0.208523563155746	2.46584999220858

0.210798318034145	2.25418975920952

0.213073072912543	2.08119603922318

0.215347827790941	1.93696568969348

0.21762258266934	1.81474552115029

0.220352288523418	1.69051492997892

0.223081994377496	1.58532383368093

0.225811700231573	1.49503988831287

0.228541406085651	1.41665890083889

0.231726062915408	1.33730303285041

0.234910719745166	1.26846529930347

0.238095376574924	1.2081659201728

0.241280033404681	1.15489848141388

0.244464690234438	1.10749556571353

0.248104298039875	1.05933144706481

0.251743905845313	1.01655104629421

0.25538351365075	0.978304535624397

0.259023121456186	0.94391447723457

0.263117680237303	0.909158413899601

0.26721223901842	0.877964667037296

0.271306797799537	0.849826030431628

0.275401356580653	0.824328857643024

0.27949591536177	0.801132314552191

0.284045425118567	0.777713485108928

0.288594934875364	0.756462721821696

0.293144444632159	0.737111579865076

0.297693954388956	0.719435294476996

0.302243464145753	0.703244204936126

0.306792973902549	0.688377132177544

0.311797434635025	0.673388638671994

0.316801895367501	0.659678934995996

0.321806356099978	0.647114183170732

0.326810816832453	0.635579488383005

0.331815277564929	0.624975665014313

0.336819738297406	0.615216647232558

0.342279150005561	0.605446027650356

0.347738561713717	0.596506829068363

0.353197973421873	0.588325298134409

0.358657385130028	0.580837039398967

0.364116796838184	0.573985601015645

0.36957620854634	0.567721312075274

0.375490571230174	0.561546845134909

0.38140493391401	0.555960938499646

0.387319296597846	0.550920371734914

0.39323365928168	0.546387024997431

0.399148021965516	0.542327191983111

0.405062384649352	0.538711003915015

0.411431698308866	0.53528244626796

0.417801011968381	0.532308383140165

0.424170325627896	0.529762949248529

0.430539639287411	0.527623311033241

0.437363903922606	0.525758294562332

0.4441881685578	0.52431421185099

0.451012433192995	0.523272026345442

0.457836697828189	0.522615148654683

0.465115913439064	0.522322981864741

0.472395129049938	0.522437817409414

0.479674344660812	0.522946906574036

0.486953560271687	0.523839560905793

0.49423277588256	0.525106966697741

0.501966942469115	0.526856293593911

0.509701109055668	0.529014137640258

0.517435275642222	0.531575822915253

0.525169442228776	0.534538427941502

0.53290360881533	0.537900687747841

0.541092726377563	0.541896711330282

0.549281843939797	0.546343387757223

0.55747096150203	0.551244798080917

0.565660079064264	0.55660684047222

0.573849196626497	0.562437213271851

0.582038314188731	0.568745416032706

0.590227431750964	0.575542767744919

0.598416549313198	0.582842441762438

0.606605666875431	0.590659517266007

0.614794784437665	0.599011047373128

0.62116409809718	0.605888340096399

0.621619049072859	0.551374528146545

0.622074000048539	0.529165679481653

0.622983901999898	0.516968085276023

0.623893803951257	0.509927353639338

0.625258656878296	0.502828680501763

0.626623509805335	0.497886005459354

0.628443313708054	0.493033289746029

0.630263117610773	0.489366022934792

0.632082921513491	0.486533855670151

0.63435767639189	0.483748749152232

0.636632431270288	0.481623954185888

0.639362137124365	0.479669942358457

0.642091842978443	0.47825193203536

0.645276499808201	0.477086722571924

0.648461156637958	0.476358715209519

0.652100764443396	0.475967626157544

0.655740372248832	0.475964341106985

0.659834931029949	0.476339584288213

0.664384440786746	0.477169427324229

0.669388901519222	0.47851661252654

0.674848313227377	0.4804402404301

0.679397822984174	0.482369875577526

0.68394733274097	0.484571724566623

0.688951793473446	0.487281729368194

0.695321107132961	0.491163174216982

0.702145371768156	0.495821847809705

0.708059734451991	0.500252317050276

0.714883999087186	0.505826806663069

0.722163214698059	0.512293310423671

0.729442430308934	0.519309952226241

0.736721645919808	0.526865909037499

0.744455812506362	0.535503260884203

0.752189979092916	0.544789133845791

0.75992414567947	0.554728939511045

0.767658312266024	0.565353862949518

0.775392478852577	0.576699565528464

0.783126645439132	0.588784176915648

0.790860812025685	0.601651919560574

0.798594978612239	0.615352042490458

0.805419243247433	0.628153415415341

0.813153409833988	0.643527593142836

0.820432625444862	0.658891453678106

0.827711841055736	0.675159134063502

0.834991056666611	0.692403940881387

0.842270272277484	0.710669399420334

0.849549487888359	0.730037172325838

0.856373752523553	0.749252450356138

0.863652968134428	0.77095264960973

0.870932183745301	0.793994488541174

0.878211399356176	0.81845014722787

0.885035663991371	0.842773314321222

0.891859928626565	0.868518221983267

0.89868419326176	0.895810342350575

0.905508457896954	0.924729180394171

0.912332722532149	0.955419968053779

0.919156987167344	0.987975562752448

0.925981251802538	1.02256322522385

0.932805516437733	1.05928893941265

0.939629781072927	1.09834309644551

0.946454045708122	1.13984187841934

0.953278310343316	1.18399678759074

0.960102574978511	1.23092532115418

0.966926839613706	1.28085068421306

0.9737511042489	1.33388008671519

0.980120417908415	1.38633003364624

0.986944682543609	1.44580017756046

0.993313996203125	1.50433378526974

1.000593211814	1.57488805440483

1.00832737840055	1.6537195046331

1.01788134888983	1.75563580647776

1.02425066254934	1.82675386731156

1.02925512328182	1.89189615551673

1.03607938791701	1.98607500266185

1.05746208377395	2.28689598129951

1.06064674060371	2.32522195644052

1.06428634840915	2.36234425438886

1.06610615231187	2.37719020105039

1.06792595621459	2.38928111055934

1.06929080914162	2.39596877741064

1.07020071109298	2.39885398639946

1.07111061304434	2.40064716095275

1.07156556402002	2.40113816937231

1.0720205149957	2.4012901300737

1.07247546597138	2.40108639546817

1.07293041694706	2.40050935078472

1.07384031889842	2.39816082054709

1.07475022084978	2.39379499349582

1.07611507377682	2.3837186661766

1.07747992670386	2.36889439049134

1.0788447796309	2.34828818074884

1.08020963255794	2.32016348905757

1.08157448548497	2.28483588675483

1.08293933841201	2.24032754061724

1.08430419133905	2.18402229144732

1.08566904426609	2.11587821615017

1.08748884816881	2.00135121700183

1.08885370109585	1.89259079112479

1.09067350499857	1.71422109936304

1.09249330890129	1.48737663157502

1.094313112804	1.19602158614065

1.09613291670672	0.827860587514024

1.09977252451216	0.0095993959369105

Computation of the ACID with the direct method. Parameters were as in fig. <ref>. While the BReT-P agrees with the Monte Carlo simulation, the direct method, computed by fixed point iteration of (<ref>), shows an inaccuracy. The domain $(\RTRoot_1, \OnValue]$ was discretized into $N = 2000$ equidistant intervals.

§ DISCUSSION

The computation of the mutual information rate of a signal and its Poisson channel output via the expression due to Liptser remains a challenge. We contributed a new computational method by exploiting that the filtering distribution forms a piecewise-deterministic Markov process.
The Markovian nature of the filtering equation was also paraphrased as ‘mysterious’ concept of recursiveness by Brémaud [16]. Back in the days its purpose of saving memory-space was emphasized.
(A priori the CI depends on the history of $\out_\ti$. With the dependence on only the current state, there is no need to record the history.) We, in contrast, use the insight on the Markovian nature to analyze the asymptotic conditional intensity distribution.
We aimed at computing the ACID in a simulation-free way, i.e., without Monte Carlo simulations. This involved the evaluation of ODEs on a grid with the dimension given by the dimension of the sufficient statistic. We considered doubly stochastic Poisson processes (signal process along Poisson channel) and self-exciting counting processes. The ACID is accessible by our method in the case of very low number of non-zero signal states. With no limitations on the number of zero states, the method enables us to analyze binary semi-Markov inputs, i.e., have non-exponential phase-type Off-times. An interesting class of examples, that are characterized by low number of states of joint Markovian progression, are the approximate filters. By their very purpose of reducing the dimension of the sufficient statistic, they suit the limitations of our computational method.

The computational method is modular. The parametrization of the sufficient statistic $\state$ can be varied. The technique to find $\idensity$ according to eq. (<ref>) or (<ref>) can be exchanged. The normalization constant can be employed from eq. (<ref>) or the resulting distribution $\density(\intime, \state)$ can be normalized. Depending on whether the entire ACID or summary statistics, such as variance or $\E[\phi(\cdot)]$, are of interest, the method can be modified. We expect that there is room for improvement: when substituting single modules in the method, the precision might be increased, computation time decreased and limitations relaxed. While our grid discretization was an ad-hoc approach, we consider our main contribution to lie in the formulation of BReT-P and the derivation of the integral boundary conditions.

We hope that the following semi-Markov perspective further guides improvements on the computational or theoretic side:
The parametrization $(\intime(\ti), \state(\ti))$ can be seen as the associated Markov process <cit.> of the semi-Markov process $\state(\ti)$ with $\intime(\ti)$ being its backward recurrence time. Conditioned on being in state $\state$, the function $\cintensity(\intime, \state)$ is the hazard and $\Density(\intime, \state)$ has an interpretation as the survival function $\Prob[\wtime(\state) > \intime]$ of the sojourn time $\wtime(\state)$. The semi-Markov kernel is degenerate in $\state$ as

\begin{equation}
\semiMarkovKernel(\statefrom, \stateto, \intime) = \Density(\intime, \statefrom)\Id(\target(\intime, \statefrom) = \stateto)
\label{eq:semiMarkovKernel}
\end{equation}

because $\target$, the target anticipated at jumps, is a deterministic function. The embedded Markov chain has transition kernel

\begin{equation}
\tKernel(\statefrom, \stateSet) = \int_{\target(\intime, \statefrom) \in \stateSet} \cintensity(\intime, \statefrom)\Density(\intime, \statefrom) \dd \statefrom.
\end{equation}

The probability density of the embedded Markov chain (EMC) is proportional to $\idensity$ because it equals the probability density of being in a state $\state$ at a jump time. Consequently, eq. (<ref>) can be interpreted as the stationarity condition for the embedded chain. Finally, the relation $\density(\intime, \state) = \idensity(\state)\Density(\intime, \state)$ reflects the fact that the EMC's stationary distribution and the sojourn time factorize in the asymptotic [53].

For the example <ref> with zero-dimensional $\state$ the semi-Markov perspective boils down to a renewal process. Then eq. (<ref>) for $\Density$ expresses the known fact that the sojourn time for the marginal is a mixture of two exponential [54].
Renewal processes in general fall within our framework with backward recurrence time $\intime(\ti)$ being sufficient without $\state(\ti)$. Their conditional intensity is $\CIntensity_\ti = -\frac{\dot\Density(\intime(\ti))}{\Density(\intime(\ti))}$ for the survival function $\Density(\intime)$. It is well established that the stationary distribution of $\intime$ has the pdf $\frac{\Density(\intime)}{\int \Density(u) \dd u}$ [43]. The transformation $\intime \mapsto -\frac{\dot\Density(\intime)}{\Density(\intime)}$ provides the ACID. This fraction is exactly the hazard.
For the special case of exponential waiting times, the hazard rate is constant, so the ACID is a delta function with peak at the constant.

The ACID can be interpreted as giving an ensemble perspective of the time-point-wise conditional intensity. Assuming that the counting process has reached the asymptotic regime, the ACID informs us how likely a certain intensity value is at a random time point.
The ACID is a static quantity. Still, it captures temporal information to some extent. For instance, different autocovariance decays showed qualitatively different ACIDs for the Hawkes process. Most prominently, the ACID contains enough temporal information to inform the asymptotic rate of the path mutual information, as the Liptser expression shows.
Limit results for the Hawkes process show that the variance function and covariance density are needed for the limit process in the central limit theorem [51]. These are not obtained from the ACID alone. The question, what properties of the counting process are uniquely determined by the ACID, remains open for future studies.

Qualitatively, the ACID can exhibit non-smooth probability density functions. The ACID can serve as a partial characteristic for counting processes. In principal, it would be desirable to compare two counting processes in terms of their path measure. This can be too complex, in particular for self-exciting counting processes. The ACID may offer an accessible lower dimensional statistic for comparison that still exceeds first- and second-order analysis. We exemplified this for the comparison of an exact or approximate filter with an approximate filter. The ACID can discriminate between them if it differs. By our current state of knowledge, the ACID is limited as a partial characteristic of $\out_\ti$ in the opposite case. It does not allow a conclusion about the distance of the path measures in case two ACIDs are close. The usefulness of the ACID might be strengthened if statements can be found that allow conclusions of the following forms. (i) If $\lambda_\infty^{(1)}$ and $\lambda_\infty^{(2)}$ are close in distribution and the deterministic dynamics $F^{(1)}$ and $F^{(2)}$ are close, then the jump updates $f^{(1)}$ and $f^{(2)}$ are close. (ii) If $\lambda_\infty^{(1)}$ and $\lambda_\infty^{(2)}$ are close in distribution and $f^{(1)}$ and $f^{(2)}$ are close, then $F^{(1)}$ and $F^{(2)}$ are close. This is subject to future research and might allow to conclude the closeness of the path measures from the closeness of two ACIDs under mild additional conditions.
The random time transform $\tilde{\out}_\to = \out(\int_0^\ti \CIntensity_s \dd s)$ is known to be a Poisson process. For instance it is used to assess goodness of fit for counting process models. In approximate marginal simulation, this approach can complement the ACID analysis. It also highlights the information content of integrals $\int_t^{t+h}\CIntensity_s \dd s$ for fixed gaps $h$. These could also be investigated asymptotically in distribution, fixing the ACIDs limitation of omitting the CI's temporal dependencies.

Finally, we contributed to information theory by tackling the capacity problem for the Poisson channel with binary stationary input under average sojourn time constraints. The simulation-free computation employing BReT-P has the advantage to make partial derivatives easily accessible, which is useful in searching for optimal system parameters with gradient methods. We illustrated qualitative features of the Markov input without and with dark current. In addition, the case study of the double On single Off model showed, that the Markov input is not optimal. It remains an open research question what sojourn time distributions are optimal for the binary Semi-Markov input with average sojourn time constraints.

§ REPARAMETRIZATION LINK

The link between (<ref>) and (<ref>) can be obtained via the transformation rule applied on the transformation $\cprob\colon [0,\infty)\to (\RTRoot, 1]$. Let $T$ satisfy $\intime(\ti) \overset{\mathrm{d.}}{\to} T $, i.e., $T$ follows the stationary distribution of $\intime(\ti)$. Then

\begin{align*}
\int_\RTRoot^1 \phi(z) \density_\condmean(z) \dd z &= \int_{\cprob^{-1}(\RTRoot)}^{\cprob^{-1}(1)} \phi(\cprob(\ti)) \density_\condmean(\cprob(\ti))\cdot \cprob'(\ti)\dd \ti \\&= \int_0^\infty \phi(\cprob(\ti)) \left[-\density_\condmean(\cprob(\ti))\cdot \cprob'(\ti)\right]\dd \ti \\&= \int_0^\infty \phi(\cprob(\ti)) \density_T(\ti) \dd \ti\\&= \idensity\int_0^\infty \phi(\cprob(\ti)) \Density(\ti) \dd \ti.
\end{align*}

§ ODE SYSTEM FOR PARTIAL DERIVATIVES

The joint evolution (for the time scaling $\intime \mapsto \gain \intime$, i.e., $\tilde f(\intime):=f(\intime/\gain)$ for $f = \density, \cprob, \density_1, \cprob_1, \cummIR_1$ and dropping the tilde again) is given by

\begin{align}
\dot \density(\intime) &= - \cprob(\intime) \density(\intime) \label{eq:rho_evolution}\\
\dot \cprob(\intime) &= \RTON - (\RTOFF + \RTON + 1) \cprob(\intime) + \cprob(\intime)^2\\
\dot \density_1(\intime) &= \partial_1\dot \density(\intime) = -\cprob_1(\intime) \density(\intime) -\cprob(\intime) \density_1(\intime)\\
\dot \cprob_1(\intime) &= \partial_1\dot \cprob(\intime) = 1 - \cprob(\intime) \nonumber\\& -(\RTOFF + \RTON + 1)\cprob_1(\intime) + 2\cprob(\intime)\cprob_1(\intime)\\
\dot \cummIR_1(\intime) &= - \phi'(\cprob(\intime))\cprob_1(\intime)\density(\intime) - \phi(\cprob(\intime))\density_1(\intime) \label{eq:Jprime_evolution}
\end{align}

with initial conditions $(\frac{\gain \RTON}{\RTON + \RTOFF}, 1, \frac{\gain \RTOFF}{(\RTON + \RTOFF)^2}, 0, 0)$.

§ DERIVATIVE WITH RESPECT TO GAIN

The scaling behaviour

\begin{equation}
\IR(\RTOn, \RTOff, \gainSc\gain) = \gainSc\IR\left(\frac{\RTOn}{\gainSc}, \frac{\RTOff}{\gainSc}, \gain \right)
\label{eq:gain_scaling}
\end{equation}

holds. Then for the derivative

\begin{align*}
&\partial_3\IR(\RTOn, \RTOff, \gain)\\ &= \lim_{h\to 0} h^{-1} (\IR(\RTOn, \RTOff, \gain + h) - \IR(\RTOn, \RTOff, \gain))\\
&= \lim_{h/\gain \to 0}\left(\frac{h}{\gain}\right)^{-1} \left(\IR\left(\frac{\RTOn}{\gain}, \frac{\RTOff}{\gain}, 1 + \frac{h}{\gain}\right) - \IR\left(\frac{\RTOn}{\gain}, \frac{\RTOff}{\gain},1\right)\right)\\
&= \partial_3\IR\left(\frac{\RTOn}{\gain}, \frac{\RTOff}{\gain}, 1\right).
\end{align*}

§ VARIANCE OF THE ACID AND ASYMPTOTIC VARIANCE OF $\OUT_\TI$

The process equation for the Hawkes process is rewritten as

\begin{equation}
\dd \CIntensity_\ti = -\HawkesAutocov(\CIntensity_\ti- \gain\HawkesMean)\dd \ti + \HawkesJump \dd \HawkesMartingale_\ti
\end{equation}

with the canonical $\sigmalg^\out_\ti$-martingale increment $\dd\HawkesMartingale_\ti = \dd \out_\ti - \CIntensity_\ti \dd \ti$. We are interested in the asymptotic behavior, i.e. $\CIntensity_\infty$. It can be brought to finite time $\ti$ under the shift [17] of the time domain $[0,\infty)$ to $(-\infty, t)$:

\begin{equation}
\CIntensity_\ti = \gain\HawkesMean + \int_{-\infty}^\ti e^{-\HawkesAutocov(\ti - s)} \HawkesJump \dd \HawkesMartingale_s.
\label{eq:HawkesIntensityMartingale}
\end{equation}

By the Ito isometry for counting processes we get (<ref>) from (<ref>) via:

\begin{align*}
\V[\CIntensity_\ti] &= \E[(\CIntensity_\ti-\gain \HawkesMean)^2]\\
&= \E\left[\left(\int_{-\infty}^\ti e^{-\HawkesAutocov(\ti - s)} \HawkesJump \dd \HawkesMartingale_s\right)^2 \right]\\
&= \int_{-\infty}^\ti  e^{-2\HawkesAutocov(\ti - s)} \HawkesJump^2 \E[\CIntensity_s] \dd s\\
&= \frac{\gain \HawkesJump^2 \mu}{2\HawkesAutocov} = \gain^2\HawkesVariance - \gain\HawkesMean\HawkesJump\\ &= \V[\intensityFct_\ti] - \E[\intensityFct_\ti]\HawkesJump.
\end{align*}

The equality $\HawkesJump$ that solves $\HawkesJump^2 + 2\HawkesAutocov \HawkesJump - \frac{2\gain\HawkesAutocov\HawkesVariance}{\HawkesMean} = 0$ yields (<ref>).
Suppose we have an intensity process $\CIntensity_\ti = \gain \GammaMean_\ti$ that depends on some auxiliary process $\GammaVar_\ti$ via

\begin{equation}
\dd \GammaMean_\ti = -\HawkesAutocov (\GammaMean_\ti - \HawkesMean) \dd \ti + \frac{\GammaVar_\ti}{\GammaMean_\ti}(\dd \out_\ti - \gain\GammaMean_\ti \dd \ti).
\label{eq:HawkesGeneralization}
\end{equation}

The process $\GammaVar_\ti$ can be regarded as a conditional variance approximation.
For a process $(\GammaMean_\ti, \GammaVar_\ti)$
that satisfies (<ref>) we obtain

\begin{align}
&\V[\out_T] = \E[(\out_T - \E\out_T)^2]\\ &= \E \left[ \left(\int_0^T c(\GammaMean_u - \GammaMu) \dd u + \int_0^T \dd \HawkesMartingale_\ti\right)^2\right]  \\
&= \E \left[ \left(\int_0^T c \int_0^ue^{-\GammaDecay(u - \ti)} \frac{\GammaVar_\ti}{\GammaMean_\ti} \dd \HawkesMartingale_\ti \dd u + \int_0^T \dd \HawkesMartingale_\ti\right)^2\right]  \\
&= \E \left[ \left(\int_0^T \frac{c}{\gamma} (1-e^{-\GammaDecay(T - \ti)})  \frac{\GammaVar_\ti}{\GammaMean_\ti} + 1 \dd \HawkesMartingale_\ti \right)^2\right] \\
&=\E \left[ \int_0^T \left(\frac{c}{\gamma} (1-e^{-\GammaDecay(T - \ti)})  \frac{\GammaVar_\ti}{\GammaMean_\ti} + 1\right)^2 c\GammaMean_\ti \dd t \right].
\label{eq:var_slope}
\end{align}

The chain rule yields the evolution equation of $\GammaMean_\ti^2$ from eq. (<ref>)

\begin{align}
\dd \GammaMean_\ti^2 &= \left\{-2\GammaDecay\GammaMean_\ti(\GammaMean_\ti - \GammaMu) - 2\gain\GammaMean_\ti \GammaVar_\ti \right\} \dd \ti \nonumber\\
&+(\GammaMean_\ti + \frac{\GammaVar_\ti}{\GammaMean_\ti})^2 - \GammaMean_\ti^2 \dd \out_\ti.
\label{eq:second_moment}
\end{align}

By applying the $\E$ and $\dd \out_\ti - c \GammaMean_\ti\dd \ti = \dd \HawkesMartingale_\ti$ we get

\begin{align*}
\gain\E\left[\frac{\GammaVar_\ti^2}{\GammaMean_\ti}\right] &= \frac{\dd}{\dd \ti} \E[\GammaMean_\ti^2 ] + 2\GammaDecay \E[\GammaMean^2] - 2\GammaDecay\GammaMu^2\\
&= \frac{\dd}{\dd \ti} \V[\GammaMean_\ti ] + 2\GammaDecay\V[\GammaMean_\ti ].
\end{align*}

Using this in eq. (<ref>), we get

\begin{align}
\lim_{T \to \infty}\frac{1}{T}  \V[\out_T] &=\gain \GammaMu + \frac{2\gain^2}{\GammaDecay}(\E[\GammaVar_\infty] + \V[ \GammaMean_\infty]).
\label{eq:asymptotic_var_slope}
\end{align}

For the Gamma filter choose $\GammaVar_\ti$ as in eq. (<ref>) and for the Hawkes choose $\GammaVar_\ti = \HawkesJump\CIntensity_\ti, \gain = 1$.

Asymptotic variance for the Gamma filter:
Taking the expectation in eq. (<ref>) and eq. (<ref>) yields

\begin{align*}
\frac{\dd}{\dd \ti}(\E[\GammaMean_\ti^2] -\GammaMu^2) &= \gain\E\left[\frac{\GammaVar_\ti^2}{\GammaMean_\ti}\right] -2\GammaDecay (\E[\GammaMean_\ti^2] - \GammaMu^2)\\
\frac{\dd}{\dd \ti}\E[\GammaVar_\ti] &= -2\GammaDecay \E[\GammaMean_\ti^2] + 2\GammaDecay\GammaSigma -c\E\left[\frac{\GammaVar_\ti^2}{\GammaMean_\ti}\right].
\end{align*}

So the derivative of the sum evolves as
\begin{align}
\frac{\dd}{\dd \ti}( \E[\GammaVar_\ti] + \V[\GammaMean_\ti]) = -2\GammaDecay (\E[\GammaVar_\ti] + \V[\GammaMean_\ti]) + 2\GammaDecay\GammaSigma.
\end{align}

Since the sum starts in the steady state $\E[\GammaVar_0] + \V[\GammaMean_0] = \GammaSigma$, it stays constant for all $\ti$ and in particular in the asymptotic.

Both Gamma filter and Hawkes process satisfy the variance decomposition

\begin{equation}
\E[\gain^2 \GammaVar_{\infty}] + \V[\gain \GammaMean_\infty] = \gain^2\GammaSigma, \quad \gain\HawkesMean\HawkesJump  + \V[\CIntensity_\infty] = \gain^2\HawkesVariance.
\label{eq:VarianceDecomposition2}
\end{equation}

Consequently both filters agree with the process they approximate in terms of the asymptotic first and second order moments.

§ WASSERSTEIN METRIC

The Gamma ACID was computed using theorem <ref> and the method in section <ref>. The $(m, s)$-plane was truncated in a way to respect the minimal value of $m$ in the progression eq. (<ref>) and to cover $99,5\%$ of the probability mass of a Gamma distribution with mean $\gain\GammaMu$ and $\gain^2\GammaSigma$. The bounds of the auxiliary $s$ were dictated by the minimum and maximum in eq. (<ref>) for the above determined range of $m$. The rectangular $(m, s)$-domain was partitioned into $100\times 50$ congruent rectangles. Denote the boundaries of the rectangles by $\intervalBound^m_{\ipart}$ and $\intervalBound^s_{\ipart}$, respectively. Similar to the Hawkes ACID, $3\times 3$ equally spaced representatives $\state_\ipart$, see eq. (<ref>), were chosen in each rectangle. Their mean function evaluations were used in eq. (<ref>) for the coefficients of $\OpI$. As for the DOnSOff numerical approximation, the times $\intime_\itime(\statefrom_\iPart)$ in (<ref>) that satisfy

\begin{equation}
\target_1(\intime, \statefrom_\iPart) = \intervalBound^m_{\ipart} \quad \text{or} \quad \target_2(\intime, \statefrom_\iPart) = \intervalBound^s_{\ipart}
\label{eq:Gamma_event}
\end{equation}

were found by evolving the ODE system (<ref>), (<ref>) and checking for the event (<ref>).

The Hawkes and Gamma filter were compared. A comparison of their ACIDs shows that they are remarkably similar. The cdfs of their ACIDs approximately agree for a range of values $\HawkesVariance, \HawkesAutocov$, while the mean and gain were kept constant $\HawkesMean = 2, \gain = 1$. The Wasserstein metric was computed for the regime $(\HawkesVariance, \HawkesAutocov) \in \{0.05, 0.1, \dot 2\} \times \{ 0.2, 0.4, \dots, 4 \}$. For the Gamma(Hawkes) filter, the numerical method yielded $93,0\%$ ($98,5\%$) of ACIDs that had mean value less than $0.01$ from the true value $\HawkesMean$ and $40,6\%$ ($53,1\%$) with a difference less than $0.001$. For the Gamma filter, most outliers (deviation $> 0.01$) were detected for $\HawkesAutocov = 0.05$ or $\HawkesVariance > 4\cdot\HawkesAutocov - 0.45$. For the Hawkes all outliers were detected at $\HawkesAutocov = 0.05$.
The Wasserstein metric values ranged from $0.0005$ to $0.13$. When neglecting $\HawkesAutocov = 0.05$ the largest value was $0.078$ for $\HawkesVariance = 4, \HawkesAutocov = 0.65$ and a decreasing trend for decreasing $\HawkesVariance$ was detected, relatively independent of $\HawkesAutocov$. The large values for $\HawkesAutocov = 0.05$ can be explained by the numerical inaccuracy in the Gamma ACID. Exemplary graphs for $\HawkesMean = 2, \HawkesAutocov = 0.65, \gain = 1$ and smaller vs. larger variance are depicted in fig. <ref>.

title=$ \HawkesVariance = 0.2$,
name = plot_left,

scale only axis,

xmin = 1,
xmax = 3,
xlabel = $\CIntensity_\infty$,
ylabel = ACID,
legend style=legend cell align=left, align=left, draw=white!15!black,
axis background/.style=fill=white

scale only axis,
xlabel style=font=,
axis background/.style=fill=white,
title style=font=,
legend style=legend cell align=left, align=left, draw=white!15!black

table[row sep=crcr]
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title=$ \HawkesVariance = 2$,
name = plot_middle,
at=(plot_left.right of south east), anchor=left of south west,
scale only axis,
xlabel = $\CIntensity_\infty$,
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axis background/.style=fill=white
[color=mycolor1, forget plot]
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[color=mycolor2, dashed, forget plot]
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title = $ \HawkesVariance = 4$,
name = plot_right,
at=(plot_left.right of south east), anchor=left of south west,
scale only axis,

xmax = 9,
xlabel = $\CIntensity_\infty$,
yticklabel = ,
axis background/.style=fill=white
[color=mycolor1, forget plot]
table[row sep=crcr]
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9.60702840244435	0.998929659623629

10.6172962508426	0.999532250899808

at=(plot_left.below south west), anchor=above north west,
scale only axis,
name = plot_below,
y tick label style=
/pgf/number format/.cd,
fixed zerofill,
xlabel style=font=,
ylabel style=font=,
ylabel=Wasserstein metric,
axis background/.style=fill=white,
legend style=at = (0.5,1), legend cell align=left, align=left, anchor = north, draw=white!15!black
[color=mycolor1, dashed, line width=1pt]
table[row sep=crcr]
0.2	0.00367544839933931

0.4	0.00853170726027663

0.600000000000001	0.012026490605562

0.8	0.0166402968695856

1	0.0200709283288871

1.2	0.022867003373424

1.4	0.0254422217137229

1.6	0.0278911416845284

1.8	0.0300267915389014

2	0.0322496100030962

2.2	0.0342544141236267

2.4	0.0364774358724747

2.6	0.038552244929531

2.8	0.0405568592550223

3	0.0425322460140034

3.2	0.0443465931539775

3.4	0.0462100246157187

3.6	0.0479336349753989

3.8	0.0498128932810387

4	0.0517466495579253

$\gamma= 0.1$

[color=mycolor2, line width=1pt]
table[row sep=crcr]
0.2	0.0013286272748374

0.4	0.00407367947775406

0.600000000000001	0.00787113772584203

0.8	0.0123184545165307

1	0.0166642001230066

1.2	0.0212119549324568

1.4	0.0256669167867294

1.6	0.0298895207387071

1.8	0.0342309538932879

2	0.0386317400368021

2.2	0.0428189337920513

2.6	0.0510464882652402

3	0.0589267460656853

3.2	0.0627587434440082

3.4	0.0668558260335255

3.6	0.0707537579799675

3.8	0.0743212169867524

4	0.0779500401123894

$\gamma= 0.65$

[color=mycolor1, dotted, line width=1pt]
table[row sep=crcr]
0.2	0.000564127200807896

0.4	0.00200033332655991

0.600000000000001	0.00421429079422619

0.8	0.00697651909171348

1	0.0103679142597297

1.2	0.0136696719087972

1.4	0.0173659018721324

1.6	0.0217948565043509

1.8	0.0239690671307216

2	0.0280373255541964

2.2	0.0323942171665195

2.4	0.0379633734181226

2.6	0.0416831789984426

2.8	0.0451174697575931

3	0.0482645529980452

3.2	0.05111941723084

3.4	0.0520207728851894

3.6	0.0556929170762146

3.8	0.0594108206844153

4	0.0623597889469831

$\gamma= 2$

[anchor = north] at (plot_left.above north west) a);
[anchor = north] at (plot_right.above north west) b);
at (plot_below.above north west) c);

Comparison of the ACID for Gamma and Hawkes. a) and b) show the cdf of the ACID for the Gamma filter and Hawkes process for $\HawkesMean = 2, \HawkesAutocov = 0.65$ and different values of $\HawkesVariance$. c) The Wasserstein metric was computed for a range of $\HawkesAutocov, \HawkesVariance$, while $\HawkesMean = 2, \gain = 1$ were fixed. In the examined regime $\HawkesAutocov$ has little effect, while the difference measure slightly increases with growing $\HawkesVariance$. The largest deviation for $\HawkesAutocov = 0.65, \HawkesVariance = 4$ was depicted in b) still showing agreement of the ACIDs.

The most prominent dissimilarity is found in base values and it becomes more pronounced for larger variance. Still the ACIDs are very similar considering the example with the largest Wasserstein metric from the
From this we conjecture, that when quantities are computed that only depend on the ACID, the optimal linear estimator might replace the Gamma filter. The mutual information rate along the Poisson channel was efficiently approximated by Monte Carlo simulation in <cit.> via the Gamma filter. The Hawkes approximation can be conjectured to be even more efficient without severe loss in accuracy. In what respect the replacement works for this case and more complicated reaction networks, must be carefully evaluated.

§ COMPUTING THE MUTUAL INFORMATION WITH GAMMA FILTER VS. HAWKES

For the birth-death input process $\msg_\ti$ with birth rate $\HawkesAutocov\HawkesMean$ and death rate $\HawkesAutocov$ (i.e., mean $\HawkesMean$ and autocovariance function $\HawkesMean e^{-\HawkesAutocov \ti}$) the gamma filter's conditional variance equation is slightly modified:

\begin{equation}
\dd\GammaVar_\ti = \{-\GammaDecay (2\GammaVar_\ti - \GammaMean_\ti - \GammaMu) - 2\gain\frac{\GammaVar_\ti^2}{\GammaMean_\ti} \} dt + \frac{\GammaVar_{\ti-}^2}{\GammaMean_{\ti-}^2}\dd \out_\ti.
\label{eq:Gamma_filter_BD}
\end{equation}

In the limit $\gamma \to \infty$ ACID is a delta distribution at $\CIntensity_\infty = \E[\intensity_\infty]$ and hence

$$ \lim_{\gamma \ti \infty}\IR(\msg, \out) =  \E[\phi(\intensity_\infty)] - \phi(\E[\intensity_\infty]).$$

Comparison of the information rate $\IR(\msg, \out)$ for a Markov-modulated Poisson process $\out$, approximately computed with the Gamma and Hawkes filter. The input $\msg$ was a birth-death process with birth rate $\gamma\mu$ and death rate $\gamma$. In the biological context this corresponds to the gene expression model with mRNA counts $\msg_\ti$ and protein translation event counts $\out_\ti$. a) The x-axis shows the average mRNA lifetime $\gamma^{-1}$. Both the Gamma and Hawkes approximate information rate were computed as Monte Carlo average using <cit.> with hyperparameters $T = 200$, sample size $10,000$. Parameters were $\mu = 10, c = 1$ and $\gamma = 1, \dots, 10$. The value at $0$ was determined analytically via $\E[\phi(\msg_\infty)] - \phi(\E[\msg_\infty])$, with $\msg_\infty \sim \Pois(\mu)$. b) shows the relative simulation time of the Gamma vs Hawkes in logarithmic scale with colors as in a).

§ DERIVATION OF THE LIOUVILLE-POISSON MASTER EQUATION

In Gardiner [21] the jump part of the differential Chapman-Kolmogorov equation requires a Kernel function $W(x \vert z,t)$ to act on the probability density, e.g. jumps are targeting a range of new values dictated by a probability kernel. In our case jumps target deterministic values, given by a function $f$.

Suppose $Z_t$ has a  trajectory-wise evolution

$$ dZ_t = A(Z_t) dt + [f(Z_t) - Z_t] dY_t $$

and jumps of $Y_t$ occur with intensity $\lambda(Z_t)$.

We derive the probability evolution equation. For this purpose let $f_{-}$ be the inverse of $f$, i.e. a jump that enters at $z$ jumped from $f_{-}(z)$.

It holds that

\begin{align*}
&\Prob[Z(t+\Delta t)\in (-\infty, z]]\\ &= \int_{-\infty}^{f_{-}(z)} \Prob[\text{jump in } [t, t+\Delta t]\vert Z(t) = z'] p(z',t) \dd z' + o(\Delta t)\\
&+ \int_{-\infty}^{z - A(z)\Delta t + o(\Delta t)} \Prob[\text{no jump in }[t, t+\Delta t] \vert Z(t) = z'] \times\\&\qquad \qquad \qquad \qquad \quad p(z',t) \dd z'\\
&= \int_{-\infty}^{f_{-}(z)} \lambda(z')\Delta t p(z',t) \dd z' + o(\Delta t)\\
&+ \int_{-\infty}^{z - A(z)\Delta t + o(\Delta t)} (1 - \lambda(z')\Delta t) p(z',t) \dd z'.
\end{align*}

Now we take the derivative with respect to $z$. This yields

\begin{align*}
&p(z, t+ \Delta t)\\ = & f_{-}'(z) \lambda(f_{-}(z)) p(f_{-}(z),t) \Delta t\\
&+ (1-\lambda(z)\Delta t)(1-A'(z)\Delta t)p(z-A(z)\Delta t,t) + o(\Delta t)\\
= & f_{-}'(z) \lambda(f_{-}(z)) p(f_{-}(z),t) \Delta t\\
&+ (1-\lambda(z)\Delta t)(1-A'(z)\Delta t)(p(z,t) - A(z)\Delta t p_z(z,t))\\ &+ o(\Delta t)\\
= & f_{-}'(z) \lambda(f_{-}(z)) p(f_{-}(z),t) \Delta t + p(z,t)\\ &- \Delta t [\lambda(z)p(z,t) + A'(z)p(z,t) + A(z)p_z(z,t)] + o(\Delta t).
\end{align*}

\begin{align*}
\lim_{\Delta t \to 0}&\frac{p(z, t+ \Delta t) - p(z, t)}{\Delta t}\\ =& - \partial_z(A(z)p(z,t)) - \lambda(z)p(z,t) \\&+ f_{-}'(z) \lambda(f_{-}(z)) p(f_{-}(z),t).
\end{align*}

§ COMPARISON OF BRET-P AND DIRECT METHOD

Figure <ref> compares the BReT-P and the direct method and shows good agreement for the three cases.

title = $\HawkesDecay = 0.3$,
name = plot_left,

scale only axis,
xlabel = $\CIntensity_\infty$,
axis background/.style=fill=white
[color=mycolor1, line width = 1pt]
table[row sep=crcr]
0.396706523493108	0.0289978038965

0.411717553273543	0.0443344905540197

0.426728583053978	0.0543757833620839

0.441739612834413	0.0632155471606231

0.501783731956151	0.0968999530184389

0.516794761736587	0.105736088444382

0.531805791517021	0.114900114762284

0.546816821297456	0.124444958045024

0.561827851077891	0.134417738428697

0.576838880858325	0.144862217926052

0.59184991063876	0.155820356173325

0.606860940419194	0.167333352909521

0.62187197019963	0.179442381460333

0.636882999980064	0.191154741057075

0.651894029760498	0.201720409751359

0.666905059540934	0.211741850197201

0.696927119101803	0.231303833899674

0.726949178662673	0.250541182123016

0.756971238223542	0.269504835027487

0.786993297784411	0.288094769040295

0.802004327564847	0.297192561134054

0.817015357345281	0.30611852335047

0.832026387125715	0.314836382149893

0.84703741690615	0.323305010256004

0.862048446686585	0.331478271397654

0.87705947646702	0.339332537374218

0.892070506247454	0.346902526318541

0.907081536027889	0.35423778105089

0.922092565808324	0.36135597537213

0.937103595588758	0.368258895633782

0.952114625369194	0.374944108545156

0.967125655149628	0.381407218841238

0.982136684930063	0.387642941487885

0.997147714710498	0.393645766453441

1.01215874449093	0.399410449503081

1.02716977427137	0.404932423166543

1.0421808040518	0.410208173209008

1.05719183383224	0.415235605489414

1.07220286361267	0.420014418469905

1.08721389339311	0.424546491740234

1.10222492317354	0.428836298293247

1.11723595295398	0.432890661610737

1.13224698273441	0.436716600844704

1.14725801251485	0.440318830522513

1.16226904229528	0.4436999052044

1.17728007207571	0.446861695636224

1.19229110185615	0.449806052951572

1.20730213163658	0.452535017699837

1.22231316141702	0.455050928882234

1.23732419119745	0.457356482316603

1.25233522097789	0.459454755431433

1.26734625075832	0.461349205420258

1.28235728053876	0.463043643399656

1.29736831031919	0.464542185039413

1.31237934009963	0.465849176851416

1.32739036988006	0.466969096461134

1.3424013996605	0.467906424528368

1.35741242944093	0.468665501629443

1.37242345922137	0.469250437757964

1.3874344890018	0.469665152717921

1.40244551878224	0.469913503250726

1.41745654856267	0.469999375948633

1.4324675783431	0.469926716511656

1.44747860812354	0.469699531873804

1.46248963790397	0.46932188237638

1.49251169746484	0.4681316126746

1.52253375702571	0.466388872449148

1.55255581658658	0.464126399234116

1.58257787614745	0.461376189488973

1.61259993570832	0.458169244048023

1.64262199526919	0.454535688685344

1.67264405483006	0.450504999883778

1.70266611439093	0.446106002768364

1.7326881739518	0.441366742916623

1.76271023351267	0.436314363698751

1.79273229307354	0.43097502830033

1.82275435263441	0.425373895671092

1.85277641219528	0.419535138478663

1.88279847175615	0.413481969983473

1.92783156109745	0.404048621070174

1.97286465043875	0.394253728240344

2.01789773978006	0.384163392151796

2.0779418589018	0.37035523551494

2.15299700780397	0.352709055490348

2.4231955438518	0.288693438885065

2.48323966297353	0.274868771394396

2.54328378209527	0.261322002836589

2.60332790121701	0.248096875762265

2.66337202033875	0.235229339985134

2.70840510968006	0.225830998968971

2.75343819902136	0.21666004226056

2.79847128836266	0.207724480335622

2.84350437770397	0.199030646948521

2.88853746704527	0.190583345700784

2.93357055638658	0.182385988518453

2.97860364572788	0.174440726073072

3.02363673506919	0.166748570275426

3.06866982441049	0.159309509025645

3.11370291375179	0.152122613430826

3.1587360030931	0.145186137726749

3.2037690924344	0.138497612168948

3.24880218177571	0.132053929179198

3.29383527111701	0.12585142304494

3.33886836045831	0.119885943475128

3.38390144979962	0.114152923320032

3.42893453914092	0.10864744076372

3.47396762848223	0.103364276296086

3.53401174760397	0.0966564098061413

3.5940558667257	0.0903205542509955

3.65409998584744	0.0843427776891899

3.71414410496918	0.078708855423276

3.77418822409092	0.0734044062836361

3.83423234321266	0.068415010480666

3.8942764623344	0.0637263107851282

3.95432058145614	0.0593240986828905

4.01436470057788	0.0551943870362006

4.07440881969962	0.0513234706714698

4.13445293882136	0.0476979762042982

4.20950808772353	0.0434914640317441

4.2845632366257	0.0396236396638754

4.35961838552788	0.0360710605116283

4.43467353443005	0.0328113883010177

4.50972868333222	0.0298234167318068

4.5847838322344	0.0270870822251128

4.67485001091701	0.0241090760923779

4.76491618959961	0.0214356692305895

4.85498236828222	0.0190385654340064

4.96005957674526	0.0165564490009693

5.06513678520831	0.0143770933307223

5.17021399367135	0.0124662752966431

5.29030223191483	0.0105719136728029

5.42540149993874	0.00876115814428324

5.56050076796265	0.00724072042308599

5.710611065767	0.00583848026224665

5.87573239335178	0.00458607708022996

6.055864750717	0.00350159196061472

6.26601916764309	0.00253003105059513

6.50619564413004	0.00171497537833787

6.7914052099583	0.00104384402267765

7.1366588949083	0.000524560740059243

7.58698978832134	0.00014436341762547

7.88721038393004	2.80399556018551e-06

table[row sep=crcr]
0.393936011805341	0.0264399662613677

0.403406018210241	0.0378299521709842

0.412876024615139	0.0454473750924009

0.422346031020039	0.0518402718547026

0.431816037424937	0.0576539629418225

0.450756050234736	0.0685066234096361

0.488636075854332	0.0896391224912989

0.507576088664131	0.100561240046922

0.526516101473929	0.111947609048013

0.545456114283729	0.123913501143692

0.554926120688627	0.130144565982873

0.564396127093527	0.136556100454564

0.573866133498425	0.143158888853012

0.583336139903325	0.1499634721583

0.592806146308224	0.156980249834779

0.602276152713124	0.164219561206483

0.611746159118022	0.171691751710364

0.621216165522922	0.179407227815263

0.63068617192782	0.18737650335728

0.64015617833272	0.194246705138017

0.659096191142519	0.207143963877131

0.687506210357217	0.225808850257911

0.725386235976814	0.250187888172213

0.75379625519151	0.268189684222079

0.782206274406208	0.285869783641481

0.801146287216007	0.297403687921713

0.820086300025805	0.308664985902618

0.839026312835603	0.31957969578057

0.857966325645402	0.330061427968447

0.8769063384552	0.340010808702488

0.895846351264998	0.349513076661863

0.914786364074796	0.358661118521665

0.933726376884595	0.367465823720126

0.952666389694393	0.37592369801386

0.971606402504191	0.384025805638009

0.990546415313991	0.391761174350314

1.00948642812379	0.399118735403022

1.02842644093359	0.406088739692057

1.04736645374339	0.412663979409809

1.06630646655318	0.418840961334705

1.08524647936298	0.424621109490973

1.10418649217278	0.430012045719117

1.12312650498258	0.435028983249151

1.14206651779238	0.439686352952874

1.16100653060218	0.443989455375274

1.17994654341197	0.44794198806377

1.19888655622177	0.451547671681498

1.21782656903157	0.454810820752149

1.23676658184137	0.457736612870674

1.25570659465117	0.460331202055196

1.27464660746097	0.462601721902292

1.29358662027077	0.464556192782091

1.31252663308056	0.466203333956463

1.33146664589036	0.467552274370835

1.35040665870016	0.468612151203343

1.36934667150996	0.469391581618302

1.38828668431976	0.469898439405442

1.40722669712956	0.470140486220872

1.42616670993935	0.470125611163743

1.44510672274915	0.469861867835677

1.46404673555895	0.469357458404161

1.48298674836875	0.468620691645445

1.50192676117855	0.467659926950315

1.52086677398835	0.466483512246565

1.53980678679814	0.465099722719982

1.55874679960794	0.463516707330415

1.58715681882264	0.460786053607086

1.61556683803734	0.457651287072871

1.64397685725204	0.454137962908144

1.67238687646673	0.450271159481662

1.70079689568143	0.446075425130275

1.72920691489613	0.441574668687053

1.75761693411083	0.436792062775966

1.78602695332552	0.431749981379696

1.81443697254022	0.426469978656538

1.85231699815982	0.419095624365017

1.89019702377941	0.411382130837573

1.92807704939901	0.403374088367812

1.97542708142351	0.393013727404831

2.022777113448	0.38233548775615

2.0795971518774	0.369201271481513

2.14588719671169	0.353575986682308

2.27846728638028	0.321934740821996

2.37316735042927	0.299471403402613

2.43945739526357	0.283999088546562

2.50574744009786	0.268827257487871

2.56256747852726	0.256112377830387

2.61938751695665	0.243701273036741

2.67620755538605	0.231622196029786

2.72355758741054	0.22182651429976

2.77090761943504	0.212287852113558

2.81825765145954	0.203014580241142

2.86560768348403	0.194013095316576

2.91295771550853	0.185288004966878

2.96030774753302	0.176842301829812

3.00765777955752	0.168677526596861

3.05500781158202	0.160793920330692

3.10235784360651	0.15319056635432

3.14970787563101	0.145865522044856

3.1970579076555	0.138815940908488

3.24440793968	0.132038185349231

3.2917579717045	0.125527930562216

3.33910800372899	0.119280259991477

3.38645803575349	0.113289752797913

3.43380806777798	0.107550563785123

3.48115809980248	0.102056496227178

3.53797813823188	0.0957780321973321

3.59479817666127	0.0898315778121148

3.65161821509067	0.0842051789005236

3.70843825352006	0.0788866636612404

3.76525829194946	0.0738637454364515

3.82207833037885	0.0691241121718136

3.87889836880825	0.0646555037766046

3.93571840723764	0.0604457785229933

4.00200845207194	0.0558456788473691

4.06829849690623	0.0515630543584127

4.13458854174053	0.0475797862426628

4.20087858657482	0.0438783180204716

4.26716863140911	0.0404417068151162

4.34292868264831	0.0368175981739789

4.4186887338875	0.0334949368885713

4.49444878512669	0.030451676816222

4.57967884277079	0.0273360700520247

4.66490890041488	0.0245198535303324

4.75013895805897	0.0219770383961642

4.84483902210797	0.0194430313695531

4.93953908615696	0.0171862230067532

5.04370915661085	0.0149905771588958

5.15734923346964	0.0128996347585719

5.27098931032843	0.0110880032307783

5.39409939359212	0.00940002503125648

5.52667948326071	0.00785783937326201

5.6781995857391	0.00639216046483959

5.83918969462238	0.00512378170198424

6.01911981631547	0.00399298319619845

6.22745995722325	0.00298342432408916

6.46421011734573	0.00213482162716971

6.73884030308781	0.00144130850210722

7.07976053366418	0.000878829029474559

7.51538082828954	0.000461159897042762

8.13093124460799	0.00017940161754737

9.1726319491469	2.93318680810728e-05

9.85447241029964	4.19412436869493e-06

title = $\HawkesDecay = 1$,
name = plot_middle,
at=(plot_left.right of south east), anchor=left of south west,
scale only axis,
xlabel = $\CIntensity_\infty$,
yticklabel = ,
axis background/.style=fill=white
[color=mycolor1, line width = 1pt, forget plot]
table[row sep=crcr]
0.772984443736583	0.88479860875823

0.791089286209328	0.6295823645754

0.809194128682073	0.566244126761044

0.82729897115482	0.532066594456809

0.845403813627565	0.510388696951587

0.86350865610031	0.495575702444857

0.881613498573055	0.485065604979575

0.8997183410458	0.477489859635451

0.917823183518545	0.472037467773648

0.935928025991291	0.468191397623723

0.954032868464036	0.465603596567689

0.972137710936781	0.464029683245432

0.990242553409526	0.463292211012778

1.00834739588227	0.463258765246351

1.02645223835502	0.463828274007795

1.04455708082776	0.464922111187267

1.06266192330051	0.46647812075752

1.08076676577325	0.468446488458863

1.098871608246	0.470786819470202

1.11697645071874	0.473466025274245

1.13508129319149	0.476456766715499

1.15318613566424	0.479736287581515

1.18939582060973	0.487088439047193

1.22560550555522	0.495403059920676

1.26181519050071	0.504594359543157

1.2980248754462	0.514600225177244

1.33423456039169	0.525375714434446

1.37044424533718	0.53688866557783

1.38854908780992	0.53406260180072

1.40665393028267	0.52409926440793

1.42475877275541	0.516637615912515

1.44286361522816	0.510438272371642

1.4609684577009	0.504968112975126

1.47907330017365	0.499988230724163

1.51528298511914	0.491000791365256

1.55149267006463	0.482838857195668

1.60580719748287	0.471432701591285

1.76875077973757	0.43809191689094

1.82306530715581	0.42613969244309

1.8592749921013	0.417722871955508

1.89548467704679	0.408873845249765

1.93169436199228	0.399532589992869

1.96790404693777	0.389640493544347

2.0584282593015	0.364001013153798

2.11274278671974	0.349558336816523

2.18516215661072	0.330966493260977

2.2575815265017	0.312895623141621

2.34810573886542	0.290883448012186

2.43862995122915	0.269500700819535

2.51104932112013	0.252930668542442

2.58346869101111	0.236970595852354

2.63778321842935	0.225501805099276

2.69209774584759	0.214489589721481

2.74641227326582	0.203896517988026

2.8188316431568	0.190389849269996

2.89125101304778	0.177567157791325

2.96367038293877	0.16541704803929

3.03608975282975	0.153933618834905

3.10850912272073	0.143110845142687

3.18092849261171	0.132937756370854

3.25334786250269	0.123393782621033

3.32576723239367	0.114447702344568

3.39818660228465	0.106068888117271

3.47060597217564	0.0982306909261013

3.54302534206662	0.0909079937523405

3.6154447119576	0.0840760082915608

3.68786408184858	0.0777097057439704

3.76028345173956	0.0717836789855841

3.85080766410329	0.0649564139394112

3.94133187646701	0.0587286658335504

4.03185608883074	0.0530549474944486

4.12238030119447	0.0478925653164843

4.21290451355819	0.0432010736163111

4.32153356839467	0.0381390856814505

4.43016262323114	0.033638247300253

4.53879167806761	0.0296416359990026

4.66552557537683	0.0255471361265904

4.79225947268604	0.0219936932976044

4.937098212468	0.0185096628013657

5.08193695224997	0.0155569745172812

5.24488053450468	0.0127750659267836

5.42592895923213	0.0102450937626646

5.62508222643233	0.00801995248346543

5.84234033610527	0.00612489217262002

6.0958081307237	0.00445784570117702

6.38548561028763	0.00308724787175407

6.72947761726979	0.00198296626366812

7.16399383661568	0.00112037337344617

7.76145363821627	0.000495748745336755

8.66669576185354	0.000124764953694623

9.80730083763649	4.0690877156635e-07

[color=mycolor2, forget plot]
table[row sep=crcr]
0.769845286858104	0.866286095051082

0.78167181557389	0.676339091876574

0.793498344289677	0.606637566131079

0.805324873005462	0.566982098357849

0.817151401721249	0.540679656110104

0.828977930437036	0.521798539679594

0.840804459152823	0.507588958324611

0.85263098786861	0.496566287863724

0.864457516584396	0.487844400499702

0.876284045300181	0.480855217445491

0.888110574015968	0.475214485546591

0.899937102731755	0.470651194355147

0.911763631447542	0.46696768421641

0.923590160163329	0.464015774299256

0.935416688879116	0.46168179741357

0.947243217594901	0.459876849158128

0.959069746310687	0.458530224038112

0.970896275026474	0.4575848718677

0.982722803742261	0.456994175681304

0.994549332458048	0.456719617954324

1.01820238988962	0.456995441748537

1.04185544732119	0.458210562015241

1.06550850475277	0.460218009404537

1.08916156218434	0.462908056792791

1.11281461961591	0.466197112551841

1.13646767704749	0.470020435067358

1.16012073447906	0.474327201825316

1.18377379191063	0.479077081202343

1.20742684934221	0.484237792772978

1.23107990677378	0.489783335269319

1.26655949292114	0.498778309258409

1.3020390790685	0.508537790092118

1.33751866521586	0.519021769732579

1.37299825136322	0.530201479230282

1.384824780079	0.534078965662252

1.39665130879479	0.525366157385184

1.40847783751058	0.519443742193454

1.42030436622636	0.514563698461247

1.43213089494215	0.510288975894854

1.44395742365793	0.506424854719382

1.46761048108951	0.499533567980558

1.49126353852108	0.493398475541166

1.51491659595265	0.487769948608531

1.55039618210001	0.479959560246948

1.59770229696316	0.470259212982281

1.68048799797367	0.454042056245154

1.75144717026838	0.439969295799834

1.79875328513153	0.430152353932622

1.84605939999468	0.419803214411347

1.88153898614204	0.411606922835116

1.9170185722894	0.402975541434376

1.95249815843676	0.393854462849337

1.98797774458411	0.384189992436493

2.04711038816305	0.367621363750269

2.09441650302619	0.355107784048853

2.15354914660513	0.339996358531172

2.22450831889985	0.322378347510769

2.30729401991035	0.302340671270585

2.39007972092086	0.28278416595076

2.47286542193136	0.263744607569675

2.54382459422608	0.247932704118806

2.60295723780501	0.235212360964381

2.66208988138395	0.223005338258552

2.72122252496288	0.21130997150655

2.78035516854181	0.200084637584123

2.83948781212074	0.189309588569403

2.89862045569968	0.178974678763584

2.9695796279944	0.167146857760608

3.04053880028911	0.155941234786393

3.11149797258383	0.145353579419314

3.18245714487855	0.135375650825258

3.25341631717327	0.125991089608855

3.32437548946799	0.11717283227715

3.39533466176271	0.108892522739128

3.46629383405743	0.101125801867257

3.53725300635215	0.09384996011058

3.60821217864687	0.0870427187108369

3.69099787965737	0.079663210648361

3.77378358066788	0.0728545855211813

3.85656928167838	0.066580082806377

3.93935498268889	0.0608036597044102

4.03396721241518	0.0547683892191753

4.12857944214147	0.0492915996958843

4.22319167186777	0.0443281470812025

4.32963043030984	0.0393047318712583

4.43606918875192	0.0348199294560914

4.542507947194	0.0308210365545829

4.66077323435187	0.0268890990581383

4.79086505022552	0.0231161784014322

4.92095686609917	0.0198517939808536

5.06287521068861	0.0167951073140831

5.21662008399383	0.0139949724493587

5.38219148601484	0.0114835948704712

5.57141594546743	0.00914563212152508

5.78429346235159	0.00706583999466304

6.02082403666732	0.00529316533831903

6.29283419713041	0.00378719427003915

6.61215047245664	0.00254815079975401

7.0024259200776	0.00156323990421292

7.51096665485642	0.000821817523023682

8.23238490651939	0.000326509802338393

9.47417042167698	6.48030937195898e-05

12.5845474739288	5.79374287212886e-07

title = $\HawkesDecay = 3$,
at=(plot_middle.right of south east), anchor=left of south west,
scale only axis,
xlabel = $\CIntensity_\infty$,
yticklabel = ,
axis background/.style=fill=white
[color=mycolor1, line width = 1pt, forget plot]
table[row sep=crcr]
1.14331513983232	6.20152105104368

1.16697960313961	1.87039427343463

1.19064406644691	1.35425380446762

1.2143085297542	1.10322538794085

1.2379729930615	0.949564563563168

1.26163745636879	0.844065161199959

1.28530191967609	0.766384442534846

1.30896638298338	0.706414812390891

1.33263084629067	0.65850650772083

1.35629530959797	0.619229170253835

1.37995977290526	0.586367656433916

1.40362423621256	0.558421536969552

1.42728869951985	0.534334915315494

1.47461762613444	0.49486875254191

1.52194655274903	0.463844121576466

1.56927547936362	0.438796894926256

1.61660440597821	0.418156287503773

1.68759779590009	0.393245356116948

1.75859118582197	0.373625256790774

1.82958457574386	0.357848982289932

1.92424242897304	0.34119605728365

2.04256474550951	0.325461970538324

2.16088706204598	0.313760869320889

2.30287384188975	0.303529262316292

2.42119615842622	0.297371431791875

2.44486062173351	0.27560581153652

2.46852508504081	0.247430803904489

2.4921895483481	0.235392842339852

2.53951847496269	0.21840972121857

2.61051186488458	0.199916084278664

2.70516971811375	0.18103268693552

2.82349203465023	0.162139251154242

2.98914327780129	0.140464613139081

3.20212344756694	0.117115426015687

3.50976147056177	0.087958764160609

3.8173994935566	0.0623452159316038

4.05404412662954	0.0492425101903677

4.36168214962437	0.0362691664576857

4.74031356254108	0.0244573249218618

5.14260943876509	0.0159698196679017

5.68689209483286	0.00901353968925278

6.3968259940517	0.00421783713372648

7.48539130618724	0.00130171946520896

9.75717978368751	0.000105272201100348

12.9518823301723	3.8941706748119e-08

[color=mycolor2, forget plot]
table[row sep=crcr]
1.13949945640905	4.70737075286788

1.15553255286982	2.38742475869722

1.17156564933058	1.74612231149344

1.18759874579134	1.42357858425621

1.20363184225211	1.22383547813568

1.21966493871287	1.08583425029097

1.23569803517364	0.983778161632856

1.2517311316344	0.904711402787207

1.26776422809517	0.841346744936303

1.28379732455593	0.789241671132149

1.2998304210167	0.745519324704741

1.31586351747746	0.708226988126388

1.33189661393823	0.675988248498676

1.34792971039899	0.647802403671616

1.36396280685975	0.622922835071289

1.39602899978128	0.580931745469279

1.42809519270281	0.546788881579332

1.46016138562434	0.518434911870337

1.49222757854587	0.49448799331449

1.5242937714674	0.473983255200803

1.55635996438893	0.456224740726359

1.60445925377122	0.433643138731451

1.65255854315351	0.414856708380942

1.70065783253581	0.399005098184933

1.76479021837886	0.381401411442504

1.82892260422192	0.366922493769099

1.90908808652575	0.352167106312567

1.98925356882957	0.340262028867652

2.08545214759415	0.328842109324594

2.1976838228195	0.318528711313775

2.32594859450562	0.309734284828046

2.43818026973097	0.304049942299205

2.45421336619173	0.280926742290806

2.4702464626525	0.268918643090299

2.50231265557403	0.252669247551342

2.53437884849556	0.240644636241367

2.58247813787785	0.226303672768129

2.64661052372091	0.210918347854403

2.72677600602473	0.195094866082727

2.82297458478931	0.179116349344422

2.95123935647543	0.160977023401838

3.11157032108308	0.141468127442067

3.32000057507301	0.119258319103306

3.60859631136677	0.0916863045679932

3.83305966181747	0.0724801298442337

4.02545681934665	0.0604267037280692

4.26595326625811	0.0482978323541658

4.55454900255187	0.0366654762908176

4.87521093176716	0.0266837246455687

5.21190595744321	0.0191170402820084

5.64479956188385	0.0124569398851158

6.17389174508908	0.00731407739518275

6.86331489290195	0.00364242764739942

7.88943306639087	0.00127618155295295

9.82943773814336	0.000171719146187854

17.1565628207127	5.56049535305192e-08

[anchor = north] at (plot_left.above north west) a);
[anchor = north] at (plot_middle.above north west) b);
[anchor = north] at (plot_right.above north west) c);

Hawkes ACID: Comparison of BReT-P and direct method. Parameters were as in <ref>. Discretization granularity was $\inum = 1000$ for the direct method.

§ ACKNOWLEDGEMENTS

M. Sinzger-D'Angelo would like to thank N. Engelmann for discussion on the semi-Markov perspective, M. Gehri for general discussion and feedback, as well as U. Henning, S. Startceva and B. Alt for advice during the writing process.

[1]
O. Macchi and B. Picinbono, “Estimation and detection of weak optical
signals,” IEEE Transactions on Information Theory, vol. 18, no. 5,
pp. 562–573, 1972.

[2]
D. R. Cox, “Some statistical methods connected with series of events,”
Journal of the Royal Statistical Society: Series B (Methodological),
vol. 17, no. 2, pp. 129–157, 1955.

[3]
D. L. Snyder, Random point processes.1em plus 0.5em minus
0.4emWiley, New York, 1975.

[4]
Y. M. Kabanov, “The capacity of a channel of the Poisson type,”
Theory of Probability & Its Applications, vol. 23, no. 1, pp.
143–147, 1978.

[5]
M. Davis, “Capacity and cutoff rate for Poisson-type channels,” IEEE
Transactions on Information Theory, vol. 26, no. 6, pp. 710–715, 1980.

[6]
D. Snyder and C. Georghiades, “Design of coding and modulation for
power-efficient use of a band-limited optical channel,” IEEE
Transactions on Communications, vol. 31, no. 4, pp. 560–565, 1983.

[7]
S. Shamai, “On the capacity of a direct-detection photon channel with
intertransition-constrained binary input,” IEEE Transactions on
Information Theory, vol. 37, no. 6, pp. 1540–1550, 1991.

[8]
S. Shamai and A. Lapidoth, “Bounds on the capacity of a spectrally
constrained Poisson channel,” IEEE Transactions on Information
Theory, vol. 39, no. 1, pp. 19–29, 1993.

[9]
R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes: II.
Applications.1em plus 0.5em minus 0.4emSpringer-Verlag, New
York, 1977-78, vol. 2.

[10]
D. Guo, S. Shamai, and S. Verdú, “Mutual information and conditional mean
estimation in Poisson channels,” IEEE Transactions on Information
Theory, vol. 54, no. 5, pp. 1837–1849, 2008.

[11]
J. Jiao, K. Venkat, and T. Weissman, “Pointwise relations between information
and estimation in the Poisson channel.”

[12]
I. Lestas, G. Vinnicombe, and J. Paulsson, “Fundamental limits on the
suppression of molecular fluctuations,” Nature, vol. 467, no. 7312,
p. 174, 2010.

[13]
L. Duso and C. Zechner, “Path mutual information for a class of
biochemical reaction networks,” in 2019 IEEE 58th Conference on
Decision and Control (CDC), 2019, pp. 6610–6615.

[14]
S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon
channel,” IEE Proceedings I - Communications, Speech and Vision, vol.
137, no. 6, pp. 424–430, 1990.

[15]
P. M. Bremaud, “A martingale approach to point-processes,” Ph.D.
dissertation, EECS Department, University of California, Berkeley, 1972.
[Online]. Available:

[16]
P. Brémaud, Point processes and queues: martingale dynamics.1em plus 0.5em minus 0.4emSpringer, 1981, vol. 50.

[17]
D. J. Daley and D. Vere-Jones, An introduction to the theory of point
processes: volume I: Elementary theory and methods.1em plus 0.5em
minus 0.4emSpringer, 2003.

[18]
K. V. Parag and O. G. Pybus, “Optimal point process filtering and estimation
of the coalescent process,” Journal of theoretical biology, vol. 421,
pp. 153–167, 2017.

[19]
A. G. Hawkes, “Spectra of some self-exciting and mutually exciting point
processes,” Biometrika, vol. 58, no. 1, pp. 83–90, 1971.

[20]
D. Oakes, “The Markovian self-exciting process,” Journal of Applied
Probability, vol. 12, no. 1, pp. 69–77, 1975.

[21]
C. Gardiner, Stochastic methods: a handbook for the natural and social
sciences, 4th ed.1em plus 0.5em minus 0.4emSpringer-Verlag,

[22]
A. Crisanti, A. de Martino, and J. Fiorentino, “Statistics of optimal
information flow in ensembles of regulatory motifs,” Physical review.
E, vol. 97, no. 2-1, p. 022407, 2018.

[23]
J. E. Purvis and G. Lahav, “Encoding and decoding cellular information through
signaling dynamics,” Cell, vol. 152, no. 5, pp. 945–956, 2013.

[24]
J. Selimkhanov, B. Taylor, J. Yao, A. Pilko, J. Albeck, A. Hoffmann,
L. Tsimring, and R. Wollman, “Accurate information transmission through
dynamic biochemical signaling networks,” Science, vol. 346, no. 6215,
pp. 1370–1373, 2014.

[25]
D. Friedrich, L. Friedel, A. Finzel, A. Herrmann, S. Preibisch, and A. Loewer,
“Stochastic transcription in the p53-mediated response to dna damage is
modulated by burst frequency,” Molecular Systems Biology, vol. 15,
no. 12, 2019.

[26]
A. Maity and R. Wollman, “Information transmission from nfkb signaling
dynamics to gene expression,” PLOS Computational Biology, vol. 16,
no. 8, p. e1008011, 2020.

[27]
Tomasz Jetka, Karol Nienałtowski, Tomasz Winarski, Sławomir
Błoński, and Michał Komorowski, “Information-theoretic analysis
of multivariate single-cell signaling responses,” PLOS Computational
Biology, vol. 15, no. 7, p. e1007132, 2019.

[28]
R. Suderman and E. J. Deeds, “Intrinsic limits of information transmission in
biochemical signalling motifs,” Interface Focus, vol. 8, no. 6, p.
20180039, 2018.

[29]
K. V. Parag, “On signalling and estimation limits for molecular
birth-processes,” Journal of theoretical biology, vol. 480, pp.
262–273, 2019.

[30]
G. Tkačik, C. G. Callan, and W. Bialek, “Information flow and
optimization in transcriptional regulation,” Proceedings of the
National Academy of Sciences, vol. 105, no. 34, pp. 12 265–12 270, 2008.

[31]
R. Suderman, J. A. Bachman, A. Smith, P. K. Sorger, and E. J. Deeds,
“Fundamental trade-offs between information flow in single cells and
cellular populations,” Proceedings of the National Academy of
Sciences, vol. 114, no. 22, pp. 5755–5760, 2017.

[32]
S. A. Pasha and V. Solo, “Computing the trajectory mutual information between
a point process and an analog stochastic process,” in 2012 Annual
International Conference of the IEEE Engineering in Medicine and Biology
Society, 2012, pp. 4603–4606.

[33]
S. A. Cepeda-Humerez, J. Ruess, and G. Tkačik, “Estimating information
in time-varying signals,” PLOS Computational Biology, vol. 15, no. 9,
p. e1007290, 2019.

[34]
F. Tostevin and P. R. ten Wolde, “Mutual information between input and output
trajectories of biochemical networks,” Phys. Rev. Lett., vol. 102,
no. 21, p. 218101, 2009.

[35]
T. S. Roy, M. Nandi, A. Biswas, P. Chaudhury, and S. K. Banik, “Information
transmission in a two-step cascade: interplay of activation and repression,”
Theory in Biosciences, vol. 140, no. 3, pp. 295–306, 2021.

[36]
H.-P. Breuer, F. Petruccione, and Petruccione, School of Pure and Applied
Physics Francesco, The Theory of Open Quantum Systems.1em plus
0.5em minus 0.4emOxford University Press, 2002.

[37]
C. Zechner and H. Koeppl, “Uncoupled analysis of stochastic reaction networks
in fluctuating environments,” PLOS Computational Biology, vol. 10,
no. 12, pp. 1–9, 2014.

[38]
Z. Cao and R. Grima, “Analytical distributions for detailed models of
stochastic gene expression in eukaryotic cells,” Proceedings of the
National Academy of Sciences, vol. 117, no. 9, pp. 4682–4692, 2020.

[39]
T. L. Lenstra, J. Rodriguez, H. Chen, and D. R. Larson, “Transcription
dynamics in living cells,” Annual review of biophysics, 2016.

[40]
P. A. W Lewis and G. S. Shedler, “Simulation of nonhomogeneous Poisson
processes by thinning,” Naval Research Logistics Quarterly, vol. 26,
no. 3, pp. 403–413, 1979.

[41]
C. Zechner, G. Seelig, M. Rullan, and M. Khammash, “Molecular circuits for
dynamic noise filtering,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 113, no. 17, pp. 4729–4734,

[42]
L. Bronstein and H. Koeppl, “Marginal process framework: A model reduction
tool for Markov jump processes,” Physical review. E, vol. 97, no.
6-1, p. 062147, 2018.

[43]
D. R. Cox, Renewal Theory, Methuen and Co, 1962.

[44]
D. Vere-Jones and Y. Ogata, “On the moments of a self-correcting process,”
Journal of Applied Probability, vol. 21, no. 2, pp. 335–342, 1984.

[45]
M. H. A. Davis, “Piecewise-deterministic Markov processes: A general class
of non-diffusion stochastic models,” Journal of the Royal Statistical
Society: Series B (Methodological), vol. 46, no. 3, pp. 353–376, 1984.

[46]
K. V. Parag and G. Vinnicombe, “Point process analysis of noise in early
invertebrate vision,” PLOS Computational Biology, vol. 13, no. 10, p.
e1005687, 2017.

[47]
J. C. Cox, J. E. Ingersoll, and S. A. Ross, “A theory of the term structure of
interest rates,” Econometrica, vol. 53, no. 2, p. 385, 1985.

[48]
I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral
Equations: I. Introduction.1em plus 0.5em minus 0.4emSpringer
Science & Business Media, 1992.

[49]
M. Sinzger, M. Gehri, and H. Koeppl, “Poisson channel with binary Markov
input and average sojourn time constraints,” 2020 IEEE International
Symposium on Information Theory: Proceedings.

[50]
H. Qiu, B. Zhang, and T. Zhou, “Influence of complex promoter structure on
gene expression,” Journal of Systems Science and Complexity, vol. 32,
no. 2, pp. 600–614, 2019.

[51]
X. Gao and L. Zhu, “Functional central limit theorems for stationary hawkes
processes and application to infinite-server queues,” Queueing
Systems, vol. 90, no. 1, pp. 161–206, 2018.

[52]
N. Limnios and G. Oprisan, Semi-Markov Processes and
Reliability.1em plus 0.5em minus 0.4emSpringer Science &
Business Media, 2001.

[53]
C. Maes, K. Netočný, and B. Wynants, “Dynamical fluctuations for
semi-Markov processes,” Journal of Physics A: Mathematical and
Theoretical, vol. 42, no. 36, p. 365002, 2009.

[54]
A. Kuczura, “The interrupted Poisson process as an overflow process,”
Bell System Technical Journal, vol. 52, no. 3, pp. 437–448, 1973.