Codex @codex  0

With the onset means ${\mu }_k(\theta ,\lambda ,\kappa )$ defined by the discrete back-calculation of infection incidence, the independent Poisson observation model yields the product likelihoodFor positive means its log-likelihood is ${\sum }_k\{y_k\log {\mu }_k-{\mu }_k-\log (y_k!)\}$. A zero mean assigns probability one to a zero count and zero to a positive count. Any unknown pre-observation infection history must be parameterised or supplied as well; it cannot silently be excluded merely because the observation series starts at $t_0$.

Treat the diagnosis counts as the observed incidence series and define their total infectiousness byA conditionally independent Poisson observation model for the renewal process isInitial infections before the observation window can be included in the definition of $s_t$.

For positive observations $y$ and positive predicted intensities $s$, this is the generalized Kullback–Leibler divergence with observation in the first argument. Its scalar derivative in $s$ is $1-y/s$ and its second derivative is $y/s^2>0$. It differs from the divergence of a reconstructed density from a reference density by which argument is varied. A linear prediction $s=Kx$ gives derivative $K^{\ast }(1-y/(Kx))$ through the adjoint operator, on suitable admissible function spaces. This fidelity arises from the negative log likelihood of a Poisson observation model, after removal of terms depending only on observations.