Why an Individual Posterior Can Have Several Useful Summaries
After individual Bayesian updating, the result is a posterior distribution for a patient's pharmacokinetic parameters, not a single uniquely privileged number. The likelihood describes how the patient's concentration data relate to candidate parameter values, while the prior represents information available before those data. In brief, for an individual parameter vector \(\theta\) and data \(y\), the posterior is proportional to the likelihood times the prior:
The posterior can be asymmetric, especially when data are limited, parameters are constrained to be positive, or the likelihood and prior combine to leave a long tail. In that setting, the maximum a posteriori estimate (MAP), posterior mean, and posterior median need not agree. Each is a different summary of the same distribution, and each answers a different question.
These definitions apply to a specified parameter and a specified posterior distribution. They do not say that one summary is always the best individual estimate. The choice depends on the purpose of the estimate, the consequences of error, and—especially for a MAP estimate—the scale on which the posterior density is expressed.
Different Summaries Correspond to Different Questions
The posterior mean and median have direct decision interpretations when a loss function is specified. If an estimate \(a\) is judged by squared error, \(L(a,\theta)=(a-\theta)^2\), the posterior expected loss is minimized by the posterior mean, provided the second moment exists. If the loss is absolute error, \(L(a,\theta)=|a-\theta|\), any posterior median minimizes expected loss. These claims concern the best action under the stated loss, not a universal ranking of summaries.
The MAP estimate instead identifies the highest point of the posterior density. It can be useful when the goal is to report the most supported local parameter value or when an estimation procedure is explicitly defined by posterior-density maximization. But it is not generally the estimate that minimizes squared or absolute error. In a continuous distribution, the probability of any one exact parameter value is zero; the MAP is a density peak, not the parameter value with a positive posterior probability of occurring.
Decide whether the quantity to summarize is a parameter such as clearance, a transformation such as log clearance, or a predicted quantity such as exposure.
Squared-error estimation points to a posterior mean; absolute-error estimation points to a posterior median. Density maximization points to a MAP estimate on a declared scale.
Check skewness, tails, boundaries, and multiple peaks before treating one summary as representative of the full individual posterior.
A Lognormal Individual Posterior
A positive parameter such as clearance is often summarized using a lognormal posterior in an illustrative calculation. Let \(C\) denote an individual's clearance, measured in L/h, and suppose the posterior for \(\eta=\log C\) is normal:
Here \(\mu\) and \(\sigma\) describe the posterior distribution of log clearance. The posterior median of \(C\) is \(\exp(\mu)\), while the mean is \(\exp(\mu+\sigma^2/2)\). The mode of the density expressed on the clearance scale is \(\exp(\mu-\sigma^2)\). Thus, for a nonzero \(\sigma\), the natural-scale MAP, median, and mean differ. The derivation below also shows why the MAP depends on parameterization.
Worked Example: Three Summaries of Individual Clearance
Suppose a hypothetical patient's posterior has \(\log C\mid y\sim N(\log 8,0.3^2)\), with \(C\) measured in L/h. Find the posterior median and mean of clearance, and the MAP when the density is expressed on the clearance scale.
Median. A normal distribution has median \(\log 8\) on the log scale. Since the exponential function is increasing, the clearance median is \( \exp(\log 8)=8.00\) L/h.
Mean. For a lognormal variable, \(E(C\mid y)=\exp(\mu+\sigma^2/2)\). Substituting \(\mu=\log 8\) and \(\sigma=0.3\) gives \( \exp(\log 8+0.3^2/2)=8\exp(0.045)=8.368\) L/h.
Clearance-scale MAP. The lognormal density on the clearance scale is maximized at \(\exp(\mu-\sigma^2)\). Therefore, \( \exp(\log 8-0.3^2)=8\exp(-0.09)=7.311\) L/h.
The posterior has a longer tail toward high clearances, so its mean lies above its median; its density peak lies below the median. These are not contradictory calculations. They summarize different features of the same skewed posterior.
Why a MAP Estimate Changes With Parameterization
A density is defined relative to its variable's scale. When changing variables, the density must include a Jacobian. That adjustment can shift the point where the density is largest. For the lognormal example, the density of \(\eta=\log C\) is normal, with its mode at \(\eta=\mu\). Exponentiating that log-scale mode gives \(\exp(\mu)\), which is the clearance median—not the clearance-scale MAP.
To find the clearance-scale mode directly, the transformed density is \(p_C(c\mid y)=p_\eta(\log c\mid y)/c\), for \(c>0\). Ignoring constants that do not depend on \(c\), its log density is
Differentiate with respect to \(c\) and set the derivative to zero:
The second derivative at this point is negative, so it is a maximum; equivalently, the lognormal density rises before this point and falls after it. Thus the clearance-scale mode is \(\exp(\mu-\sigma^2)\). By contrast, maximizing the density of \(\eta\) gives \(\mu\) and then transforming gives \(\exp(\mu)\). A report of “the MAP” should therefore state the scale on which the posterior density was maximized.
Worked Example: A Gamma Posterior With Strong Right Skew
A second distribution makes the separation among summaries clear without relying on a lognormal formula. Suppose a hypothetical individual clearance has posterior density \(p(c\mid y)=\beta^2c\exp(-\beta c)\) for \(c>0\), where \(\beta=0.5\) h/L. This is a Gamma distribution with shape 2 and rate 0.5; it is a proper, positive, right-skewed posterior.
The posterior mean for a Gamma distribution with shape \(k\) and rate \(\beta\) is \(k/\beta\), so the mean is \(2/0.5=4.00\) L/h. The density is proportional to \(c\exp(-\beta c)\). Differentiating its log gives \(1/c-\beta\), which is zero at \(c=1/\beta=2.00\) L/h; the density increases before and decreases after this value, so the MAP is 2.00 L/h.
The median is defined by cumulative probability 0.5. For this shape-2 Gamma distribution, the cumulative distribution function is \(1-\exp(-\beta c)(1+\beta c)\). The median \(m\) therefore satisfies
Solving this one-dimensional equation gives \(m\approx3.357\) L/h. For example, at \(m=3.3567\), \(\exp(-1.67835)(1+1.67835)\approx0.5000\), so the cumulative probability is approximately 0.5000. The three values—MAP 2.00 L/h, median 3.357 L/h, and mean 4.00 L/h—reflect the posterior's tail rather than three competing calculations of one identical feature.
Parameter Summaries Are Not Automatically Forecast Summaries
Individualization is often performed to forecast concentrations or exposure, rather than to report a parameter for its own sake. A nonlinear function of a parameter generally does not have the same posterior summary as the function evaluated at a parameter summary. Under linear pharmacokinetics with known bioavailability equal to 1, dose \(D\), and clearance \(C\), AUC is \(D/C\). Consequently, the posterior mean AUC depends on \(E(1/C\mid y)\), not on \(1/E(C\mid y)\).
Worked Example: Predicting AUC From a Clearance Posterior
Use the hypothetical lognormal clearance posterior from the earlier calculation and a hypothetical dose \(D=400\) mg. Under the stated linear model, calculate the posterior mean AUC and compare it with AUC obtained by substituting the posterior mean clearance.
For \(\log C\mid y\sim N(\mu,\sigma^2)\), the reciprocal has mean \(E(1/C\mid y)=\exp(-\mu+\sigma^2/2)\). With \(\mu=\log 8\), \(\sigma=0.3\), and \(D=400\) mg:
Substituting the posterior mean clearance instead gives \(400/8.368=47.80\) mg·h/L. Substituting the posterior median clearance gives \(400/8=50.00\) mg·h/L. The posterior mean AUC is not the dose divided by the posterior mean clearance because reciprocal transformation is nonlinear. These calculations assume the stated dose-to-AUC relationship and known bioavailability; a different model or uncertain bioavailability changes the forecast.
Diagnostics, Alternatives, and Common Pitfalls
Before using a single individual estimate for a dose or forecast, inspect whether the posterior is concentrated and whether its shape makes the selected summary representative. A useful check is to compare the mean, median, and mode on the declared scale and examine posterior quantiles or a plot. Large separation among the summaries signals skewness; a peak near a parameter boundary or multiple peaks also makes a single mode potentially unrepresentative. A numerical MAP can depend on grid resolution or optimizer behavior, so verify that it is a genuine density maximum rather than an artifact of the search.
| Summary | What it describes | Typical decision interpretation | Important caution |
|---|---|---|---|
| MAP | Location of a highest posterior density value | Density maximization on a declared parameter scale | Can change under reparameterization; can be unstable with multiple peaks or boundaries |
| Posterior mean | Posterior expectation | Minimizes posterior expected squared-error loss | May be pulled toward a long tail; may not represent a typical patient value |
| Posterior median | Posterior midpoint by probability | Minimizes posterior expected absolute-error loss | Does not by itself describe tail size or uncertainty |
A common mistake is to say, “The log-clearance MAP is \(\log 8\), so the clearance MAP is 8 L/h.” That reasoning maximizes the density on the log scale and then transforms the answer. If the requested MAP is for clearance, the transformed density includes a \(1/c\) Jacobian; in the worked example its mode is \(8\exp(-0.09)=7.311\) L/h. The value 8.00 L/h is the clearance median in that example, not its natural-scale MAP.
Another pitfall is to assume that the posterior mean is always the best number for individual dosing. It is optimal for squared-error loss on the stated parameter scale, but clinical decisions may concern a nonlinear forecast or have asymmetric consequences for under- and overexposure. A point estimate also omits posterior uncertainty. In practice, state the target and scale, justify the summary by its purpose, and retain the rest of the posterior information when assessing the forecast.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If a positive parameter has \(\log\theta\mid y\sim N(\log 5,0.2^2)\), what is its posterior median?
- For the same posterior, calculate the posterior mean and the MAP on the \(\theta\) scale.
- Which summary minimizes posterior expected absolute-error loss, and what property defines it?
- Why can exponentiating a MAP estimate computed on the log scale fail to give the MAP on the original scale?
Solutions. (1) The lognormal median is \(\exp(\log 5)=5.00\), in the units of \(\theta\). (2) The mean is \(5\exp(0.2^2/2)=5\exp(0.02)=5.101\). The original-scale MAP is \(5\exp(-0.2^2)=5\exp(-0.04)=4.804\). (3) A posterior median minimizes expected absolute-error loss; at a median, at least half the posterior probability is at or below the value and at least half is at or above it. (4) A nonlinear transformation changes the density by a Jacobian. The transformed density can therefore have its maximum at a different point from the transformed maximum of the original density.