,
\(p) brm(
y ~ x,
data = dat_bigger,
prior = set_prior(p, class = "b", coef = "x"),
silent = 0
)
)Compiling Stan program...
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Chain 4: toc()272.3 sec elapsedbeep(3)names(res_bigger) <- priors$prior
grab_posteriors_bigger <- function(prior) {
res_bigger[[prior]] |>
spread_draws(b_x) |>
mutate(prior = prior)
}
all_posteriors_bigger <- map(names(res_bigger), grab_posteriors_bigger) |> bind_rows()Make some pictures:
all_posteriors |>
ggplot(aes(x = b_x)) +
geom_histogram(bins = 75) +
facet_wrap(vars(prior), scales = "free_y") +
theme_few() +
labs(x = ~ beta[x],
y = "Count",
title = "Impact of prior distribution on posterior (N = 15)")
all_posteriors_bigger |>
ggplot(aes(x = b_x)) +
geom_histogram(bins = 75) +
facet_wrap(vars(prior), scales = "free_y") +
theme_few() +
labs(x = ~ beta[x],
y = "Count",
title = "Impact of prior distribution on posterior (N = 1500)")
And all glued together:
all_posteriors_15_1500 <- bind_rows(
all_posteriors |> mutate(n = 15),
all_posteriors_bigger |> mutate(n = 1500)
)final_plot <- all_posteriors_15_1500 |>
ggplot(aes(x = b_x)) +
geom_density() +
facet_grid(cols = vars(prior), rows = vars(paste("n =", n)), scales = "free_y") +
theme_few() +
labs(x = ~ beta[x],
y = "Density") +
xlim(2 - 3, 2 + 3)
final_plotWarning: Removed 3066 rows containing non-finite outside the scale range (`stat_density()`).
ggsave("final_plot_bayes.png", final_plot, height = 3, width = 7, dpi = 300)Warning: Removed 3066 rows containing non-finite outside the scale range (`stat_density()`).Let’s see if we can drag the posterior to cover 2 when the prior is N(0, 0.1), with a huge n = 1,000,000.
set.seed(1335)
the_n <- 1e6
dat_biggest <- tibble(x = rnorm(the_n, 0, 1),
y = 2 * x + rnorm(the_n, 0, 10))
ggplot(dat_bigger, aes(x, y)) +
geom_point()
tic()
big_mod <- brm(
y ~ x,
data = dat_biggest,
prior = set_prior("normal(0, 0.1)", class = "b", coef = "x"),
silent = 0
)Compiling Stan program...
Start sampling
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Chain 4: toc()2168.81 sec elapsedbeep(3)summary(big_mod) Family: gaussian
Links: mu = identity; sigma = identity
Formula: y ~ x
Data: dat_biggest (Number of observations: 1000000)
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 -0.01 0.01 -0.02 0.01 1.00 1844 1707
x 1.99 0.01 1.97 2.01 1.00 2029 1821
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 9.99 0.01 9.97 10.00 1.00 5050 3015
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).big_mod_pic <- big_mod |>
spread_draws(b_x) |>
ggplot(aes(x = b_x)) +
geom_histogram(bins = 101) +
theme_few() +
labs(x = ~ beta[x], y = "Count") +
xlim(2 - .1, 2 + .1)
big_mod_pic Warning: Removed 2 rows containing missing values or values outside the scale range (`geom_bar()`).
It’s getting there – you need a big sample size to undo a silly prior.
ggsave("big_mod_pic.png", big_mod_pic, height = 3, width = 7, dpi = 300)Warning: Removed 2 rows containing missing values or values outside the scale range (`geom_bar()`).lm(y ~ x, data = dat_biggest) |> confint() 2.5 % 97.5 %
(Intercept) -0.02607613 0.01307723
x 1.98706536 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