When Likelihood-Free Calibration Is Needed

Why the Likelihood Can Be the Obstacle

Bayesian calibration combines a prior distribution for mechanistic parameters with the probability model for the observations. In many QSP applications, that probability model is available: the system of ODEs predicts biomarker values, and a specified observation-error model assigns a density to the measured data. The resulting likelihood may be expensive to evaluate, but it is still evaluable. In other applications, a simulator can generate plausible data under a parameter setting while the probability of the observed data cannot be calculated. That distinction is central to deciding whether likelihood-free calibration is worth considering.

“Likelihood-free” does not mean that there is no probability model, that the model has no assumptions, or that a Bayesian analysis can proceed without data. It means that the likelihood needed for conventional Bayesian computation is unavailable or impractical to evaluate, while the model can still generate simulated observations. Simulation-based approximate Bayesian computation (ABC) is one family of approaches for this setting. The decision to use it should follow from the structure of the model and the observation process, not simply from a difficult calibration or an unsatisfactory MCMC run.

Definition: For observed data \(y\) and parameters \(\theta\), the likelihood is \(L(\theta;y)=p(y\mid\theta)\), the probability mass or probability density assigned to the observed data by the model at \(\theta\). A likelihood is intractable for a calibration task when this quantity cannot be evaluated accurately and practically enough for the intended inference, even though the model can be simulated at specified parameter values. For continuous observations, a likelihood is a density, not the probability of observing one exact value.

When it is available, the likelihood enters Bayes’ rule as \(p(\theta\mid y)\propto p(y\mid\theta)p(\theta)\), where \(p(\theta)\) is the prior density. Conventional likelihood-based algorithms use likelihood values, or ratios of them, to explore the posterior. For example, a Metropolis-Hastings update generally needs a likelihood ratio between a proposed and a current parameter value. If the likelihood cannot be evaluated, that calculation cannot be carried out in its usual form. This does not establish that likelihood-free calibration is automatically suitable; it establishes the computational obstacle that must be addressed.

What Counts as Intractable?

There are several distinct situations that are easily conflated. A deterministic QSP model may require a costly ODE solve, yet the predicted measurements and their observation densities can still be calculated. The likelihood is then expensive, not unavailable. A posterior may be strongly correlated, multimodal, or difficult for a particular sampler to explore; posterior geometry is a computational challenge, not evidence that the likelihood is intractable. Likewise, a poor fit or an implausible posterior predictive distribution is a concern about the model or calibration, not a reason by itself to discard an evaluable likelihood.

A more substantive obstacle arises when the observed-data likelihood requires integrating over latent processes, random effects, or unobserved states, and that integration has no practical exact or sufficiently accurate numerical solution. It may also arise when a stochastic simulator specifies how to generate complete trajectories but does not provide the probability density or mass of an observed trajectory. In either case, simulation is possible while likelihood evaluation is not. Numerical integration, analytic marginalization, or a validated likelihood approximation may still solve the problem; these should be considered before committing to an approximate likelihood-free analysis.

Worked Example: A Likelihood That Is Available Despite a Nonlinear ODE

Suppose a deterministic QSP model predicts two biomarker measurements of \(10\) units each at a proposed parameter setting. The hypothetical observations are \(11\) and \(9\) units. Assume independent normal measurement errors with known standard deviation \(\sigma=2\) units, as specified by the observation model. The likelihood is the product of the two normal densities:

L(\theta;y)=\prod_{i=1}^{2}\frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(y_i-\mu_i(\theta))^2}{2\sigma^2}\right].

The residuals are \(11-10=1\) unit and \(9-10=-1\) unit. Substitution gives

L(\theta;y)= \left(\frac{1}{\sqrt{2\pi}\,2}\right)^2 \exp\left[-\frac{1^2+(-1)^2}{2(2^2)}\right] \approx 0.0397887\,e^{-0.25} \approx 0.03099\ \text{units}^{-2}.

The normalizing factor \(0.0397887\) is already the product of the two individual density factors; it is not squared again. The result is a joint density with units \(\text{units}^{-2}\). Even if obtaining each predicted mean requires solving a nonlinear ODE, this likelihood is evaluable once the predictions are available. If the ODE solve is slow, the immediate issue is computational cost, not a need for likelihood-free calibration.

When Simulation Does Not Supply the Likelihood

A simulator can define a full generative model without offering a convenient formula for \(p(y\mid\theta)\). For instance, it may generate a stochastic cell-population trajectory through many interacting events, and the recorded clinical outcomes may be a complicated function of that trajectory. To evaluate the likelihood of the observed outcomes, one would need to account for all latent paths that could have produced them. If the simulator can draw those paths but cannot calculate their probabilities or sum or integrate over them, direct likelihood evaluation may be impractical.

A likelihood-free method uses the ability to simulate as a substitute for directly evaluating \(p(y\mid\theta)\). In ABC, the observed data are compared with simulated data, often through selected summaries and a discrepancy rule. This tutorial concerns the decision point: whether the likelihood really is the obstacle and whether the simulator and observations make such a strategy plausible. The specific ABC algorithms and the choices of summaries and tolerance require separate treatment.

Worked Example: A Simulator Without an Observed-Data Density

Consider a hypothetical QSP model in which a stochastic immune-cell simulator generates a full trajectory for each virtual patient. The observed clinical data are a complicated combination of event counts, event timing, and biomarker measurements. The software can simulate those quantities under a parameter vector \(\theta\), but it does not calculate the probability of the observed combination, because doing so would require summing or integrating over many possible latent trajectories.

Suppose calibration focuses on two prespecified summaries: the mean biomarker at week 8 and the number of participants with a relapse by week 12. The hypothetical observed summaries are \(8.0\) units and \(7\) relapses. A simulated data set under one candidate parameter setting gives \(7.4\) units and \(9\) relapses. If the calibration team has chosen scales of \(1.0\) biomarker unit and \(2\) relapses for these summaries, the scaled discrepancies are

d_1=\frac{|7.4-8.0|}{1.0}=0.6,\qquad d_2=\frac{|9-7|}{2}=1.0.

For illustration, the Euclidean distance between these scaled summaries is \(\sqrt{0.6^2+1.0^2}=\sqrt{1.36}\approx1.166\). This calculation does not produce a likelihood or establish that the parameter setting is supported by the full data. It shows only that a simulator-generated data set can be compared with the observations using specified summaries. A likelihood-free analysis is plausible only if these summaries preserve information relevant to the calibration goals, the simulator represents the observation process, and enough simulations can be run. If important features of the data are omitted, a close summary match can still conceal a poor fit.

Check Whether the Obstacle Can Be Removed

Before adopting a likelihood-free approach, diagnose why likelihood evaluation fails. The following questions help separate a genuine likelihood obstacle from a problem that calls for a different remedy.

Write down the observation model.
State what is observed, what is latent, and how the measurements or outcomes are generated. A simulator alone is not a complete description of the inferential target unless its observation process is specified.

Locate the missing calculation.
Determine whether the problem is an unavailable density, an intractable integral over latent quantities, or merely expensive prediction and density evaluation. Consider whether analytic marginalization or numerical integration is feasible.

Check what the simulator can generate.
It must generate data comparable to the actual observations under specified parameter values, including relevant variability and measurement error. Simulating only latent biological states may not suffice if the observed data arise through an additional measurement process.

Assess the calibration target and computational budget.
Ask whether informative summaries can be defined for the scientific questions and whether enough simulations are feasible. If the required summaries discard important information or simulation is prohibitively costly, a likelihood-free approach may not be useful.

The justification for using ABC is therefore conditional. If a simulator reproduces the data-generating process and a chosen summary \(S(y)\) retains the information needed for the target, then comparing \(S(y)\) with summaries of simulated data can provide an approximation to inference from the full observations. The approximation becomes less defensible when the summary is uninformative, the simulation model omits relevant measurement variation, or the number of simulations is too small to explore plausible parameter regions. No summary comparison can repair a misspecified biological or observation model.

A Latent Integral Is Not Automatically Intractable

The presence of latent variables does not by itself require likelihood-free calibration. For example, if \(z\) is a latent quantity, the observed-data likelihood is obtained by integrating it out:

p(y\mid\theta)=\int p(y\mid z,\theta)\,p(z\mid\theta)\,dz.

The integral may be analytically available, numerically tractable, or genuinely impractical, depending on its dimension and structure. The right question is whether the required calculation can be done with enough accuracy at a feasible cost—not whether the model contains unobserved states.

Worked Example: A Non-Closed-Form Likelihood That Can Still Be Evaluated

Suppose a hypothetical latent biological state \(Z\) is uniformly distributed between \(0\) and \(1\), and, conditional on \(Z=z\) at \(\theta=0\), a continuous measurement \(Y\) has distribution \(N(z^2,1)\). The observation is \(y=0\). The observed-data likelihood is the one-dimensional integral

p(y=0\mid\theta=0)= \int_0^1 \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{(0-z^2)^2}{2}\right)\,dz.

This integral has no simple elementary antiderivative, but that fact alone does not make the likelihood unavailable. For example, composite Simpson quadrature with four equal subintervals has step size \(h=0.25\). The integrand values at \(z=0,0.25,0.50,0.75,1\) are approximately \(0.398942,\ 0.398164,\ 0.386668,\ 0.34056,\ 0.241971\), respectively. Thus

p(y=0\mid\theta=0)\approx \frac{0.25}{3}\left[ 0.398942+0.241971+ 4(0.398164+0.34056)+2(0.386668) \right] \approx 0.3641\ \text{measurement}^{-1}.

This is an approximation to a density, with units inverse to those of \(Y\). Its numerical accuracy should be checked by refining the quadrature or using another validated integration method. If that check succeeds and the computation is feasible across parameter values, likelihood-based calibration remains an option. Likelihood-free calibration would not be justified merely because an elementary closed form is absent.

Alternatives, Diagnostics, and Limits

The main alternatives depend on the diagnosed obstacle. If the likelihood is available but an ODE solve is slow, improving numerical implementation or using a carefully validated computational approximation may be appropriate. If latent-variable integration is the bottleneck, analytic marginalization or numerical integration may help. If the likelihood is evaluable but the posterior is difficult to explore, the earlier material on posterior geometry and MCMC diagnostics is more directly relevant. Likelihood-free methods become candidates when the data can be simulated but the needed likelihood cannot practically be obtained, and when a simulation-based approximation is scientifically adequate.

Situation	What is difficult?	Initial response
Known observation model and computable predictions	Possibly an expensive ODE solve	Evaluate the likelihood; address computational cost or sampler performance
Latent quantities with a manageable integral	Marginalization over latent states	Use validated analytic or numerical integration
Forward simulator available, observed-data density unavailable	Likelihood evaluation itself	Assess whether a simulator-based approximation is suitable

Diagnostics for a proposed likelihood-free analysis should include checks that simulated data have the same form as observed data, that measurement error and censoring are handled consistently, and that the summaries capture features relevant to the scientific target. Compare simulated and observed summaries across plausible parameter settings rather than relying on a single illustrative draw. If important observed patterns cannot be reproduced, the problem may be model inadequacy rather than a need for a different computational method. Posterior predictive checks remain important after calibration; changing how parameters are inferred does not validate the model.

A central limitation is that likelihood-free inference based on summaries generally targets an approximation, not automatically the posterior based on all the raw data. The result can depend on which summaries are used and how discrepancies are measured. Computational expense can also be severe when simulations are slow or only a small fraction resemble the observed data. These limitations do not make the approach unsuitable in every case, but they require reporting what was simulated, what information was compared, and why that comparison answers the calibration question.

Common Mistakes

Calling a slow likelihood intractable. A costly ODE solve with a known observation density still yields an evaluable likelihood. Diagnose the cost before changing inferential targets.
Assuming latent variables require ABC. Latent quantities can often be integrated out analytically or numerically. Check the integral’s practical tractability first.
Treating simulation as a likelihood value. A simulated data set is a random draw, not the probability density of the observed data. Simulation can support an approximation, but it does not directly supply \(p(y\mid\theta)\).
Assuming a summary match proves model fit. A discrepancy based on selected summaries says nothing directly about features those summaries omit. Assess whether the summaries address the scientific target and inspect other relevant predictive features.

Key takeaway: Consider likelihood-free calibration when the observed-data likelihood is genuinely unavailable or impractical to evaluate, a faithful forward simulator is available, and an informative simulation-based approximation is feasible. A difficult posterior, slow ODE, or missing closed-form expression alone is not sufficient justification.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A QSP model requires a slow ODE solve, but its measurement errors are independent normal variables with known standard deviations. Is the likelihood necessarily intractable?
A model contains a latent random effect, but its one-dimensional likelihood integral can be evaluated accurately by quadrature. Does the latent effect alone justify a likelihood-free method?
A stochastic simulator can generate event histories but cannot calculate the density of the observed history. Name two conditions that should be checked before considering ABC.
Why does a small discrepancy between observed and simulated summaries not establish that the full observed-data likelihood is large?

Solutions. (1) No. Once the ODE predictions are obtained, the stated normal observation model defines an evaluable likelihood. The solve may be expensive, but that is not the same as an unavailable likelihood. (2) No. If quadrature evaluates the integral accurately at feasible cost, the likelihood remains usable; latent structure is not itself a reason to avoid it. (3) Check that the simulator reproduces the observed-data generation process, including measurement variability, and that feasible summaries retain information relevant to the calibration target. (4) The discrepancy compares only the chosen summaries. The raw data may differ in features those summaries omit, so a close summary match is not equivalent to evaluating or maximizing the full-data likelihood.