Why posterior geometry matters
An individual PK posterior is not fully described by a list of parameter estimates and marginal intervals. Two parameters can each have substantial uncertainty while their joint values lie along a narrow trade-off; alternatively, both can be uncertain with little linear association. The shape of the joint posterior helps distinguish these situations and indicates which combinations of parameters the observed concentrations constrain.
Posterior correlation is a useful summary, but it is not itself a test of identifiability. It can reveal a linear trade-off in the posterior draws, yet miss curved relationships or other complex shapes. A defensible interpretation considers marginal uncertainty, joint geometry, the parameter scale, and the PK model and observations that produced the posterior.
Let \(\theta_j\) and \(\theta_k\) be two individual parameters, or transformed versions of them, and let posterior draws be \((\theta_j^{(m)},\theta_k^{(m)})\) for \(m=1,\ldots,M\). Their posterior correlation is the covariance divided by the product of their posterior standard deviations:
Here \(y\) denotes the individual concentration data and other observations included in the model. Correlation is dimensionless and lies between \(-1\) and \(1\). A value near \(-1\) or \(1\) indicates strong posterior linear association on the analyzed scale; it does not say whether either marginal distribution is narrow. Correlation is also scale-specific: for positive parameters, the association on the log scale can be more relevant and more nearly linear than on the original scale.
Reading marginal uncertainty and joint shape together
A marginal posterior interval describes uncertainty about one parameter after integrating over the other parameters in the model. A joint posterior plot instead shows which parameter combinations have appreciable posterior support. If the joint draws form a long, narrow ellipse, many combinations along its long axis are compatible with the observations, even though movement across its short axis is constrained. The long direction is the weakly constrained combination; it need not coincide with either parameter axis.
For an approximately bivariate normal posterior, the covariance matrix summarizes this ellipse. In the special case of equal marginal standard deviations \(s\) and correlation \(\rho\), the covariance matrix is \(s^2\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}\). Its eigenvectors are \((1,1)\) and \((1,-1)\), with variances \(s^2(1+\rho)\) and \(s^2(1-\rho)\), respectively. This follows by multiplying the matrix by each vector: the first component direction is scaled by \(s^2(1+\rho)\), and the second by \(s^2(1-\rho)\). Thus, when \(\rho\) is negative and close to \(-1\), the \((1,-1)\) direction has much larger variance than the \((1,1)\) direction.
This derivation describes a local Gaussian geometry, not a universal property of PK posteriors. A posterior may be skewed, curved, heavy-tailed, or multimodal. In those cases the covariance and correlation compress important features into two numbers. Scatterplots of posterior draws, marginal density plots, and summaries on clinically meaningful parameter scales provide complementary evidence.
Worked Example: A Strong Negative Posterior Correlation
Suppose hypothetical posterior draws for two positive individual PK parameters are summarized on the log-ratio scale, relative to fixed reference values. The posterior means of the log ratios are 0.20 and \(-0.10\), and each has posterior standard deviation 0.20. Their posterior correlation is \(-0.95\). Find the covariance, approximate marginal 95% intervals, and the principal-axis variances.
The covariance is the correlation multiplied by the two standard deviations:
Using a normal approximation for these marginals, the interval is the mean plus or minus \(1.96\) posterior standard deviations. For the first log ratio, \(0.20\pm1.96(0.20)=0.20\pm0.392\), giving \([-0.192,0.592]\). For the second, \(-0.10\pm0.392\), giving \([-0.492,0.292]\). These are approximate posterior intervals on the log-ratio scale, not intervals for the untransformed parameters.
With equal standard deviations, the principal-axis variances are \(s^2(1+\rho)\) and \(s^2(1-\rho)\). Substitution gives \(0.0400(0.05)=0.00200\) and \(0.0400(1.95)=0.0780\), respectively. The posterior is therefore much more spread along the \((1,-1)\) direction than along \((1,1)\). The parameter values trade off strongly in the posterior. Neither the correlation nor these intervals establish that the model has identified each parameter precisely; the intervals show notable marginal uncertainty, and the joint plot would show its direction.
Correlation is not a complete identifiability diagnostic
A high absolute posterior correlation is a warning that the parameters may trade off under the fitted model and data. It is not proof of non-identifiability: a strong association can occur even when both marginal posteriors are sufficiently narrow for the intended decision. Conversely, low correlation is not evidence that each parameter is well identified. Marginal uncertainty must be assessed separately, relative to a scientifically meaningful scale or decision threshold.
Pearson correlation measures linear association. If a posterior occupies a curved ridge, positive and negative parts of that curve can cancel in the covariance, leaving a correlation near zero even when the parameters are strongly dependent. A scatterplot or conditional summaries can reveal such geometry. Likewise, a multimodal posterior may have an overall correlation that conceals distinct modes with different parameter combinations.
The following example uses coordinates \(u\) and \(v\) representing two transformed, dimensionless parameter deviations. It illustrates why a correlation summary must be checked against the joint draws.
Worked Example: A Near-Zero Correlation on a Curved Ridge
Consider four representative hypothetical posterior draws in transformed coordinates: \((-1,1)\), \((1,1)\), \((-2,4)\), and \((2,4)\). The pattern is symmetric in \(u\), while \(v\) follows \(u^2\). Calculate the sample correlation and interpret the pattern.
The sample means are \(\bar{u}=0\) and \(\bar{v}=(1+1+4+4)/4=2.5\). The sum of cross-products about the means is \( (-1)(-1.5)+(1)(-1.5)+(-2)(1.5)+(2)(1.5)=0\). Thus the sample covariance, which divides this sum by \(n-1=3\), is zero. The sample variances are \(s_u^2=(1+1+4+4)/3=10/3\) and \(s_v^2=(2.25+2.25+2.25+2.25)/3=3.00\). Therefore:
A correlation of zero here does not indicate posterior independence. The four draws lie exactly on the curved relation \(v=u^2\); in a larger posterior sample, a narrow cloud around a similar curve would still indicate a constrained joint relationship. The calculation is a warning against interpreting correlation without inspecting the joint distribution. Four illustrative draws are not a reliable estimate of a real posterior correlation; the point is the geometry they demonstrate.
A practical diagnostic workflow
Posterior geometry should be interpreted only after checking that the draws adequately represent the posterior. Poorly mixed or unconverged chains can produce misleading marginal intervals and correlations. Conversely, apparently stable sampling diagnostics do not mean that the parameters are practically identifiable: a sampler can accurately explore a broad ridge. Numerical adequacy and parameter information are related but distinct questions.
Use the scale on which the model is parameterized and interpreted. For positive PK parameters, inspect whether log-scale summaries are useful, and translate results back to clinical units when appropriate.
Review posterior intervals and compare their width with a scientifically meaningful range. A narrow interval can still be inadequate for a decision, and a broad one need not make every prediction uninformative.
Use pair plots and, where helpful, conditional summaries to look for tilted ellipses, curved ridges, separated modes, boundaries, or outliers. Do not rely on a correlation matrix alone.
Check whether chains explore the visible regions and whether sampling diagnostics support reliable summaries. If draws do not explore a ridge or modes, improve the computation before drawing an identifiability conclusion.
Evaluate whether the uncertain parameter combinations materially change the concentration forecasts or dosing decision. Parameter uncertainty and predictive uncertainty are not interchangeable.
A local Gaussian approximation, such as a covariance estimate near a posterior mode, can efficiently summarize a smooth, unimodal posterior. It may be misleading when the posterior is skewed, bounded, multimodal, or curved. Direct posterior draws are more informative about non-Gaussian geometry if sampling is adequate. Likelihood-based profiles can help distinguish data support from the influence of the prior, while posterior summaries describe uncertainty under the specified Bayesian model. These views answer related but different questions; none should be treated as a universal substitute for the others.
Worked Example: Broad Marginals With Little Linear Association
Suppose two hypothetical positive individual parameters have log-ratio posterior means of 0 and posterior standard deviations 0.50 and 0.45. Their posterior correlation is 0.08. Using a normal approximation, calculate their marginal 95% intervals and explain what the summaries do and do not show.
The covariance is \(0.08(0.50)(0.45)=0.0180\). The approximate intervals are \(0\pm1.96(0.50)=[-0.980,0.980]\) and \(0\pm1.96(0.45)=[-0.882,0.882]\), both on the log-ratio scale. Exponentiating the endpoints gives approximate ratio intervals of \([e^{-0.980},e^{0.980}]=[0.375,2.66]\) and \([e^{-0.882},e^{0.882}]=[0.414,2.42]\).
The small correlation says only that the posterior has little linear association on this scale. The wide intervals indicate substantial marginal uncertainty: each parameter can vary over a broad range relative to its reference value. The joint draws still need inspection for nonlinear structure, and the interval widths must be judged against the intended clinical decision. This example does not establish whether the uncertainty comes from limited data, the prior, or both.
Common misinterpretations and limitations
- “A correlation near one means neither parameter is identifiable.” Not necessarily. It indicates a strong linear posterior association. Inspect each marginal interval and the relevant parameter combinations before judging practical adequacy.
- “A correlation near zero means the parameters are independent.” This is false in general. Zero Pearson correlation rules out only linear association as summarized by covariance; curved dependence and multimodality can remain.
- “A converged sampler proves that the data identify the parameters.” Convergence concerns whether computation has represented the specified posterior. A well-sampled posterior can still be broad or ridge-shaped.
- “A narrow posterior interval proves that the likelihood is informative.” The posterior also depends on the prior. A narrow interval alone cannot assign the source of information.
- “Uncertain parameters imply useless forecasts.” Not always. If uncertain parameter combinations produce similar predictions for the decision-relevant times and doses, predictive uncertainty can be smaller than parameter uncertainty suggests. Conversely, a modest parameter uncertainty can matter greatly for a sensitive prediction.
Posterior geometry is conditional on the model, prior, observations, and parameterization. A pair plot cannot diagnose every model misspecification, and a single posterior correlation cannot establish structural identifiability. Conclusions should therefore be appropriately scoped: describe what the fitted posterior shows, which parameter combinations remain uncertain, and whether that uncertainty matters for the intended forecast or dose adaptation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If two log-parameter posterior marginals are narrow but their correlation is \(-0.90\), what does the correlation establish, and what does it not establish?
- Can two posterior parameters have correlation zero while being dependent? Give the reason.
- A sampler explores a long, narrow posterior ridge reliably. Is this evidence of failed computation, weak practical identification, both, or neither?
- Why should a broad marginal posterior interval be considered relative to a clinical or decision-relevant scale?
Solutions. (1) The value indicates strong negative linear association on the log scale, or a trade-off in the posterior. It does not by itself establish that either marginal is imprecise or that the data fail to identify every relevant combination. (2) Yes. Nonlinear dependence, such as a symmetric curved relation, can have zero covariance and therefore zero Pearson correlation. (3) Reliable exploration argues against failed computation as the explanation for the sampled ridge, but the ridge is evidence that some parameter combinations remain weakly constrained, so practical identifiability may be limited. (4) Statistical width alone does not determine whether uncertainty affects use: the same interval may be negligible for one decision and consequential for another. Interpret it against the parameter’s units, plausible range, and impact on the intended forecast or dosing decision.