From a Continuous-Time Model to an Observation Likelihood
A diffusion model describes how a pharmacokinetic state evolves continuously, but data are usually recorded at finitely many times. If the state is observed at \(t_0<t_1<\cdots<t_n\), likelihood construction asks a specific question: what is the joint density of those observations under a proposed parameter value? For a Markov diffusion with available one-step transition densities, the answer is a product of those densities, one for each successive observation interval.
This construction relies on the Markov property, not on observations being equally spaced. The transition over an interval depends on its length and, in general, on its starting time and starting state. It also matters whether the initial state is conditioned upon or modeled as random. These choices determine which factors belong in the likelihood.
The transition density must be understood relative to a measure and on the model's state space. For a one-dimensional positive concentration, for example, the density is with respect to concentration on \((0,\infty)\), not with respect to log concentration. If a transformation is used to obtain a density, its Jacobian must be included when expressing the result on the original observation scale.
Why the Product Is the Joint Density
Assume the observation times are fixed, the process is Markov under each parameter value being considered, and the relevant joint and conditional densities exist. The ordinary chain rule for densities gives $$ f_\theta(x_{0:n}) =f_\theta(x_0)\prod_{i=1}^{n} f_\theta(x_i\mid x_0,\ldots,x_{i-1}). $$ The Markov property says that, conditional on \(X_{t_{i-1}}=x_{i-1}\), the distribution of \(X_{t_i}\) does not depend on the earlier states. Thus each conditional factor is the one-step transition density.
Write the joint density as the density of \(X_{t_0}\) multiplied by the conditional densities of \(X_{t_i}\) given all preceding observations.
For every \(i\geq1\), replace the conditional density given \(x_0,\ldots,x_{i-1}\) by the conditional density given only \(x_{i-1}\).
Each remaining conditional density is \(p_\theta(t_{i-1},x_{i-1};t_i,x_i)\). Multiplying them with the initial density gives the stated joint density; conditioning on \(x_0\) removes that initial factor. \(\square\)
The likelihood is a function of \(\theta\) after the observed values and times have been fixed. If every factor is positive and finite, taking logarithms turns the product into a sum:
For a joint likelihood, add \(\log p_{0,\theta}(x_0)\). This algebra does not assert that the observation intervals are independent without conditioning: their dependence is handled by the successive conditional transitions.
Worked Example: A Scalar Positive-State Diffusion
Consider a proportional-noise model for a positive concentration: $$ dC_t=-kC_t\,dt+\sigma C_t\,dW_t,\qquad C_t>0,\quad k\in\mathbb R,\quad \sigma>0, $$ where \(W_t\) is standard Brownian motion. This example assumes the state itself is observed without measurement error. We first derive the transition density, then use it to form a likelihood.
Apply Itô's formula to \(Y_t=\log C_t\). Since the first and second derivatives of \(\log c\) are \(1/c\) and \(-1/c^2\), respectively, $$ dY_t=\frac{1}{C_t}dC_t-\frac{1}{2C_t^2}(dC_t)^2 =-(k+\sigma^2/2)\,dt+\sigma\,dW_t. $$ Consequently, for an interval of length \(h>0\), $$ \log C_{t+h}\mid C_t=x \sim N\left(\log x-(k+\sigma^2/2)h,\ \sigma^2h\right). $$ Transforming back to concentration gives the density on \(y>0\): $$ p_\theta(t,x;t+h,y) =\frac{1}{y\sigma\sqrt{2\pi h}} \exp\left[ -\frac{\left\{\log(y/x)+(k+\sigma^2/2)h\right\}^2}{2\sigma^2h} \right], \qquad \theta=(k,\sigma). $$ The factor \(1/y\) is the change-of-variable Jacobian. The density is zero for \(y\leq0\), as the model has positive state space.
Take observations \(C_0=0.20\), \(C_1=0.16\), \(C_2=0.12\) at times \(0,0.5,1\), and evaluate the conditional likelihood at \(k=0.40\), \(\sigma=0.30\). For either interval, \(h=0.5\), the conditional mean of the log ratio is \(-(0.40+0.30^2/2)(0.5)=-0.2225\), and its variance is \(0.30^2(0.5)=0.045\). The two observed log ratios and their residuals from this mean are $$ r_1=\log(0.16/0.20)-(-0.2225) =\log(0.8)+0.2225 \approx-0.00064355, $$ $$ r_2=\log(0.12/0.16)-(-0.2225) =\log(0.75)+0.2225 \approx-0.06518207. $$ Each concentration-scale log-density is \(-\log C_i-\tfrac12\log(2\pi(0.045))-r_i^2/(2(0.045))\). Here \(-\tfrac12\log(2\pi(0.045))\approx0.63160786\), so the contributions are $$ \log p(C_0;C_1) =-\log(0.16)+0.63160786-\frac{(0.00064355)^2}{0.09} \approx2.46418, $$ $$ \log p(C_1;C_2) =-\log(0.12)+0.63160786-\frac{(0.06518207)^2}{0.09} \approx2.70466. $$ Thus the conditional log-likelihood is approximately \(2.46418+2.70466=5.16885\), and the conditional likelihood is \(\exp(5.16885)\approx175.7\). It is a density, not a probability, so it need not be at most one. If the initial concentration is modeled with a parameter-dependent density, its log-density must also be added.
Worked Example: Including a Random Initial State
Let \(X_0\sim N(0,1)\), and suppose the process between observations is driftless Brownian motion. Observe \(x_0=0.2\) and \(x_1=0.2\) after an interval of length \(0.04\). The initial density is $$ p_0(0.2)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{0.2^2}{2}\right) =\frac{1}{\sqrt{2\pi}}\exp(-0.02) \approx0.391043. $$ The transition is normal with mean \(x_0=0.2\) and variance \(0.04\), giving $$ p(0.2;0.2) =\frac{1}{\sqrt{2\pi(0.04)}} \exp\left(-\frac{(0.2-0.2)^2}{2(0.04)}\right) \approx1.994711. $$ Therefore the joint density is $$ p(x_0=0.2,x_1=0.2) \approx0.391043(1.994711) \approx0.780018. $$ If inference conditions on the observed \(x_0\), only the transition factor \(1.994711\) is used. If inference instead treats the initial distribution as part of the model, both factors are included. This distinction is especially important when the initial distribution contains parameters shared with the transition model.
Likelihood Checks and Common Failure Modes
Before optimizing a transition likelihood, check that each factor describes the correct interval and scale. Unequal time gaps require their own transition laws; copying one fixed-step density across all observations changes the model. Known dosing or other inputs must also be reflected in the transition from the state before an interval to the state after it. A transition density that ignores an input is not repaired by multiplying it correctly.
- Support check: If the model is positive-state and an observed concentration is nonpositive, the proposed exact transition density assigns that observation zero density. Do not silently substitute a density on the real line.
- Initial-state check: Decide whether inference is conditional on \(x_0\). Including an initial density in a conditional likelihood double-counts the initial observation; omitting a parameter-dependent initial density from a joint likelihood changes the likelihood.
- Scale check: A density for \(\log C_i\) is not the density for \(C_i\). On the concentration scale, the lognormal transition includes the \(1/C_i\) Jacobian factor.
- Observation check: The product above applies to directly observed states. If measured concentrations include separate assay error, the observed-data likelihood must account for that error rather than treating each measurement as the latent state.
- Numerical check: Compute the log-likelihood as a sum of log-densities. This avoids underflow from multiplying many small factors, and makes it easier to inspect unusually large negative contributions.
A common mistake is to call successive observations independent and then multiply their marginal densities. The correction is to use conditional densities: \(p_\theta(x_i\mid x_{i-1})\) for each interval, with the initial density added only for a joint likelihood. The Markov property justifies this factorization; it does not make the observed states marginally independent.
What This Likelihood Does—and Does Not—Assume
When the transition densities are exact for the stated diffusion, their product is the exact likelihood for the discretely observed state path under the assumptions above. In practice, a transition density may be unavailable in closed form. Replacing it with a density derived from a time-discretization scheme produces an approximate likelihood, not the exact diffusion likelihood. The same product structure can be used, but conclusions then depend on the approximation as well as on the model.
This approach is most direct when the state is observed, the process is Markov, and usable transition densities are available. It does not by itself handle unobserved intermediate states or measurement error: in either case, the observed data are not simply the complete state sequence in the product. Nor does factorization guarantee that the parameter is identifiable, that a numerical optimizer will find a maximum, or that the assumed diffusion adequately describes the data. Those questions require separate modeling and diagnostic work.
Check Your Understanding
Answer each question, then compare with the solutions.
- For three observations at \(t_0<t_1<t_2\), write the conditional likelihood given \(x_0\).
- When should the initial density appear in the likelihood?
- Why does the concentration-scale geometric-diffusion density contain a factor \(1/y\)?
- Suppose one observed concentration is negative under the positive-state model. What does the model assign to the observed path?
Solutions. (1) The conditional likelihood is \(p_\theta(t_0,x_0;t_1,x_1)p_\theta(t_1,x_1;t_2,x_2)\). (2) Include \(p_{0,\theta}(x_0)\) when modeling the joint distribution of the initial state and later observations; omit it when conditioning on \(x_0\). (3) The normal density is first obtained for \(\log C_{t+h}\); changing variables from log concentration to concentration has absolute derivative \(1/y\). (4) The exact positive-state model assigns zero density to a path containing that observation, so its log-likelihood is \(-\infty\).