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Aki Vehtari develops computational Bayesian statistics, including probabilistic programming, model diagnostics, model assessment and Gaussian processes. His author research page describes these as connected parts of a Bayesian workflow: specify a model, compute its implications, check the computation and examine predictive adequacy.^[ref]

He coauthored Bayesian Data Analysis by Andrew Gelman and others. A posterior distribution summarizes uncertainty conditional on the model and data; checking whether that model reproduces important observations is a further task. This distinction connects Bayesian inference with posterior predictive checks, rather than treating successful computation as sufficient evidence that the model is suitable.^[ref]

Andrew Gelman works on Bayesian statistics, statistical modelling and the interpretation of data in scientific and social questions. His research and teaching connect hierarchical Bayesian models, regression, computation and statistical graphics. His Columbia research page describes applications ranging from survey inference and elections to environmental and medical measurements.^[ref]

A characteristic modelling problem is learning about many related groups: completely separate estimates can be noisy, while a single common estimate can hide differences. A hierarchical Bayesian model estimates group parameters together and lets the data influence their degree of pooling. Model checking, including posterior predictive checks, then asks whether simulated observations reproduce scientifically relevant features of the actual data. This connects fitting a model with examining the adequacy of its assumptions.

For data $D$, parameters $\theta$ and model $M$, conditional probability givesThe likelihood describes the measurement process, the prior encodes parameter information before these data, and the Bayesian evidence normalizes the posterior. A prior is a probability density with respect to a specified measure; a uniform prior in distance is not uniform in parallax. Bayes' theorem combines conditional probabilities and does not itself select an appropriate likelihood or prior.

An astronomical application starts with a forward model. For measured fluxes $d_i$ with predictions $f_i(\theta )$ and known Gaussian covariance $C$, $L\propto |C{|}^{-1/2}\exp [-(d-f{)}^{\mathsf{T}}{C}^{-1}(d-f)/2]$. Correlated calibration errors belong in $C$ or in explicit calibration nuisance parameters; assuming every filter or epoch is independent can give falsely narrow uncertainties. For a faint photon-counting source, the more appropriate likelihood can instead be $p(N\mid s,b)=e^{-(s+b)}(s+b{)}^N/N!$, with nonnegative expected source and background counts. Background measurements constrain $b$, which is then marginalized rather than set to an exact subtracted number. The observed background-subtracted count can be negative even though the physical expected source count is nonnegative.

For a simple flux example with $d\mid F\sim \mathcal{N}(F,{\sigma }^2)$ and a Gaussian calibration prior $F\sim \mathcal{N}(F_0,{\sigma }_0^2)$, completing the square givesThis explicitly shows how independent information combines. A physical nonnegative-flux prior would truncate or replace that illustrative Gaussian prior. Photometric-redshift fitting similarly compares predicted galaxy colours with measured fluxes, then integrates over luminosity, dust and other nuisance parameters to obtain $p(z\mid D)$. Multimodal posteriors should be retained when different spectral features produce similar colours; a best fit alone hides that ambiguity. The same framework applies to orbit parameters, stellar temperatures, cosmological parameters and cluster SZ amplitudes.

credible intervals state posterior probability conditional on the model and prior. posterior predictive checks compare replicated observations with the data and can reveal a deficient noise or source model. Sensitivity to plausible priors is especially important for weak signals and inverse problems. For competing models, posterior odds equal prior odds times the Bayes factor $Z_1/Z_2$; the evidence averages fit quality over the prior volume and therefore depends on proper, physically justified prior ranges. Detection thresholds and catalogue selection must enter the sampling model when inferring a population, otherwise a carefully calculated posterior can still describe a biased sample. The practical gain is a joint uncertainty distribution with nuisance parameters propagated, not merely a single fitted number.

Model the count relative to exposure, for exampleHere $E_i$ is a measured exposure such as patient-days, $x_i$ contains hospital predictors, and $b_i$ is a random effect. This Poisson regression separates hospital size from rate variation and supplies partial pooling. Use proper, scale-appropriate priors on coefficients and variation; add regional or temporal effects if the data demand them. A gamma mixing law with an estimated shape is another way to allow more flexible overdispersion. Compare the fitted model's counts and dispersion through posterior predictive checks.

Fit both hierarchies to the historical data and compare their posterior predictive checks. Use discrepancies sensitive to between-individual tails, such as the maximum standardized personal mean, tail counts and the spread of individual sample means. Simulate new individual means as well as repeated observations; conditioning only on fitted personal means would conceal a poor population model.

Leave whole individuals out for cross-validation, rather than just measurements, when the aim is prediction for a new individual. Compare held-out predictive log densities or interval calibration, with uncertainty in the comparison. Check both fits' Markov chain Monte Carlo convergence diagnostics and sensitivity to scale priors. Better tail prediction and adequate calibration would support the heavier-tailed model; the data are required to decide, so no empirical winner can be asserted here.

For each plate use its conditional fitted intensity, including the fitted random effect. At a posterior draw $s$ form the Pearson residualA simpler plot uses ${\hat{\mu }}_{ij}$ in this expression, but posterior draws also display uncertainty in the fitted values. Look for residuals centred around zero, unexpectedly large tails or isolated outliers, and patterns with dose, fitted intensity or plate grouping. For known conditional parameters the Poisson distribution gives mean zero and variance one for this residual; approximate normality is reasonable only at adequate count sizes. Fitting the same data and estimating latent effects changes its calibration, so compare the diagnostics with residuals from conditional posterior predictive checks, rather than treating them as independent exact standard normals.

This checks the plate-level Poisson sampling layer given the effects and can reveal remaining unmodelled variation. It does not by itself establish the population dose-response curve: fitted plate effects can absorb departures from that curve.

Fit two hierarchical Bayesian models to the same observations, one with the bounded uniform distribution sampling density and the other with an exponential distribution density. Specify rate versus mean in the exponential model and assign appropriate, scientifically comparable proper prior distributions; that parameter is no longer a literal maximum size. Obtain posterior distributions, for example by Markov chain Monte Carlo, and check convergence.

At the same observational level in both models use the Bayesian deviance $D(\varphi )=-2\log p(\mathbf{y}\mid \varphi )$, excluding prior densities. Retain the same likelihood constants and consistently either condition on breed effects or integrate them out. Estimate $\overline{D}=\mathbb{E}[D(\varphi )\mid \mathbf{y}]$, evaluate $D(\overline{\varphi })$ at the posterior mean, and compute the effective parameter count in DIC:Smaller DIC favours the fit–complexity tradeoff. Supplement it with posterior predictive checks; the uniform model's parameter-dependent support and the hierarchical structure make the criterion a diagnostic rather than an automatic definitive decision.

Fit both the normal and Student t random-effect model with comparable proper prior distributions. Compare priors on the same spread measure: a normal distribution scale $\tau$ is a standard deviation, whereas the $t_4$ standard deviation is $\sqrt{2}\psi$.

Use a posterior predictive check: draw study effects and binomial counts from each fitted hierarchy and compare replicated dispersion and extreme study contrasts with the observations. For predicting a new study, generate a new effect from the hierarchy rather than reusing an existing fitted effect. A Leave-one-out cross-validation with entire studies held out can compare integrated predictive probabilities for both arms of each omitted trial, averaging over hyperparameters and its unobserved study effect. Leave-one-study-out influence analysis also reveals whether the difference is driven by a single trial.

Prefer the heavier-tailed hierarchy if it improves the relevant predictive checks and held-out study predictions robustly to reasonable prior choices. The deviance information criterion can supplement the comparison, but its effective parameter count can depend on the latent-variable representation, and six studies give limited information about tail shape. A small numerical criterion difference alone is insufficient evidence.

The likelihood is $L(\theta ;d)=p(d\mid \theta )$ considered as a function of parameters at the observed data. It is not itself a probability distribution over $\theta$. Bayes' theorem givesThe prior represents parameter information before these data; the denominator normalizes the posterior and can compare fully specified models with compatible prior assumptions. Detector response, backgrounds, dependence and selection must be included in the data model.

For independent measurements $x_i\sim N(\theta ,{\sigma }_i^2)$ with known errors, $\log L=\mathrm{constant}-\frac{1}{2}{\sum }_i(x_i-\theta {)}^2/{\sigma }_i^2$. Maximum likelihood minimizes this chi-squared and gives $\hat{\theta }={\sum }_i{w}_i{x}_i/{\sum }_i{w}_i$, $w_i={\sigma }_i^{-2}$. A Gaussian prior $N({\mu }_0,s_0^2)$ instead yieldsThis explicitly combines data and prior precision, approaching the weighted mean when prior information is weak. Unknown calibration parameters should be jointly inferred and marginalized, not silently fixed. posterior intervals are conditional on the assumed model and prior; posterior predictive checks and sensitivity to plausible priors help expose poor fits and weakly identified parameters. The maximum-likelihood estimate and a Bayesian posterior answer different inferential questions even when their numerical centres agree.

Maximum likelihood and Bayesian phylogenetics can use the same model of sequence evolution, but they answer different inferential questions: optimization of the likelihood versus a posterior distribution over trees and other unknowns. Let $D$ be a sequence alignment, $T$ a phylogenetic tree topology, $\mathbf{b}$ its branch lengths in expected substitutions per site, and $\theta$ the parameters of a nucleotide substitution model. Both methods can useThe likelihood function sums over unobserved ancestral states, rather than choosing one ancestral sequence and treating it as observed. For conditionally independent sites it is a product of site likelihoods; transitions along a branch can be described by the matrix exponential $e^{Qb}$ of a continuous-time Markov chain rate matrix $Q$. The phylogenetic pruning algorithm computes the ancestral sums efficiently.^[ref]

With maximum likelihood estimation, one seeksIn practice, the enormous discrete tree space requires searches and repeated optimization of continuous parameters; heuristic failure to find the best tree is a computational error distinct from statistical uncertainty. A tree's optimized likelihood is not the probability that it is correct. bootstrap analyses resample alignment columns and repeat inference to assess support under the resampling procedure; their frequencies are not automatically posterior probabilities. Independence assumptions matter, because linked sites and heterogeneous loci can make simple column resampling an imperfect representation of replication.

In Bayesian phylogenetic inference,The prior distribution supplies weights on topologies, branch lengths and model parameters. Integrating over nuisance parameters givesThus a very narrow likelihood peak can lose posterior support to a broader region with a slightly lower maximum. maximum a posteriori estimation is another optimization and does not replace this integration. Markov chain Monte Carlo provides dependent samples approximating the posterior distribution; a clade's posterior probability is the posterior mass of trees containing that clade. The early likelihood-based Bayesian formulation explicitly distinguished these probabilities from bootstrap support.^[ref]

Neither approach is protected against systematic error. Poor sequence alignment, incorrect orthology assignments, compositionally heterogeneous sequences, and an inadequate substitution process can support a wrong tree with high apparent confidence. A gene tree can differ from a species tree because of incomplete lineage sorting or introgression; extra sequence from that gene does not automatically fix the mismatch. Model comparison, biological examination of the data and sensitivity to locus sampling are needed in both approaches. In Bayesian analyses, agreement among short chains is not proof of convergence: independent runs, adequate mixing between topologies and an appropriate effective sample size of a Markov chain should be checked. A credible interval and a confidence interval also have different interpretations even if their numerical endpoints happen to resemble one another.

Time inference adds a particularly strong role for assumptions. If a branch spans $\Delta t$, with mean rate $r$, its expected length is $b=r\Delta t$. Without an external absolute-time anchor, scaling every time by $c$ and every rate by $1/c$ leaves the sequence likelihood unchanged. Even a very long alignment can identify branch lengths accurately without identifying an absolute calendar scale. For example, a lineage length $b=0.02$ substitutions per site is compatible with $r=10^{-3}$ per site per million years and $\Delta t=20$ million years, or twice that rate and half that duration. This is an identifiability issue, not merely a shortage of sampled bases.

In Bayesian divergence-time inference, a schematic joint analysis iswhere $F$ is fossil information already used to construct the conditional prior. An alternative generative analysis includes a fossil-sampling likelihood explicitly; the same fossil evidence must not be counted twice. A strict molecular clock equates rates across branches; a relaxed molecular clock allows variation, whose distribution and correlations also require specification. A narrow rate prior favors particular absolute ages, and a strong rate-smoothing assumption can redistribute inferred time among branches. A birth-death process prior on the tree encodes assumptions about lineage origination, extinction and sampling; it is not a neutral choice just because it is mathematically convenient.

fossil calibration must separate a fossil's geological age from the age of the lineage split it constrains. A confidently placed crown fossil gives a minimum age for the relevant crown, whereas a stem fossil does not automatically set that crown minimum. Uncertain placement and the incompleteness of the record complicate upper bounds. hard calibration bounds assign zero probability outside the permitted range; enough sequence cannot overcome a zero prior. soft calibration bounds allow small tail probabilities, exposing rather than eliminating the possibility of conflict. Multiple-calibration analyses demonstrate that changing the treatment of bounds changes inferred times and uncertainty.^[ref]

Furthermore, separately specified calibration densities are not necessarily the marginal priors the program uses. The rule that an ancestor must be older than its descendant truncates their joint possibilities. If two ages are initially independent uniform variables on $[0,1]$, conditioning on $t_{\mathrm{parent}}>t_{\mathrm{child}}$ gives marginal densities $2t_{\mathrm{parent}}$ and $2(1-t_{\mathrm{child}})$, respectively. Neither remains uniform. Combining tree-process priors, calibration densities and ordering constraints can therefore introduce substantial information unintentionally.^[ref]

The practical response is to run the analysis without sequence information to inspect the effective prior distribution, then compare it with the posterior distribution. Repeat under defensible calibration tails, rate models, tree priors and fossil placements; report which ages change and whether the data sharpen relative ages more than absolute ages. Examine prior predictive checks and posterior predictive checks, and keep numerical mixing diagnostics separate from biological model adequacy. Priors can legitimately supply the time information missing from sequences, but narrow posterior ages are persuasive only when that information and its sensitivity are explicit.

A posterior predictive check compares a summary of the observed data with the same summary in replicated data drawn from a model's posterior predictive distribution. The replication must match the target: replicating an existing group's observations conditional on its effect differs from predicting a new group with a newly drawn effect. Such checks expose mismatches in dispersion, tails or other features; using the observations to fit and check the model means they are not automatically calibrated frequentist tests.

A measurable summary of a possible dataset is chosen to detect disagreement with a specified predictive model. Compare its observed value with its distribution under replicated datasets. Large-discrepancy statistics can be checked with $\mathbb{P}(T(X^{\mathrm{rep}})\ge T(x))$. A posterior predictive check integrates over posterior parameter uncertainty; a fully specified null instead supplies a fixed reference law.

A prior predictive check simulates parameters from the prior distribution and observations from their sampling model, then examines whether those observations are plausible before conditioning on data. This reveals unintended information from apparently broad priors, especially after nonlinear transformations or when scales and correlations are coupled. It differs from a posterior predictive check, which uses the posterior distribution.