plotting-mcmc-draws.Rmd
This vignette focuses on plotting parameter estimates from MCMC draws. MCMC diagnostic plots are covered in the separate vignette Visual MCMC diagnostics, and graphical posterior predictive model checking is covered in the vignette Graphical posterior predictive checks.
In addition to bayesplot we’ll load the following packages:
library("bayesplot")
library("ggplot2")
library("rstanarm")
The bayesplot package provides various plotting functions for visualizing Markov chain Monte Carlo (MCMC) draws from the posterior distribution of the parameters of a Bayesian model.
In this vignette we’ll use draws obtained using the stan_glm
function in the rstanarm package (Gabry and Goodrich, 2017), but MCMC draws from using any package can be used with the functions in the bayesplot package. See, for example, brms, which, like rstanarm, calls the rstan package internally to use Stan’s MCMC sampler.
head(mtcars) # see help("mtcars")
mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
# linear regression model using stan_glm
# using '~ .' to include all variables
fit <- stan_glm(mpg ~ ., data = mtcars, seed = 1111)
print(fit)
stan_glm
family: gaussian [identity]
formula: mpg ~ .
observations: 32
predictors: 11
------
Median MAD_SD
(Intercept) 12.6 20.1
cyl -0.1 1.1
disp 0.0 0.0
hp 0.0 0.0
drat 0.8 1.7
wt -3.7 1.9
qsec 0.8 0.8
vs 0.3 2.1
am 2.5 2.1
gear 0.6 1.6
carb -0.2 0.9
Auxiliary parameter(s):
Median MAD_SD
sigma 2.7 0.4
Sample avg. posterior predictive distribution of y:
Median MAD_SD
mean_PPD 20.1 0.7
------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
To use the posterior draws with the functions in the bayesplot package we’ll extract them from the fitted model object:
posterior <- as.array(fit)
dim(posterior)
[1] 1000 4 12
dimnames(posterior)
$iterations
NULL
$chains
[1] "chain:1" "chain:2" "chain:3" "chain:4"
$parameters
[1] "(Intercept)" "cyl" "disp" "hp" "drat"
[6] "wt" "qsec" "vs" "am" "gear"
[11] "carb" "sigma"
We’ve used as.array
above (as opposed to as.matrix
) because it keeps the Markov chains separate (stan_glm
runs four chains by default). Most of the plots don’t actually need the chains to be separate, but for a few of the plots we make in this vignette we’ll want to show the chains individually.
For models fit using MCMC we can compute posterior uncertainty intervals (sometimes called “credible intervals”) in various ways. bayesplot currently provides plots of central intervals based on quantiles, although additional options may be provided in future releases (e.g., HDIs, which can be useful in particular cases).
Documentation:
help("MCMC-intervals")
Central posterior uncertainty intervals can be plotted using the mcmc_intervals
function.
color_scheme_set("red")
mcmc_intervals(posterior, pars = c("cyl", "drat", "am", "sigma"))
The default is to show 50% intervals (the thick segments) and 90% intervals (the thinner outer lines). These defaults can be changed using the prob
and prob_outer
arguments, respectively. The points in the above plot are posterior medians. The point_est
argument can be used to select posterior means instead or to omit the point estimates.
To show the uncertainty intervals as shaded areas under the estimated posterior density curves we can use the mcmc_areas
function.
mcmc_areas(
posterior,
pars = c("cyl", "drat", "am", "sigma"),
prob = 0.8, # 80% intervals
prob_outer = 0.99, # 99%
point_est = "mean"
)
bayesplot provides functions for looking at histograms or kernel density estimates of marginal posterior distributions, either with all Markov chains combined or with the chains separate.
Documentation:
help("MCMC-distributions")
The mcmc_hist
function plots marginal posterior distributions (combining all chains):
color_scheme_set("green")
mcmc_hist(posterior, pars = c("wt", "sigma"))
If we want to plot log(sigma)
rather than sigma
we can either transform the draws in advance or use the transformations
argument.
color_scheme_set("blue")
mcmc_hist(posterior, pars = c("wt", "sigma"),
transformations = list("sigma" = "log"))
Most of the other functions for plotting MCMC draws also have a transformations
argument.
To view separate histograms of each of the four Markov chains we can use mcmc_hist_by_chain
, which plots each chain in a separate facet in the plot.
color_scheme_set("brightblue")
mcmc_hist_by_chain(posterior, pars = c("wt", "sigma"))
The mcmc_dens
function is similar to mcmc_hist
but plots kernel density estimates instead of histograms.
color_scheme_set("purple")
mcmc_dens(posterior, pars = c("wt", "sigma"))
Like mcmc_hist_by_chain
, the mcmc_dens_overlay
function separates the Markov chains. But instead of plotting each chain individually, the density estimates are overlaid.
mcmc_dens_overlay(posterior, pars = c("wt", "sigma"))
The mcmc_violin
function plots the density estimates of each chain as violins and draws horizontal line segments at user-specified quantiles.
color_scheme_set("teal")
mcmc_violin(posterior, pars = c("wt", "sigma"), probs = c(0.1, 0.5, 0.9))
Various functions are available for plotting bivariate marginal posterior distributions. Some of these functions also take optional arguments for adding MCMC diagnostic information to the plots. That additional functionality is discussed in the separate Visual MCMC diagnostics vignette.
Documentation:
help("MCMC-scatterplots")
The mcmc_scatter
function creates a simple scatterplot of two parameters.
color_scheme_set("gray")
mcmc_scatter(posterior, pars = c("(Intercept)", "wt"),
size = 1.5, alpha = 0.5)
The mcmc_hex
function creates a similar plot but using hexagonal binning, which can be useful to avoid overplotting.
mcmc_hex(posterior, pars = c("(Intercept)", "wt"))
In addition to mcmc_scatter
and mcmc_hex
, bayesplot now provides an mcmc_pairs
function for creating pairs plots with more than two parameters.
color_scheme_set("pink")
mcmc_pairs(posterior, pars = c("(Intercept)", "wt", "sigma"),
off_diag_args = list(size = 1.5))
The univariate marginal posteriors are shown along the diagonal as histograms, but this can be changed to densities by setting diag_fun="dens"
. Bivariate plots are displayed above and below the diagonal as scatterplots, but it is also possible to use hex plots by setting off_diag_fun="hex"
. By default, mcmc_pairs
shows some of the Markov chains (half, if an even number of chains) above the diagonal and the others below. There are many more options for controlling how the draws should be split between the plots above and below the diagonal (see the documentation for the condition
argument), but they are more useful when MCMC diagnostic information is included. This is discussed in the Visual MCMC diagnostics vignette.
Trace plots are time series plots of Markov chains. In this vignette we show the standard trace plots that bayesplot can make. For models fit using any Stan interface (or Hamiltonian Monte Carlo in general), the Visual MCMC diagnostics vignette provides an example of also adding information about divergences to trace plots.
Documentation:
help("MCMC-traces")
The mcmc_trace
function creates standard trace plots:
color_scheme_set("blue")
mcmc_trace(posterior, pars = c("wt", "sigma"))
If it’s hard to see the difference between the chains we can change to a mixed color scheme, for example:
color_scheme_set("mix-blue-red")
mcmc_trace(posterior, pars = c("wt", "sigma"),
facet_args = list(ncol = 1, strip.position = "left"))
The code above also illustrates the use of the facet_args
argument, which is a list of parameters passed to facet_wrap
in ggplot2. Specifying ncol=1
means the trace plots will be stacked in a single column rather than placed side by side, and strip.position="left"
moves the facet labels to the y-axis (instead of above each facet).
The "viridis"
color scheme is also useful for trace plots because it is comprised of very distinct colors:
color_scheme_set("viridis")
mcmc_trace(posterior, pars = "(Intercept)")
The mcmc_trace_highlight
function uses points instead of lines and reduces the opacity of all but a single chain (which is specified using the highlight
argument).
mcmc_trace_highlight(posterior, pars = "sigma", highlight = 3)
Warning: Using alpha for a discrete variable is not advised.
Gabry, J., and Goodrich, B. (2017). rstanarm: Bayesian Applied Regression Modeling via Stan. R package version 2.15.3. http://mc-stan.org/rstanarm, https://CRAN.R-project.org/package=rstanarm
Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., Gelman, A. (2018). Visualization in Bayesian workflow. Journal of the Royal Statistical Society Series A, accepted for publication. arXiv preprint: https://arxiv.org/abs/1709.01449
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. Chapman & Hall/CRC Press, London, third edition.
Stan Development Team. (2017). Stan Modeling Language Users Guide and Reference Manual. http://mc-stan.org/users/documentation/