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:

**ggplot2**, in case we want to customize the ggplot objects created by**bayesplot****rstanarm**, for fitting the example models used throughout the vignette

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 demonstrate a few of these functions. Example usage of the
functions not demonstrated here can be found in the package
documentation.

For demonstration we will 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.

```
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) 12.0 19.3
cyl -0.1 1.1
disp 0.0 0.0
hp 0.0 0.0
drat 0.8 1.6
wt -3.7 1.9
qsec 0.8 0.8
vs 0.3 2.1
am 2.6 2.1
gear 0.6 1.5
carb -0.2 0.9
Auxiliary parameter(s):
Median MAD_SD
sigma 2.7 0.4
------
* 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:

`[1] 1000 4 12`

```
$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")`

- mc-stan.org/bayesplot/reference/MCMC-intervals

Central posterior uncertainty intervals can be plotted using the
`mcmc_intervals`

function.

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")`

- mc-stan.org/bayesplot/reference/MCMC-distributions

The `mcmc_hist`

function plots marginal posterior
distributions (combining all chains):

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.

The `mcmc_dens`

function is similar to
`mcmc_hist`

but plots kernel density estimates instead of
histograms.

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.

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")`

- mc-stan.org/bayesplot/reference/MCMC-scatterplots

The `mcmc_scatter`

function creates a simple scatterplot
of two parameters.

The `mcmc_hex`

function creates a similar plot but using
hexagonal binning, which can be useful to avoid overplotting.

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")`

- mc-stan.org/bayesplot/reference/MCMC-traces

The `mcmc_trace`

function creates standard trace
plots:

If it’s hard to see the difference between the chains we can change to a mixed color scheme, for example: