How To Explore And Summarize A Mcmc Draw

Plotting MCMC draws using the bayesplot package

2022-03-09

• Introduction
  • Setup
  • Example model
• Posterior uncertainty intervals
• Univariate marginal posterior distributions
• Bivariate plots
• Trace plots
• References

Introduction

This vignette focuses on plotting parameter estimates from MCMC draws. MCMC diagnostic plots are covered in the divide vignette Visual MCMC diagnostics, and graphical posterior predictive model checking is covered in the vignette Graphical posterior predictive checks.

Setup

In addition to bayesplot we'll load the following packages:

• ggplot2, in case nosotros want to customize the ggplot objects created by bayesplot
• rstanarm, for plumbing equipment the example models used throughout the vignette

                                                library("bayesplot")                                  library("ggplot2")                                  library("rstanarm")                                          

Example model

The bayesplot package provides diverse plotting functions for visualizing Markov concatenation Monte Carlo (MCMC) draws from the posterior distribution of the parameters of a Bayesian model.

In this vignette nosotros'll use draws obtained using the stan_glm function in the rstanarm parcel (Gabry and Goodrich, 2017), but MCMC draws from using any packet can be used with the functions in the bayesplot packet. Encounter, for example, brms, which, similar 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   six  160 110 iii.90 2.620 xvi.46  0  i    4    four Mazda RX4 Wag     21.0   vi  160 110 3.90 2.875 17.02  0  1    4    four Datsun 710        22.8   4  108  93 3.85 ii.320 18.61  i  1    4    i Hornet 4 Drive    21.4   6  258 110 three.08 3.215 nineteen.44  i  0    3    one Hornet Sportabout xviii.7   eight  360 175 3.15 iii.440 17.02  0  0    3    two Valiant           eighteen.1   6  225 105 ii.76 3.460 20.22  ane  0    three    1          

                                                # linear regression model using stan_glm                                                  # using '~ .' to include all variables                                fit                  <-                  stan_glm(mpg                  ~                  .,                  information =                  mtcars,                  seed =                  1111)                                  print(fit)                          

            stan_glm  family:       gaussian [identity]  formula:      mpg ~ .  observations: 32  predictors:   11 ------             Median MAD_SD (Intercept) 12.0   19.3   cyl         -0.1    i.1   disp         0.0    0.0   hp           0.0    0.0   drat         0.8    ane.6   wt          -3.7    one.nine   qsec         0.8    0.8   vs           0.iii    2.1   am           two.half-dozen    2.1   gear         0.vi    one.5   carb        -0.2    0.ix    Auxiliary parameter(s):       Median MAD_SD sigma two.seven    0.four     ------ * For assist interpreting the printed output meet ?print.stanreg * For info on the priors used see ?prior_summary.stanreg          

To use the posterior draws with the functions in the bayesplot package nosotros'll extract them from the fitted model object:

                              posterior                  <-                  every bit.array(fit)                                  dim(posterior)                          

            [1] m    iv   12          

            $iterations NULL  $chains [1] "concatenation:1" "concatenation:ii" "chain:three" "concatenation:4"  $parameters  [one] "(Intercept)" "cyl"         "disp"        "hp"          "drat"         [half dozen] "wt"          "qsec"        "vs"          "am"          "gear"        [11] "carb"        "sigma"                      

We've used every bit.array above (as opposed to as.matrix) because it keeps the Markov chains divide (stan_glm runs four bondage by default). Most of the plots don't actually need the chains to be divide, only for a few of the plots we make in this vignette nosotros'll want to show the chains individually.

Posterior dubiety intervals

For models fit using MCMC we can compute posterior uncertainty intervals (sometimes chosen "apparent intervals") in various ways. bayesplot currently provides plots of key intervals based on quantiles, although boosted options may be provided in future releases (e.g., HDIs, which tin be useful in item cases).

Documentation:

• help("MCMC-intervals")
• mc-stan.org/bayesplot/reference/MCMC-intervals

---

mcmc_intervals, mcmc_areas

Central posterior uncertainty intervals can be plotted using the mcmc_intervals function.

                                                color_scheme_set("ruby-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 tin can be changed using the prob and prob_outer arguments, respectively. The points in the above plot are posterior medians. The point_est argument tin 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 nosotros tin use the mcmc_areas part.

                                                mcmc_areas(                                  posterior,                                                  pars =                  c("cyl",                  "drat",                  "am",                  "sigma"),                                  prob =                  0.8,                  # 80% intervals                                                  prob_outer =                  0.99,                  # 99%                                                  point_est =                  "mean"                                )                          

Univariate marginal posterior distributions

bayesplot provides functions for looking at histograms or kernel density estimates of marginal posterior distributions, either with all Markov bondage combined or with the bondage split up.

Documentation:

• help("MCMC-distributions")
• mc-stan.org/bayesplot/reference/MCMC-distributions

---

mcmc_hist

The mcmc_hist office 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 nosotros can either transform the draws in advance or utilize the transformations statement.

                                                color_scheme_set("blueish")                                  mcmc_hist(posterior,                  pars =                  c("wt",                  "sigma"),                                  transformations =                  list("sigma"                  =                  "log"))                          

Most of the other functions for plotting MCMC draws as well have a transformations argument.

mcmc_hist_by_chain

To view separate histograms of each of the four Markov bondage we can employ mcmc_hist_by_chain, which plots each concatenation in a separate facet in the plot.

                                                color_scheme_set("brightblue")                                  mcmc_hist_by_chain(posterior,                  pars =                  c("wt",                  "sigma"))                          

mcmc_dens

The mcmc_dens function is similar to mcmc_hist simply plots kernel density estimates instead of histograms.

                                                color_scheme_set("purple")                                  mcmc_dens(posterior,                  pars =                  c("wt",                  "sigma"))                          

mcmc_dens_overlay

Like mcmc_hist_by_chain, the mcmc_dens_overlay role separates the Markov bondage. Only instead of plotting each chain individually, the density estimates are overlaid.

                                                mcmc_dens_overlay(posterior,                  pars =                  c("wt",                  "sigma"))                          

mcmc_violin

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))                          

Bivariate plots

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:

• assist("MCMC-scatterplots")
• mc-stan.org/bayesplot/reference/MCMC-scatterplots

---

mcmc_scatter

The mcmc_scatter office creates a simple scatterplot of 2 parameters.

                                                color_scheme_set("grayness")                                  mcmc_scatter(posterior,                  pars =                  c("(Intercept)",                  "wt"),                                                  size =                  1.5,                  blastoff =                  0.five)                          

mcmc_hex

The mcmc_hex function creates a similar plot but using hexagonal binning, which tin be useful to avert overplotting.

                                                # requires hexbin package                                                  if                  (requireNamespace("hexbin",                  quietly =                  True)) {                                  mcmc_hex(posterior,                  pars =                  c("(Intercept)",                  "wt"))                }                          

mcmc_pairs

In addition to mcmc_scatter and mcmc_hex, bayesplot now provides an mcmc_pairs function for creating pairs plots with more than 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, only this tin can exist changed to densities by setting diag_fun="dens". Bivariate plots are displayed above and below the diagonal every bit scatterplots, just it is also possible to apply hex plots by setting off_diag_fun="hex". By default, mcmc_pairs shows some of the Markov chains (one-half, if an even number of bondage) above the diagonal and the others beneath. There are many more options for decision-making how the draws should be split up between the plots above and below the diagonal (see the documentation for the status argument), but they are more than useful when MCMC diagnostic information is included. This is discussed in the Visual MCMC diagnostics vignette.

Trace plots

Trace plots are fourth dimension series plots of Markov bondage. In this vignette we evidence the standard trace plots that bayesplot tin brand. For models fit using whatsoever Stan interface (or Hamiltonian Monte Carlo in general), the Visual MCMC diagnostics vignette provides an example of also adding data nearly divergences to trace plots.

Documentation:

• help("MCMC-traces")
• mc-stan.org/bayesplot/reference/MCMC-traces

---

mcmc_trace

The mcmc_trace part creates standard trace plots:

                                                color_scheme_set("blue")                                  mcmc_trace(posterior,                  pars =                  c("wt",                  "sigma"))                          

If information technology's hard to meet the difference betwixt the chains nosotros tin can modify to a mixed color scheme, for example:

                                                color_scheme_set("mix-blue-red")                                  mcmc_trace(posterior,                  pars =                  c("wt",                  "sigma"),                                                  facet_args =                  listing(ncol =                  i,                  strip.position =                  "left"))                          

The code above also illustrates the employ 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 singled-out colors:

                                                color_scheme_set("viridis")                                  mcmc_trace(posterior,                  pars =                  "(Intercept)")                          

mcmc_trace_highlight

The mcmc_trace_highlight office 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 =                  iii)                          

References

Gabry, J., and Goodrich, B. (2017). rstanarm: Bayesian Applied Regression Modeling via Stan. R package version 2.15.3. https://mc-stan.org/rstanarm/, https://CRAN.R-project.org/packet=rstanarm

Gabry, J., Simpson, D., Vehtari, A., Betancourt, Yard. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. :10.1111/rssa.12378. (journal version, arXiv preprint, lawmaking on GitHub)

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Assay. Chapman & Hall/CRC Press, London, tertiary edition.

Stan Development Team. (2017). Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/

Source: https://cran.r-project.org/web/packages/bayesplot/vignettes/plotting-mcmc-draws.html

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