Ridgelines in bayesplot 1.5.0

At the end of March, Jonah Gabry and I released bayesplot 1.5.0. The major additions to the package were visualizations using ridgelines and a new plot for PIT diagnostics from LOO validation. I don’t know what that LOO PIT thing is yet, so I’ll talk about how ridgelines have been incorporated into bayesplot instead.

Unknown Pleasures cover

Ridgelines display the distributions of some variables—think of a series of bell curves—but allow the curves to vary in height or overlap each other. Several curves packed together look like a mountain ridge (hence the name). The canonical example of this visualization are the cold, austere lines from the cover of Joy Division’s Unknown Pleasures.

In this post, I briefly describe where ridgelines appear in the package now. At the end of the post, I show a more involved example of fitting a model and plotting results from it.

Behind the scenes of mcmc_areas()

I’m going to start with a less obvious use of ridgelines. mcmc_areas() visualizes the posterior distribution of model parameters. It helps us see where the peak values are, and occasionally, whether there is more than one peak. By default, it will plot the complete range of the distribution, shade the middle 50% of the distribution, and draw a line at the median.

For this release, I rewrote the function so that it uses the ggridges package. I had noticed that the shapes it produced had different areas. Here’s an example of how the plots have changed.

library(ggplot2)

# Temporarily install the previous version of bayesplot and 
# make an mcmc_areas() plot.
withr::with_temp_libpaths({
  versions::install.versions("bayesplot", "1.4.0", quiet = TRUE)
  
  example_data <- bayesplot::example_mcmc_draws()
  p_a <- bayesplot::mcmc_areas(example_data, pars = c("beta[1]", "beta[2]")) +
    ggtitle("Version 1.4.0") +
    bayesplot::theme_default()
})
#> package 'bayesplot' successfully unpacked and MD5 sums checked

library(bayesplot)
#> This is bayesplot version 1.5.0
#> - Plotting theme set to bayesplot::theme_default()
#> - Online documentation at mc-stan.org/bayesplot
p_b <- mcmc_areas(example_data, c("beta[1]", "beta[2]")) + 
  ggtitle("Version 1.5.0") +
  bayesplot::theme_default()

# Show the two together
cowplot::plot_grid(p_a, p_b)

Comparison of mcmc_areas() in bayesplot 1.4.0 and 1.5.0

In the plot on the left, the shapes have different areas. Eyeballing it, it looks like beta[2] would fit inside beta[1]. Area is width times height. These shapes have different widths because they cover different values—that makes sense. Therefore, the problem is the height—namely, the fact that the shapes have the same height. To make the areas more similar, we adjust the heights, and that’s what’s happening in the plot on the right. The distribution for beta[2] is concentrated to a more narrow range, and that feature manifests in that shape’s much higher peak.

Three plots in one

This plot is built using the ggridges packages which handles drawing the shapes at different heights. It took a bit of trickery, however, to do the annotation for the shaded 50% interval and the median. Here is a high-level overview of what’s happening.

mcmc_areas_data() tidies the MCMC samples into a dataframe suitable for plotting. It computes the density of the parameter values, and it returns the densities over three intervals: 1) an outer interval, 2) an inner interval to be shaded, and 3) a narrow “point” interval.

library(dplyr, warn.conflicts = FALSE)

d <- mcmc_areas_data(example_data, c("beta[1]", "beta[2]")) %>% 
  mutate(
    interval = factor(interval, c("outer", "inner", "point")))

d
#> # A tibble: 4,238 x 5
#>    parameter interval interval_width     x density
#>    <fct>     <fct>             <dbl> <dbl>   <dbl>
#>  1 beta[1]   inner             0.500 0.136   0.986
#>  2 beta[1]   inner             0.500 0.137   0.988
#>  3 beta[1]   inner             0.500 0.137   0.989
#>  4 beta[1]   inner             0.500 0.138   0.990
#>  5 beta[1]   inner             0.500 0.138   0.991
#>  6 beta[1]   inner             0.500 0.139   0.992
#>  7 beta[1]   inner             0.500 0.139   0.993
#>  8 beta[1]   inner             0.500 0.139   0.994
#>  9 beta[1]   inner             0.500 0.140   0.995
#> 10 beta[1]   inner             0.500 0.140   0.997
#> # ... with 4,228 more rows

Internally, mcmc_areas() uses ggridges::geom_density_ridges() to draw three different ridgeline plots.

# Separate the full range so it can be drawn on each facet.
full_range <- d %>% 
  filter(interval == "outer") %>% 
  select(-interval)

p_base <- ggplot(d) + 
  aes(x = x, y = parameter, fill = interval, height = density) + 
  # For reference include the full-range. 
  # `stat = "identity"` means that ggridges should not scale the heights
  ggridges::geom_density_ridges(
    stat = "identity", data = full_range, color = "grey80",
    size = 1, fill = NA) + 
  ggridges::geom_density_ridges(stat = "identity") + 
  scale_fill_brewer(type = "seq") + 
  guides(fill = FALSE)

p_base + 
  facet_wrap("interval")

The three layers behind an mcmc_areas() plot drawn separately

These intervals are layered to achieve the desired look.

The layers from the previous plot flattened together

There’s a bit more fussing to the actual function. For example, the width of the intervals and whether a point is calculated are options that have to be handled. The tips of the topmost peaks are not cut off in those plots as well. But that’s the basic idea.

Plots with the classic, overlapping Unknown Pleasures look are supported by mcmc_areas_ridges(). I like these plots for hierarchical models where the parameters represent similar units. In fact, part of the reason Jonah approached me about contributing to bayesplot was a proof-of-concept demo I wrote to visualize hierarchical effects using ridgeline plots, back when they were popularly called “joyplots”.

Below is the eight schools example where there is a treatment effect for each school (thetas) and an overall average effect (mu).

m <- shinystan::eight_schools@posterior_sample

mcmc_areas_ridges(m, pars = "mu", regex_pars = "theta") +
  ggplot2::ggtitle("Treatment effect on eight schools (Rubin, 1981)")

Overlapping ridgelines in a typical looking ridgeline plot

If we really want to go for the Unknown Pleasures look, we can manually plot the densities in a white-on-black theme. Most of the work here is padding zero values onto the ridgelines so that they all have the same width.

d <- mcmc_areas_ridges_data(m, pars = "mu", regex_pars = "theta") %>% 
  filter(interval == "outer") 

zeroes <- d %>%
  # Get range of each parameter
  group_by(parameter) %>% 
  summarise(pmin = min(x), pmax = max(x)) %>% 
  mutate(
    min = min(pmin), 
    max = max(pmax),
    # On each row, make a data-frame of zeroes from absolute minimum 
    # value to this parameter's minimum value
    lower_df = purrr::map2(
      min, pmin, 
      ~ data.frame(x = seq(.x, .y, length.out = 20), density = 0)),
    # Make a dataframe of zeroes from each parameter's max to the 
    # overall max
    upper_df = purrr::map2(
      pmax, max, 
      ~ data.frame(x = seq(.x, .y, length.out = 20), density = 0)),
    # Gather the zeroes together
    df = purrr::map2(lower_df, upper_df, bind_rows)) %>% 
  select(parameter, df) %>% 
  tidyr::unnest(df)

ggplot(bind_rows(d, zeroes)) + 
  aes(x = x, y = parameter, height = density) +
  ggridges::geom_density_ridges(
    stat = "identity", fill = NA, 
    color = "grey70", scale = 5) + 
  theme_void() + 
  theme(
    plot.background = element_rect(fill = "black"), 
    plot.margin = unit(c(.25, .25, .25, .25), units = "npc"),
    axis.ticks = element_blank())

The above plot rethemed to look like Unknown Pleausres

Densities of different sampling chains

Finally, ridgelines appear in by-chain diagnostics using mcmc_dens_chains().

mcmc_dens_chains(m, pars = c("theta[1]", "theta[2]", "theta[3]"))

A ridgeline plot where MCMC chains are drawn in different colors

There really isn’t much interesting to say about this plot, except that it follows Mike DeCrescenzo’s early example of using ridgelines on Bayesian models.

An example of fitting and plotting a model

The cake dataset in lme4 reports the angle of breakage of chocolate cakes for 3 different recipes at 6 temperatures tested 15 times each (3 × 6 × 15 = 270 observations). The data are shown below, ignoring temperature details:

set.seed(20180405)
cake <- lme4::cake

theme_set(theme_grey())

ggplot(cake) + 
  # aes(x = temp, y = angle, color = recipe) + 
  aes(x = replicate, y = angle, color = recipe) + 
  geom_text(aes(label = recipe), position = position_jitter(.15), size = 3) + 
  stat_summary(color = "grey10", size = .4) + 
  guides(color = FALSE, label = FALSE) +
  labs(
    x = "Replicate", 
    y = "Angle of breakage", 
    caption = "Letters: Recipe. Points: Replicate Mean ± SE")
#> No summary function supplied, defaulting to `mean_se()

The cake dataset

It looks like the first few replications showed larger breakage angles compared to later ones. From a modeling perspective, we would want to capture variation due to replication. There could be variation due to recipe × replication effects, so we will model the data at that level of detail.

library(rstanarm)
#> Loading required package: Rcpp
#> Warning: package 'Rcpp' was built under R version 3.4.4
#> Loading required package: methods
#> rstanarm (Version 2.17.3, packaged: 2018-02-17 05:11:16 UTC)
#> - Do not expect the default priors to remain the same in future rstanarm versions.
#> Thus, R scripts should specify priors explicitly, even if they are just the defaults.
#> - For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores())
#> - Plotting theme set to bayesplot::theme_default().
m <- stan_glmer(
  angle ~ recipe * temp + (1 | recipe:replicate), 
  family = gaussian(), 
  data = cake,
  prior = normal(0, 1),
  seed = 20180405)

This models predicts the breakage angle using an average recipe, temperature, and recipe × temperate effects and adjusting the prediction for each recipe × replication.

Below, we use mcmc_areas_ridges() to plot the distributions of replication effects for recipe A. The plotting functions in bayesplot take a set of MCMC samples from a model. We can extract the samples using as.matrix(). We can narrow the parameters to be plotted by naming them individually with pars or by selecting all the parameters that match a regular expression pattern with regex_pars.

# Revert to the bayesplot default theme
theme_set(bayesplot::theme_default())

mcmc_areas_ridges(
  as.matrix(m), 
  regex_pars = "recipe:replicate:A") 

Replications effects for recipe A

The effects describe deviations from the recipe average. A faint line is drawn at x = 0 represents the average value. Ridges to the left of the line are below average and those to the right are above average. We can see that the model captured how the first three replications had particular high breakage angles.