Communication and interpretation: posterior summaries, predictive uncertainty, decision reporting, and reproducibility

What you should be able to do after this chapter

define posterior summaries as functionals of posterior draws rather than as isolated scalar estimates;
distinguish equal-tail intervals, HPD sets, and predictive intervals;
separate parameter uncertainty from outcome uncertainty mathematically;
report results on scientifically meaningful transformed scales;
construct decision-oriented posterior summaries that remain transparent about assumptions.

1 Reporting is part of inference

Bayesian inference is not finished when posterior draws are obtained. The analyst must decide which posterior functionals matter for the scientific question. That step is substantive, not cosmetic.

If $\theta$ is the model parameter but the scientific estimand is

\[ \psi = g(\theta), \]

then the real inferential target is the posterior of $\psi$:

\[ p(\psi \mid y). \]

Reporting $\theta$ directly when collaborators care about $g(\theta)$ is a failure of translation.

2 Posterior summaries as functionals

For any posterior distribution $p(\theta \mid y)$, common summaries are functionals:

posterior mean \[ E(\theta \mid y); \]
posterior median, defined coordinatewise or marginally;
posterior mode \[ \arg\max_\theta p(\theta \mid y); \]
posterior standard deviation \[ \sqrt{\operatorname{Var}(\theta \mid y)}; \]
posterior tail probabilities such as \[ p(\theta > 0 \mid y). \]

None of these is universally best. The choice depends on loss function, interpretability, and audience.

3 Credible sets

A set $C_\alpha(y)$ is a $(1-\alpha)$ posterior credible set if

\[ p(\theta \in C_\alpha(y) \mid y)=1-\alpha. \]

3.1 Equal-tail intervals

For a scalar parameter,

\[ C_\alpha^{\mathrm{ET}}(y) = \left[ F^{-1}_{\theta \mid y}\left(\frac{\alpha}{2}\right), F^{-1}_{\theta \mid y}\left(1-\frac{\alpha}{2}\right) \right]. \]

3.2 Highest posterior density sets

An HPD set satisfies:

it has posterior probability $1-\alpha$;
every point inside has density at least as large as every point outside.

HPD sets are natural for skewed posteriors, but they can be awkward or nonunique in multimodal settings.

3.3 Credible versus confidence intervals

A Bayesian credible interval is a posterior probability statement conditional on the observed data. A frequentist confidence interval is a procedure with long-run coverage properties. The two objects may be numerically similar in regular models, but they are conceptually different.

4 Transformed and derived quantities

Many scientific questions are about transformations:

odds ratios in logistic regression;
risk differences;
contrasts between treatment arms;
elasticities;
predicted mean differences;
threshold exceedance probabilities.

If posterior draws are $\theta^{(1)},\dots,\theta^{(S)}$, then transformed draws

\[ \psi^{(s)} = g(\theta^{(s)}) \]

automatically approximate the posterior of $\psi$.

This is one of the strongest practical advantages of Bayesian computation: nonlinear uncertainty propagation is handled by transforming draws rather than linearizing everything analytically.

5 Parameter uncertainty versus predictive uncertainty

Analysts frequently conflate uncertainty about parameters with uncertainty about future outcomes. The law of total variance separates them:

\[ \operatorname{Var}(\tilde y \mid y) = E\left[\operatorname{Var}(\tilde y \mid \theta,y)\mid y\right] + \operatorname{Var}\left(E[\tilde y \mid \theta,y]\mid y\right). \]

The first term is irreducible observational noise under the model. The second is uncertainty induced by not knowing $\theta$ exactly.

5.1 Why the distinction matters

If one reports only credible intervals for coefficients, collaborators may think the study says more about future observables than it actually does. Predictive intervals are often wider and more decision-relevant.

6 Decision-oriented summaries

Suppose actions lie in $\mathcal{A}$ and the loss is $L(a,\theta)$. The posterior risk is

\[ \rho(a \mid y)=E[L(a,\theta)\mid y]. \]

The Bayes action is

\[ a^\star(y)=\arg\min_{a \in \mathcal{A}} \rho(a \mid y). \]

Reporting should then include:

the recommended action;
the posterior risk of that action;
the posterior risks of close alternatives;
how the recommendation changes under plausible alternative losses.

This is more honest than pretending that one posterior mean alone determines the decision.

7 Worked example: Logistic regression communication

Suppose

\[ \Pr(Y=1 \mid x,\beta)=\operatorname{logit}^{-1}(x^\top \beta). \]

The coefficient $\beta_j$ is a log-odds change. Many collaborators do not think in log odds. More meaningful summaries may include:

the posterior odds ratio \[ \exp(\beta_j); \]
the risk difference at representative covariate values \[ \operatorname{logit}^{-1}(x_1^\top \beta) - \operatorname{logit}^{-1}(x_0^\top \beta); \]
the posterior probability the intervention improves risk by more than a practically relevant threshold $\delta$: \[ p\left( \operatorname{RD} > \delta \mid y \right). \]

These are more directly aligned with scientific or policy decisions.

8 Visualization as statistical argument

Plots are not decorations. They are inferential objects that emphasize some posterior functionals and suppress others.

Strong Bayesian graphics typically show:

raw uncertainty, not only point summaries;
prior-to-posterior movement when prior sensitivity matters;
predictive distributions, not only parameter intervals;
differences between groups on substantively relevant scales.

Visualization is part of model criticism as well as communication (Gabry et al. 2019).

9 Reproducibility and literate reporting

A credible Bayesian report should make the following traceable:

the model formula;
prior specification and rationale;
computation engine and tuning choices;
diagnostics;
predictive checks;
sensitivity analysis;
code and environment.

Literate tools such as Quarto make it possible to couple derivation, code, diagnostics, and narrative in one document.

10 `R` implementation

The companion lab is posterior_reporting_summary.R. It:

constructs posterior draws for a simple regression example;
reports coefficient summaries, transformed-scale summaries, and predictive intervals;
separates parameter uncertainty from predictive noise;
prints a compact decision-oriented summary table.

Posterior summaries from draws are straightforward:

summarize_draws <- function(x) {
  c(
    mean = mean(x),
    sd = sd(x),
    q05 = unname(quantile(x, 0.05)),
    q50 = unname(quantile(x, 0.50)),
    q95 = unname(quantile(x, 0.95))
  )
}

11 Diagnostics and pitfalls

11.1 Reporting only posterior means

This collapses uncertainty, skewness, and multimodality into a single number.

11.2 Reporting parameter scales that nobody uses

If collaborators need absolute risk, reporting only log odds is not enough.

11.3 Conflating statistical and practical significance

A large posterior probability of a tiny effect does not imply practical importance.

11.4 Hiding model assumptions

Posterior summaries should be accompanied by enough model context that readers know what assumptions produced them.

12 Exercises

Define a posterior credible set and explain how it differs from a confidence interval.
Use the law of total variance to separate predictive uncertainty into noise and parameter-uncertainty components.
For a logistic model, describe three transformed quantities more interpretable than raw coefficients.
Explain why posterior tail probabilities may be more useful than means for threshold-based decisions.
Draft a one-paragraph posterior summary for a nontechnical collaborator that preserves uncertainty honestly.

13 Summary

Communication is not an afterthought. It determines whether the posterior is translated into the right scientific object:

posterior summaries are loss-dependent functionals;
credible sets and predictive intervals answer different questions;
transformed quantities often matter more than primitive parameters;
decision reporting requires posterior risk, not only coefficient tables;
reproducible reporting keeps the inferential chain auditable.

14 Further reading

Gabry et al. (2019) is the reference for visualisation in a Bayesian workflow, and treats graphics as part of the inference rather than as decoration added at the end.

McElreath (2020) is the best model for how to explain Bayesian results to a reader who does not want the machinery, and is particularly good on why a credible interval should not be described as though it were a confidence interval.

Gelman et al. (2013) chapter 9 covers decision-relevant summaries, and Berger (1985) the underlying point that which summary to report depends on the loss the reader faces. Gelman et al. (2020) covers the reproducibility side, where the argument is less about statistics than about making the analysis inspectable.

References

Berger, James O. 1985. Statistical Decision Theory and Bayesian Analysis. 2nd ed. Springer.

Gabry, Jonah, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman. 2019. “Visualization in Bayesian Workflow.” Journal of the Royal Statistical Society: Series A 182 (2): 389–402.

Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. CRC Press.

Gelman, Andrew, Aki Vehtari, Daniel Simpson, et al. 2020. “Bayesian Workflow.” arXiv Preprint arXiv:2011.01808.

McElreath, Richard. 2020. Statistical Rethinking. 2nd ed. CRC Press.

--- title: "Communication and interpretation: posterior summaries, predictive uncertainty, decision reporting, and reproducibility" --- ::: {.callout-note} ## What you should be able to do after this chapter - define posterior summaries as functionals of posterior draws rather than as isolated scalar estimates; - distinguish equal-tail intervals, HPD sets, and predictive intervals; - separate parameter uncertainty from outcome uncertainty mathematically; - report results on scientifically meaningful transformed scales; - construct decision-oriented posterior summaries that remain transparent about assumptions. ::: ## Reporting is part of inference Bayesian inference is not finished when posterior draws are obtained. The analyst must decide which posterior functionals matter for the scientific question. That step is substantive, not cosmetic. If $\theta$ is the model parameter but the scientific estimand is $$ \psi = g(\theta), $$ then the real inferential target is the posterior of $\psi$: $$ p(\psi \mid y). $$ Reporting $\theta$ directly when collaborators care about $g(\theta)$ is a failure of translation. ## Posterior summaries as functionals For any posterior distribution $p(\theta \mid y)$, common summaries are functionals: - posterior mean $$ E(\theta \mid y); $$ - posterior median, defined coordinatewise or marginally; - posterior mode $$ \arg\max_\theta p(\theta \mid y); $$ - posterior standard deviation $$ \sqrt{\operatorname{Var}(\theta \mid y)}; $$ - posterior tail probabilities such as $$ p(\theta > 0 \mid y). $$ None of these is universally best. The choice depends on loss function, interpretability, and audience. ## Credible sets A set $C_\alpha(y)$ is a $(1-\alpha)$ posterior credible set if $$ p(\theta \in C_\alpha(y) \mid y)=1-\alpha. $$ ### Equal-tail intervals For a scalar parameter, $$ C_\alpha^{\mathrm{ET}}(y) = \left[ F^{-1}_{\theta \mid y}\left(\frac{\alpha}{2}\right), F^{-1}_{\theta \mid y}\left(1-\frac{\alpha}{2}\right) \right]. $$ ### Highest posterior density sets An HPD set satisfies: 1. it has posterior probability $1-\alpha$; 2. every point inside has density at least as large as every point outside. HPD sets are natural for skewed posteriors, but they can be awkward or nonunique in multimodal settings. ### Credible versus confidence intervals A Bayesian credible interval is a posterior probability statement conditional on the observed data. A frequentist confidence interval is a procedure with long-run coverage properties. The two objects may be numerically similar in regular models, but they are conceptually different. ## Transformed and derived quantities Many scientific questions are about transformations: - odds ratios in logistic regression; - risk differences; - contrasts between treatment arms; - elasticities; - predicted mean differences; - threshold exceedance probabilities. If posterior draws are $\theta^{(1)},\dots,\theta^{(S)}$, then transformed draws $$ \psi^{(s)} = g(\theta^{(s)}) $$ automatically approximate the posterior of $\psi$. This is one of the strongest practical advantages of Bayesian computation: nonlinear uncertainty propagation is handled by transforming draws rather than linearizing everything analytically. ## Parameter uncertainty versus predictive uncertainty Analysts frequently conflate uncertainty about parameters with uncertainty about future outcomes. The law of total variance separates them: $$ \operatorname{Var}(\tilde y \mid y) = E\left[\operatorname{Var}(\tilde y \mid \theta,y)\mid y\right] + \operatorname{Var}\left(E[\tilde y \mid \theta,y]\mid y\right). $$ The first term is irreducible observational noise under the model. The second is uncertainty induced by not knowing $\theta$ exactly. ### Why the distinction matters If one reports only credible intervals for coefficients, collaborators may think the study says more about future observables than it actually does. Predictive intervals are often wider and more decision-relevant. ## Decision-oriented summaries Suppose actions lie in $\mathcal{A}$ and the loss is $L(a,\theta)$. The posterior risk is $$ \rho(a \mid y)=E[L(a,\theta)\mid y]. $$ The Bayes action is $$ a^\star(y)=\arg\min_{a \in \mathcal{A}} \rho(a \mid y). $$ Reporting should then include: - the recommended action; - the posterior risk of that action; - the posterior risks of close alternatives; - how the recommendation changes under plausible alternative losses. This is more honest than pretending that one posterior mean alone determines the decision. ## Worked example: Logistic regression communication Suppose $$ \Pr(Y=1 \mid x,\beta)=\operatorname{logit}^{-1}(x^\top \beta). $$ The coefficient $\beta_j$ is a log-odds change. Many collaborators do not think in log odds. More meaningful summaries may include: - the posterior odds ratio $$ \exp(\beta_j); $$ - the risk difference at representative covariate values $$ \operatorname{logit}^{-1}(x_1^\top \beta) - \operatorname{logit}^{-1}(x_0^\top \beta); $$ - the posterior probability the intervention improves risk by more than a practically relevant threshold $\delta$: $$ p\left( \operatorname{RD} > \delta \mid y \right). $$ These are more directly aligned with scientific or policy decisions. ## Visualization as statistical argument Plots are not decorations. They are inferential objects that emphasize some posterior functionals and suppress others. Strong Bayesian graphics typically show: - raw uncertainty, not only point summaries; - prior-to-posterior movement when prior sensitivity matters; - predictive distributions, not only parameter intervals; - differences between groups on substantively relevant scales. Visualization is part of model criticism as well as communication [@gabry2019]. ## Reproducibility and literate reporting A credible Bayesian report should make the following traceable: - the model formula; - prior specification and rationale; - computation engine and tuning choices; - diagnostics; - predictive checks; - sensitivity analysis; - code and environment. Literate tools such as Quarto make it possible to couple derivation, code, diagnostics, and narrative in one document. ## `R` implementation The companion lab is [posterior_reporting_summary.R](../../labs/r/posterior_reporting_summary.R). It: - constructs posterior draws for a simple regression example; - reports coefficient summaries, transformed-scale summaries, and predictive intervals; - separates parameter uncertainty from predictive noise; - prints a compact decision-oriented summary table. Posterior summaries from draws are straightforward: ```r summarize_draws <- function(x) { c( mean = mean(x), sd = sd(x), q05 = unname(quantile(x, 0.05)), q50 = unname(quantile(x, 0.50)), q95 = unname(quantile(x, 0.95)) ) } ``` ## Diagnostics and pitfalls ### Reporting only posterior means This collapses uncertainty, skewness, and multimodality into a single number. ### Reporting parameter scales that nobody uses If collaborators need absolute risk, reporting only log odds is not enough. ### Conflating statistical and practical significance A large posterior probability of a tiny effect does not imply practical importance. ### Hiding model assumptions Posterior summaries should be accompanied by enough model context that readers know what assumptions produced them. ## Exercises 1. Define a posterior credible set and explain how it differs from a confidence interval. 2. Use the law of total variance to separate predictive uncertainty into noise and parameter-uncertainty components. 3. For a logistic model, describe three transformed quantities more interpretable than raw coefficients. 4. Explain why posterior tail probabilities may be more useful than means for threshold-based decisions. 5. Draft a one-paragraph posterior summary for a nontechnical collaborator that preserves uncertainty honestly. ## Summary Communication is not an afterthought. It determines whether the posterior is translated into the right scientific object: - posterior summaries are loss-dependent functionals; - credible sets and predictive intervals answer different questions; - transformed quantities often matter more than primitive parameters; - decision reporting requires posterior risk, not only coefficient tables; - reproducible reporting keeps the inferential chain auditable. ## Further reading @gabry2019 is the reference for visualisation in a Bayesian workflow, and treats graphics as part of the inference rather than as decoration added at the end. @mcelreath2020 is the best model for how to explain Bayesian results to a reader who does not want the machinery, and is particularly good on why a credible interval should not be described as though it were a confidence interval. @gelman2013 chapter 9 covers decision-relevant summaries, and @berger1985 the underlying point that which summary to report depends on the loss the reader faces. @gelman2020 covers the reproducibility side, where the argument is less about statistics than about making the analysis inspectable.