A trace plot shows one parameter across iterations. But what if the computational problem lies in combinations of parameter values? A pairs plot shows two parameters jointly across posterior draws. Its shape can reveal posterior dependence that slows computation or makes a particular parameterization difficult to interpret.
Constructing two parameterizations
Suppose a phonetic model has mean vowel durations \mu_A and \mu_B in two experimental conditions. Duration is the elapsed acoustic time assigned to the vowel interval, and the interval boundaries must be defined by the same segmentation rule in both conditions. Define the overall location and condition contrast as
m=\frac{\mu_A+\mu_B}{2}
and
d=\mu_A-\mu_B.
The inverse transformation is
\mu_A=m+\frac{d}{2},
\qquad
\mu_B=m-\frac{d}{2}.
If posterior uncertainty about the overall location is much larger than uncertainty about the contrast, \mu_A and \mu_B move together. Their pair plot forms a positive diagonal band. The (m,d) parameterization separates shared location from the theoretically relevant difference.
Simulating the geometry
Code
set.seed(774)draw_count <-5000m <-rnorm(draw_count, mean =200, sd =8)d <-rnorm(draw_count, mean =-10, sd =3)mu_a <- m + d /2mu_b <- m - d /2cor_means <-cor(mu_a, mu_b)cor_location_contrast <-cor(m, d)old_par <-par(mfrow =c(1, 2))plot(mu_a, mu_b, pch =16, cex = .25,xlab ="condition A mean", ylab ="condition B mean")plot(m, d, pch =16, cex = .25,xlab ="overall location", ylab ="condition contrast")par(old_par)stopifnot(cor_means > .8)stopifnot(abs(cor_location_contrast) < .05)c(condition_means = cor_means,location_and_contrast = cor_location_contrast)
The two displays show the same posterior in different coordinates. The (m,d) coordinates separate uncertainty about the overall location from uncertainty about the linguistic contrast, while leaving fitted predictions unchanged.
Reading common shapes
A roughly circular cloud suggests weak dependence for that pair, a diagonal ellipse suggests that combinations along one direction remain plausible, and a curved or funnel-shaped cloud suggests that uncertainty in one parameter changes with another.
The shape alone does not identify its cause. Limited data, scale parameters, parameter constraints, and redundant parameterizations can each produce posterior dependence.
Overlaying computational warnings
Pair plots can mark iterations associated with divergent transitions. If marked draws cluster in a narrow end of a joint distribution, the display locates a region that the sampler may not explore reliably.
A matrix containing hundreds of parameter pairs is unreadable. Select the pairs implicated by the model structure or by another diagnostic.
Avoiding the parameter-causation reading
A diagonal cloud shows that the observed data and model permit parameter combinations along a shared direction. It does not show that changing one parameter would cause the other to change.
Nor does a compact cloud establish model fit. It describes uncertainty under the fitted model. Observed-data adequacy requires comparing model predictions with relevant linguistic summaries.
Check your understanding
Why are \mu_A and \mu_B positively associated in the constructed example?
What distinct quantities do m and d represent?
Why does reparameterization leave the underlying posterior information unchanged?
What can a divergence overlay show that a plain pair plot cannot?
Source Code
---title: "Reading pairs plots of posterior draws"---A [trace plot](checking-markov-chains.qmd) shows one parameter across iterations. But what if the computational problem lies in combinations of parameter values? A [**pairs plot**](https://mc-stan.org/bayesplot/articles/plotting-mcmc-draws.html) shows two parameters jointly across posterior draws. Its shape can reveal posterior dependence that slows computation or makes a particular parameterization difficult to interpret.## Constructing two parameterizationsSuppose a phonetic model has mean vowel durations $\mu_A$ and $\mu_B$ in two experimental conditions. Duration is the elapsed acoustic time assigned to the vowel interval, and the interval boundaries must be defined by the same segmentation rule in both conditions. Define the overall location and condition contrast as$$m=\frac{\mu_A+\mu_B}{2}$$and$$d=\mu_A-\mu_B.$$The inverse transformation is$$\mu_A=m+\frac{d}{2},\qquad\mu_B=m-\frac{d}{2}.$$If posterior uncertainty about the overall location is much larger than uncertainty about the contrast, $\mu_A$ and $\mu_B$ move together. Their pair plot forms a positive diagonal band. The $(m,d)$ parameterization separates shared location from the theoretically relevant difference.## Simulating the geometry```{r}#| label: mean-contrast-pair-geometry#| echo: trueset.seed(774)draw_count <-5000m <-rnorm(draw_count, mean =200, sd =8)d <-rnorm(draw_count, mean =-10, sd =3)mu_a <- m + d /2mu_b <- m - d /2cor_means <-cor(mu_a, mu_b)cor_location_contrast <-cor(m, d)old_par <-par(mfrow =c(1, 2))plot(mu_a, mu_b, pch =16, cex = .25,xlab ="condition A mean", ylab ="condition B mean")plot(m, d, pch =16, cex = .25,xlab ="overall location", ylab ="condition contrast")par(old_par)stopifnot(cor_means > .8)stopifnot(abs(cor_location_contrast) < .05)c(condition_means = cor_means,location_and_contrast = cor_location_contrast)```The two displays show the same posterior in different coordinates. The $(m,d)$ coordinates separate uncertainty about the overall location from uncertainty about the linguistic contrast, while leaving fitted predictions unchanged.## Reading common shapesA roughly circular cloud suggests weak dependence for that pair, a diagonal ellipse suggests that combinations along one direction remain plausible, and a curved or funnel-shaped cloud suggests that uncertainty in one parameter changes with another.The shape alone does not identify its cause. Limited data, scale parameters, parameter constraints, and redundant parameterizations can each produce posterior dependence.## Overlaying computational warningsPair plots can mark iterations associated with [divergent transitions](divergent-transitions.qmd). If marked draws cluster in a narrow end of a joint distribution, the display locates a region that the sampler may not explore reliably.A matrix containing hundreds of parameter pairs is unreadable. Select the pairs implicated by the model structure or by another diagnostic.## Avoiding the parameter-causation readingA diagonal cloud shows that the observed data and model permit parameter combinations along a shared direction. It does not show that changing one parameter would cause the other to change.Nor does a compact cloud establish model fit. It describes uncertainty under the fitted model. Observed-data adequacy requires comparing model predictions with relevant linguistic summaries.## Check your understanding1. Why are $\mu_A$ and $\mu_B$ positively associated in the constructed example?2. What distinct quantities do $m$ and $d$ represent?3. Why does reparameterization leave the underlying posterior information unchanged?4. What can a divergence overlay show that a plain pair plot cannot?