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Many probabilistic inference problems such as stochastic filtering or the computation of rare event probabilities require model analysis under initial and terminal constraints. We propose a solution to this bridging problem for the widely used class of population-structured Markov jump processes. The method is based on a state-space lumping scheme that aggregates states in a grid structure. The resulting approximate bridging distribution is used to iteratively refine relevant and truncate irrelevant parts of the state-space. This way, the algorithm learns a well-justified finite-state projection yielding guaranteed lower bounds for the system behavior under endpoint constraints. We demonstrate the method's applicability to a wide range of problems such as Bayesian inference and the analysis of rare events.
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Analysis of Markov Jump Processes under
Terminal Constraints
Michael Backenk¨ohler1,B, Luca Bortolussi2,3, Gerrit Großmann1, Verena
Wolf1,3
1Saarbr¨ucken Graduate School of Computer Science, Saarland University, Saarland
Informatics Campus E1 3, Saarbr¨ucken, Germany
Bmichael.backenkoehler@uni-saarland.de
2Univeristy of Trieste, Trieste, Italy
3Saarland University, Saarland Informatics Campus E1 3, Saarbr¨ucken, Germany
Abstract. Many probabilistic inference problems such as stochastic fil-
tering or the computation of rare event probabilities require model anal-
ysis under initial and terminal constraints. We propose a solution to
this bridging problem for the widely used class of population-structured
Markov jump processes. The method is based on a state-space lumping
scheme that aggregates states in a grid structure. The resulting approxi-
mate bridging distribution is used to iteratively refine relevant and trun-
cate irrelevant parts of the state-space. This way, the algorithm learns
a well-justified finite-state projection yielding guaranteed lower bounds
for the system behavior under endpoint constraints. We demonstrate the
method’s applicability to a wide range of problems such as Bayesian
inference and the analysis of rare events.
Keywords: Bayesian Inference ·Bridging problem ·Smoothing ·Lump-
ing ·Rare Events.
1 Introduction
Discrete-valued continuous-time Markov Jump Processes (MJP) are widely used
to model the time evolution of complex discrete phenomena in continuous time.
Such problems naturally occur in a wide range of areas such as chemistry [16],
systems biology [49,46], epidemiology [36] as well as queuing systems [10] and
finance [39]. In many applications, an MJP describes the stochastic interaction
of populations of agents. The state variables are counts of individual entities of
different populations.
Many tasks, such as the analysis of rare events or the inference of agent
counts under partial observations naturally introduce terminal constraints on
the system. In these cases, the system’s initial state is known, as well as the
system’s (partial) state at a later time-point. The probabilities corresponding
to this so-called bridging problem are often referred to as bridging probabilities
[17,19]. For instance, if the exact, full state of the process Xthas been observed
at time 0 and T, the bridging distribution is given by
Pr(Xt=x|X0=x0, XT=xg)
2 M. Backenk¨ohler et al.
for all states xand times t[0, T ]. Often, the condition is more complex, such
that in addition to an initial distribution, a terminal distribution is present.
Such problems typically arise in a Bayesian setting, where the a priori behavior
of a system is filtered such that the posterior behavior is compatible with noisy,
partial observations [11,25]. For example, time-series data of protein levels is
available while the mRNA concentration is not [1,25]. In such a scenario our
method can be used to identify a good truncation to analyze the probabilities
of mRNA levels.
Bridging probabilities also appear in the context of rare events. Here, the rare
event is the terminal constraint because we are only interested in paths contain-
ing the event. Typically researchers have to resort to Monte-carlo simulations in
combination with variance reduction techniques in such cases [14,26].
Efficient numerical approaches that are not based on sampling or ad-hoc
approximations have rarely been developed.
Here, we combine state-of-the-art truncation strategies based on a forward
analysis [28,4] with a refinement approach that starts from an abstract MJP with
lumped states. We base this lumping on a grid-like partitioning of the state-space.
Throughout a lumped state, we assume a uniform distribution that gives an
efficient and convenient abstraction of the original MJP. Note that the lumping
does not follow the classical paradigm of Markov chain lumpability [12] or its
variants [15]. Instead of an approximate block structure of the transition-matrix
used in that context, we base our partitioning on a segmentation of the molecule
counts. Moreover, during the iterative refinement of our abstraction, we identify
those regions of the state-space that contribute most to the bridging distribution.
In particular, we refine those lumped states that have a bridging probability
above a certain threshold δand truncate all other macro-states. This way, the
algorithm learns a truncation capturing most of the bridging probabilities. This
truncation provides guaranteed lower bounds because it is at the granularity of
the original model.
In the rest of the paper, after presenting related work (Section 2) and back-
ground (Section 3), we discuss the method (Section 4) and several applications,
including the computation of rare event probabilities as well as Bayesian smooth-
ing and filtering (Section 5).
2 Related Work
The problem of endpoint constrained analysis occurs in the context of Bayesian
estimation [41]. For population-structured MJPs, this problem has been ad-
dressed by Huang et al. [25] using moment closure approximations and by Wild-
ner and K¨oppl [48] further employing variational inference. Golightly and Sher-
lock modified stochastic simulation algorithms to approximatively augment gen-
erated trajectories [17]. Since a statistically exact augmentation is only possible
for few simple cases, diffusion approximations [18] and moment approximations
[35] have been employed. Such approximations, however, do not give any guaran-
tees on the approximation error and may suffer from numerical instabilities [43].
Analysis of Markov Jump Processes under Terminal Constraints 3
The bridging problem also arises during the estimation of first passage times
and rare event analysis. Approaches for first-passage times are often of heuristic
nature [42,22,8]. Rigorous approaches yielding guaranteed bounds are currently
limited by the performance of state-of-the-art optimization software [6]. In bi-
ological applications, rare events of interest are typically related to the reach-
ability of certain thresholds on molecule counts or mode switching [45]. Most
methods for the estimation of rare event probabilities rely on importance sam-
pling [26,14]. For other queries, alternative variance reduction techniques such
as control variates are available [5]. Apart from sampling-based approaches, dy-
namic finite-state projections have been employed by Mikeev et al. [34], but are
lacking automated truncation schemes.
The analysis of countably infinite state-spaces is often handled by a pre-
defined truncation [27]. Sophisticated state-space truncations for the (uncondi-
tioned) forward analysis have been developed to give lower bounds and rely on a
trade-off between computational load and tightness of the bound [37,28,4,24,31].
Reachability analysis, which is relevant in the context of probabilistic veri-
fication [8,38], is a bridging problem where the endpoint constraint is the visit
of a set of goal states. Backward probabilities are commonly used to compute
reachability likelihoods [2,50]. Approximate techniques for reachability, based
on moment closure and stochastic approximation, have also been developed in
[8,9], but lack error guarantees. There is also a conceptual similarity between
computing bridging probabilities and the forward-backward algorithm for com-
puting state-wise posterior marginals in hidden Markov models (HMMs) [40].
Like MJPs, HMMs are a generative model that can be conditioned on obser-
vations. We only consider two observations (initial and terminal state) that are
not necessarily noisy but the forward and backward probabilities admit the same
meaning.
3 Preliminaries
3.1 Markov Jump Processes with Population Structure
A population-structured Markov jump process (MJP) describes the stochastic
interactions among agents of distinct types in a well-stirred reactor. The assump-
tion of all agents being equally distributed in space, allows to only keep track
of the overall copy number of agents for each type. Therefore the state-space is
S ⊆ NnSwhere nSdenotes the number of agent types or populations. Interac-
tions between agents are expressed as reactions. These reactions have associated
gains and losses of agents, given by non-negative integer vectors v
jand v+
jfor
reaction j, respectively. The overall effect is given by vj=v+
jv
j. A reaction
between agents of types S1, . . . , SnSis specified in the following form:
nS
X
`=1
v
j` S`
αj(x)
nS
X
`=1
v+
j` S`.(1)
4 M. Backenk¨ohler et al.
The propensity function αjgives the rate of the exponentially distributed firing
time of the reaction as a function of the current system state x∈ S. In population
models, mass-action propensities are most common. In this case the firing rate
is given by the product of the number of reactant combinations in xand a rate
constant cj, i.e.
αj(x):=cj
nS
Y
`=1 x`
v
j` .(2)
In this case, we give the rate constant in (1) instead of the function αj. For a
given set of nRreactions, we define a stochastic process {Xt}t0describing the
evolution of the population sizes over time t. Due to the assumption of exponen-
tially distributed firing times, Xis a continuous-time Markov chain (CTMC) on
Swith infinitesimal generator matrix Q, where the entries of Qare
Qx,y =(Pj:x+vj=yαj(x),if x6=y,
PnR
j=1 αj(x),otherwise. (3)
The probability distribution over time can be analyzed as an initial value prob-
lem. Given an initial state x0, the distribution1
π(xi, t) = Pr(Xt=xi|X0=x0), t 0 (4)
evolves according to the Kolmogorov forward equation
d
dtπ(t) = π(t)Q , (5)
where π(t) is an arbitrary vectorization (π(x1, t), π(x2, t), . . . , π(x|S |, t)) of the
states.
Let xg∈ S be a fixed goal state. Given the terminal constraint Pr(XT=xg)
for some T0, we are interested in the so-called backward probabilities
β(xi, t) = Pr(XT=xg|Xt=xi), t T . (6)
Note that β(·, t) is a function of the conditional event and thus is no probability
distribution over the state-space. Instead β(·, t) gives the reaching probabilities
for all states over the time span of [t, T ]. To compute these probabilities, we can
employ the Kolmogorov backward equation
d
dtβ(t) = (t)>,(7)
where we use the same vectorization to construct β(t) as we used for π(t). The
above equation is integrated backwards in time and yields the reachability prob-
ability for each state xiand time t<T of ending up in xgat time T.
1In the sequel, xidenotes a state with index iinstead of its i-th component.
Analysis of Markov Jump Processes under Terminal Constraints 5
The state-space of many MJPs with population structure, even simple ones,
is countably infinite. In this case, we have to truncate the state-space to a rea-
sonable finite subset. The choice of this truncation heavily depends on the goal of
the analysis. If one is interested in the most “common” behavior, for example, a
dynamic mass-based truncation scheme is most appropriate [32]. Such a scheme
truncates states with small probability during the numerical integration. How-
ever, common mass-based truncation schemes are not as useful for the bridging
problem. This is because trajectories that meet the specific terminal constraints
can be far off the main bulk of the probability mass. We solve this problem by
a state-space lumping in connection with an iterative refinement scheme.
Consider as an example a birth-death process. This model can be used to
model a wide variety of phenomena and often constitutes a sub-module of larger
models. For example, it can be interpreted as an M/M/1 queue with service rates
being linearly dependent on the queue length. Note, that even for this simple
model, the state-space is countably infinite.
Model 1 (Birth-Death Process). The model consists of exponentially dis-
tributed arrivals and service times proportional to queue length. It can be ex-
pressed using two mass-action reactions:
10
Xand X.1
.
The initial condition X0= 0 holds with probability one.
3.2 Bridging Distribution
The process’ probability distribution given both initial and terminal constraints
is formally described by the conditional probabilities
γ(xi, t) = Pr(Xt=xi|X0=x0, XT=xg),0tT(8)
for fixed initial state x0and terminal state xg. We call these probabilities the
bridging probabilities. It is straight-forward to see that γadmits the factorization
γ(xi, t) = π(xi, t)β(xi, t)(xg, T ) (9)
due to the Markov property. The normalization factor, given by the reachability
probability π(xg, T ) = β(x0,0), ensures that γ(·, t) is a distribution for all time
points t[0, T ]. We call each γ(·, t) a bridging distribution. From the Kolmogorov
equations (5) and (7) we can obtain both the forward probabilities π(·, t) and
the backward probabilities β(·, t) for t<T.
We can easily extend this procedure to deal with hitting times constrained
by a finite time-horizon by making the goal state xgabsorbing.
In Figure 1 we plot the forward, backward, and bridging probabilities for
Model 1. The probabilities are computed on a [0,100] state-space truncation. The
approximate forward solution ˆπshows how the probability mass drifts upwards
towards the stationary distribution Poisson(100). The backward probabilities
6 M. Backenk¨ohler et al.
Fig. 1. Forward, backward, and bridging probabilities for Model 1 with initial con-
straint X0= 0 and terminal constraint X10 = 40 on a truncated state-space. Proba-
bilities over 0.1 in ˆπand ˆ
βare given full intensity for visual clarity. The lightly shaded
area (60) indicates a region being more relevant for the forward than for the bridging
probabilities.
are highest for states below the goal state xg= 40. This is expected because
upwards drift makes reaching xgmore probable for “lower” states. Finally, the
approximate bridging distribution ˆγcan be recognized to be proportional to the
product of forward ˆπand backward probabilities ˆ
β.
4 Bridge Truncation via Lumping Approximations
We first discuss the truncation of countably infinite state-spaces to analyze back-
ward and forward probabilities (Section 4.1). To identify effective truncations we
employ a lumping scheme. In Section 4.2, we explain the construction of macro-
states and assumptions made, as well as the efficient calculation of transition
rates between them. Finally, in Section 4.3 we present an iterative refinement
algorithm yielding a suitable truncation for the bridging problem.
4.1 Finite State Projection
Even in simple models such as a birth-death Process (Model 1), the reachable
state-space is countably infinite. Direct analyzes of backward (6) and forward
equations (4) are often infeasible. Instead, the integration of these differential
equations requires working with a finite subset of the infinite state-space [37]. If
states are truncated, their incoming transitions from states that are not trun-
cated can be re-directed to a sink state. The accumulated probability in this
sink state is then used as an error estimate for the forward integration scheme.
Consequently, many truncation schemes, such as dynamic truncations [4], aim
to minimize the amount of “lost mass” of the forward probability. We use the
same truncation method but base the truncation on bridging probabilities rather
than the forward probabilities.
4.2 State-Space Lumping
When dealing with bridging problems, the most likely trajectories from the initial
to the terminal state are typically not known a priori. Especially if the event in
Analysis of Markov Jump Processes under Terminal Constraints 7
question is rare, obtaining a state-space truncation adapted to its constraints is
difficult. We devise a lumping scheme that groups nearby states, i.e. molecule
counts, into larger macro-states. A macro-state is a collection of states treated
as one state in a lumped model, which can be seen as an abstraction of the
original model. These macro-states form a partitioning of the state-space. In this
lumped model, we assume a uniform distribution over the constituent micro-
states inside each macro-state. Thus, given that the system is in a particular
macro-state, all of its micro-states are equally likely. This partitioning allows us
to analyze significant regions of the state-space efficiently albeit under a rough
approximation of the dynamics. Iterative refinement of the state-space after each
analysis moves the dynamics closer to the original model. In the final step of the
iteration, the considered system states are at the granularity of the original model
such that no approximation error is introduced by assumptions of the lumping
scheme. Computational efficiency is retained by truncating in each iteration
step those states that contribute little probability mass to the (approximated)
bridging distributions.
We choose a lumping scheme based on a grid of hypercube macro-states whose
endpoints belong to a predefined grid. This topology makes the computation
of transition rates between macro-states particularly convenient. Mass-action
reaction rates, for example, can be given in a closed-form due to the Faulhaber
formulae. More complicated rate functions such as Hill functions can often be
handled as well by taking appropriate integrals.
Our choice is a scheme that uses nS-dimensional hypercubes. A macro-state
¯xi(`(i), u(i)) (denoted by ¯xifor notational ease) can therefore be described by
two vectors `(i)and u(i). The vector `(i)gives the corner closest to the origin,
while u(i)gives the corner farthest from the origin. Formally,
¯xi= ¯xi(`(i), u(i)) = {xNnS|`(i)xu(i)},(10)
where ’’ stands for the element-wise comparison. This choice of topology makes
the computation of transition rates between macro-states particularly conve-
nient: Suppose we are interested in the set of micro-states in macro-state ¯xithat
can transition to macro-state ¯xkvia reaction j. It is easy to see that this set is
itself an interval-defined macro-state ¯xij
k. To compute this macro-state we can
simply shift ¯xiby vj, take the intersection with ¯xkand project this set back.
Formally,
¯xij
k= ((¯xi+vj)¯xk)vj,(11)
where the additions are applied element-wise to all states making up the macro-
states. For the correct handling of the truncation it is useful to define a general
exit state
¯xij
= ((¯xi+vj)\¯xi)vj.(12)
This state captures all micro-states inside ¯xithat can leave the state via reaction
j. Note that all operations preserve the structure of a macro-state as defined in
(10). Since a macro-state is based on intervals the computation of the transition
rate is often straight-forward. Under the assumption of polynomial rates, as
8 M. Backenk¨ohler et al.
Fig. 2. A lumping approximation of Model 1 on the state-space truncation to [0,200]
on t[0,50]. On the left-hand side solutions of a regular truncation approximation
and a lumped truncation (macro-state size is 5) are given. On the right-hand side the
respective terminal distributions Pr(X50 =xi) are contrasted.
it is the case for mass-action systems, we can compute the sum of rates over
this transition set efficiently using Faulhaber’s formula. We define the lumped
transition function
¯αj(¯x) = X
x¯x
αj(x) (13)
for macro-state ¯xand reaction j. As an example consider the following mass-
action reaction 2Xc
.For macro-state ¯x={0, . . . , n}we can compute the
corresponding lumped transition rate
¯α(¯x) = c
2
n
X
i=1
i(i1) = c
2
n
X
i=1
(i2i) = c
22n3+ 3n2+n
6n2+n
2
eliminating the explicit summation in the lumped propensity function.
For polynomial propensity functions αsuch formulae are easily obtained au-
tomatically. For non-polynomial propensity functions, we can use the continuous
integral as an approximation. This is demonstrated on a case study in Section 5.2.
Using the transition set computation (11) and the lumped propensity func-
tion (13) we can populate the Q-matrix of the finite lumping approximation:
¯
Q¯xi,¯xk=
PnR
j=1 ¯αj¯xij
k/vol (¯xi),if ¯xi6= ¯xk
PnR
j=1 ¯αj¯xij
/vol (¯xi),otherwise
(14)
In addition to the lumped rate function over the transition state ¯xij
k, we need to
divide by the total volume of the lumped state ¯xi. This is due to the assumption
of a uniform distribution inside the macro-states. Using this Q-matrix, we can
compute the forward and backward solution using the respective Kolmogorov
equations (5) and (7).
Interestingly, the lumped distribution tends to be less concentrated. This is
due to the assumption of a uniform distribution inside macro-states. This effect
Analysis of Markov Jump Processes under Terminal Constraints 9
is illustrated by the example of a birth-death process in Figure 2. Due to this
effect, an iterative refinement typically keeps an over-approximation in terms of
state-space area. This is a desirable feature since relevant regions are less likely
to be pruned due to lumping approximations.
4.3 Iterative Refinement Algorithm
The iterative refinement algorithm (Alg. 1) starts with a set of large macro-states
that are iteratively refined, based on approximate solutions to the bridging prob-
lem. We start by constructing square macro-states of size 2min each dimension
for some mNsuch that they form a large-scale grid S(0). Hence, each initial
macro-state has a volume of (2m)nS. This choice of grid size is convenient be-
cause we can halve states in each dimension. Moreover, this choice ensures that
all states have equal volume and we end up with states of volume 20= 1 which
is equivalent to a truncation of the original non-lumped state-space.
An iteration of the state-space refinement starts by computing both the for-
ward and backward probabilities (lines 2 and 3) via integration of (5) and (7),
respectively, using the lumped ˆ
Q-matrix. Based on the resulting approximate
forward and backward probabilities, we compute an approximation of the bridg-
ing distributions (line 4). This is done for each time-point in an equispaced grid
on [0, T ]. The time grid granularity is a hyper-parameter of the algorithm. If
the grid is too fine, the memory overhead of storing backward ˆ
β(i)and forward
solutions ˆπ(i)increases.2If, on the other hand, the granularity is too low, too
much of the state-space might be truncated. Based on a threshold parameter
δ > 0 states are either removed or split (line 7), depending on the mass assigned
to them by the approximate bridging probabilities ˆγ(i)
t. A state can be split by
the split-function which halves the state in each dimension. Otherwise, it is
removed. Thus, each macro-state is either split into 2nSnew states or removed
entirely. The result forms the next lumped state-space S(i+1). The Q-matrix is
adjusted (line 10) such that transition rates for S(i+1) are calculated accord-
ing to (14). Entries of truncated states are removed from the transition matrix.
Transitions leading to them are re-directed to a sink state (see Section 4.1). Af-
ter miterations (we started with states of side lengths 2m) we have a standard
finite state projection scheme on the original model tailored to computing an
approximation of the bridging distribution.
In Figure 3 we give a demonstration of how Algorithm 1 works to refine the
state-space iteratively. Starting with an initial lumped state-space S(0) covering
a large area of the state-space, repeated evaluations of the bridging distributions
are performed. After five iterations the remaining truncation includes all states
that significantly contribute to the bridging probabilities over the times [0, T ].
It is important to realize that determining the most relevant states is the
main challenge. The above algorithm solves this problem by considering only
2We denote the approximations with a hat (e.g. ˆπ) rather than a bar (e.g. ¯π) to
indicate that not only the lumping approximation but also a truncation is applied
and similarly for the Q-matrix.
10 M. Backenk¨ohler et al.
Algorithm 1: Iterative refinement for the bridging problem
input : Initial partitioning S(0) , truncation threshold δ
output: approximate bridging distribution ˆγ
1for i= 1,...,m do
2ˆπ(i)
tsolve approximate forward equation on S(i);
3ˆ
β(i)
tsolve approximate backward equation on S(i);
4ˆγ(i)
tˆ
β(i)ˆπ(i)/ˆπ(xg, T ); /* approximate bridging distribution */
5S(i+1) ← ∅;
6foreach ¯x∈ S(i)do
7if t.ˆγ(i)
tx)δ;/* refine based on bridging probabilities */
8then
9S(i+1) ← S(i+1) split( ¯x);
10 update ˆ
Q-matrix;
11 return ˆγ(i);
Fig. 3. The state-space refinement algorithm on two parallel unit-rate arrival processes.
The bridging problem from (0,0) to (64,64) and T= 10 and truncation threshold
δ= 5e-3. States with a bridging probability below δare light grey. The macro-state
containing the goal state is marked in black. The initial macro-states are of size 16×16.
those parts of the state-space that contribute most to the bridging probabilities.
The truncation is tailored to this condition and might ignore regions that are
likely in the unconditioned case. For instance, in Fig. 1 the bridging probabili-
ties mostly remain below a population threshold of #X= 60 (as indicated by
the lighter/darker coloring), while the forward probabilities mostly exceed this
bound. Hence, in this example a significant portion of the forward probabilities
ˆπ(i)
tis captured by the sink state. However, the condition in line 7 in Algorithm 1
ensures that states contributing significantly to ˆγ(i)
twill be kept and refined in
the next iteration.
5 Results
We present four examples in this section to evaluate our proposed method.
A prototype was implemented in Python 3.8. For numerical integration we
Analysis of Markov Jump Processes under Terminal Constraints 11
threshold δ1e-2 1e-3 1e-4 1e-5
truncation size 1154 2354 3170 3898
overall states 2074 3546 4586 5450
estimate 8.8851e-30 1.8557e-29 1.8625e-29 1.8625e-29
rel. error 5.2297e-01 3.6667e-03 3.7423e-05 9.5259e-08
Table 1. Estimated reachability probabilities based on varying truncation thresholds
δ: The true probability is 1.8625e-29. We also report the size of the final truncation and
the accumulated size of all truncations during refinement iterations (overall states).
used the Scipy implementation [47] of the implicit method based on backward-
differentiation formulas [13]. The analysis as a Jupyter notebook is made avail-
able online3.
5.1 Bounding Rare Event Probabilities
We consider a simple model of two parallel Poisson processes describing the
production of two types of agents. The corresponding probability distribution
has Poisson product form at all time points t0 and hence we can compare
the accuracy of our numerical results with the exact analytic solution. We use
the proposed approach to compute lower bounds for rare event probabilities. 4
Model 2 (Parallel Poisson Processes). The model consists of two parallel
independent Poisson processes with unit rates.
1
Aand 1
B .
The initial condition X0= (0,0) holds with probability one. After ttime units
each species abundance is Poisson distributed with rate λ=t.
We consider the final constraint of reaching a state where both processes exceed
a threshold of 64 at time 20. Without prior knowledge, a reasonable truncation
would have been 160×160. But our analysis shows that just 20% of the states are
necessary to capture over 99.6% of the probability mass reaching the target event
(cf. Table 1). Decreasing the threshold δleads to a larger set of states retained
after truncation as more of the bridging distribution is included (cf. Figure 4).
We observe an increase in truncation size that is approximately logarithmic in δ,
which, in this example, indicates robustness of the method with respect to the
choice of δ.
3https://www.github.com/mbackenkoehler/mjp bridging
4These bounds are rigorous up to the approximation error of the numerical inte-
gration scheme. However, the forward solution could be replaced by an adaptive
uniformization approach [3] for a more rigorous integration error control.
12 M. Backenk¨ohler et al.
Fig. 4. State-space truncation for varying values of the threshold parameter δ: Two
parallel Poisson processes under terminal constraints X(A)
20 64 and X(B)
20 64. The
initial macro-states are 16 ×16 such that the final states are regular micro states.
Comparison to other methods The truncation approach that we apply is similar
to the one used by Mikeev et al. [34] for rare event estimation. However, they used
a given linearly biased MJP model to obtain a truncation. A general strategy
to compute an appropriate biasing was not proposed. It is possible to adapt
our truncation approach to the dynamic scheme in Ref. [34] where states are
removed in an on-the-fly fashion during numerical integration.
A finite state-space truncation covering the same area as the initial lumping
approximation would contain 25,600 states.5The standard approach would be
to build up the entire state-space for such a model [27]. Even using a conser-
vative truncation threshold δ= 1e-5, our method yields an accurate estimate
using only about a fifth (5450) of this accumulated over all intermediate lumped
approximations.
5.2 Mode Switching
Mode switching occurs in models exhibiting multi-modal behavior [44] when a
trajectory traverses a potential barrier from one mode to another. Often, mode
switching is a rare event and occurs in the context of gene regulatory networks
where a mode is characterized by the set of genes being currently active [30].
Similar dynamics also commonly occur in queuing models where a system may
for example switch its operating behavior stochastically if traffic increases above
or decreases below certain thresholds. Using the presented method, we can get
both a qualitative and quantitative understanding of switching behavior without
resorting to Monte-Carlo methods such as importance sampling.
Exclusive Switch The exclusive switch [7] has three different modes of opera-
tion, depending on the DNA state, i.e. on whether a protein of type one or two
is bound to the DNA.
5Here, the goal is not treated as a single state. Otherwise, it consists of 24,130 states.
Analysis of Markov Jump Processes under Terminal Constraints 13
Model 3 (Exclusive Switch). The exclusive switch model consists of a pro-
moter region that can express both proteins P1and P2. Both can bind to the
region, suppressing the expression of the other protein. For certain parameteri-
zations, this leads to a bi-modal or even tri-modal behavior.
Dρ
D+P1Dρ
D+P2P1
λ
P2
λ
D+P1
β
D.P1D.P 1γ
D+P1D.P1
α
D.P1+P1
D+P2
β
D.P2D.P 2γ
D+P2D.P2
α
D.P2+P2
The parameter values are ρ=1e-1, λ=1e-3, β=1e-2, γ=8e-3, and α=1e-1.
Since we know a priori of the three distinct operating modes, we adjust the
method slightly: The state-space for the DNA states is not lumped. Instead
we “stack” lumped approximations of the P1-P2phase space upon each other.
Special treatment of DNA states is common for such models [28].
To analyze the switching, we choose the transition from (variable order: P1,
P2,D,D.P1,D.P2)x1= (32,0,0,0,1) to x2= (0,32,0,1,0) over the time
interval t[0,10]. The initial lumping scheme covers up to 80 molecules of P1
and P2for each mode. Macro-states have size 8×8 and the truncation threshold
is δ= 1e-4.
In the analysis of biological switches, not only the switching probability but
also the switching dynamics is a central part of understanding the underlying
biological mechanisms. In Figure 5 (left), we therefore plot the time-varying
probabilities of the gene state conditioned on the mode. We observe a rapid un-
binding of P2, followed by a slow increase of the binding probability for P1. These
dynamics are already qualitatively captured by the first lumped approximation
(dashed lines).
Toggle Switch Next, we apply our method to a toggle switch model exhibiting
non-polynomial rate functions. This well-known model considers two proteins A
and Binhibiting the production of the respective other protein [29].
Model 4. Toggle Switch (Hill functions) We have population types Aand B
with the following reactions and reaction rates.
α1(·)
A , where α1(x) = ρ
1 + xB
, A λ
α1(·)
B , where α1(x) = ρ
1 + xA
, B λ
The parameterization is ρ= 10,λ= 0.1.
Due to the non-polynomial rate functions α1and α2, the transition rates between
macro-states are approximated by using the continuous integral
¯α1(¯x)Zb+0.5
a0.5
ρ
1 + xdx =ρ(log (b+ 1.5) log (a+ 0.5))
14 M. Backenk¨ohler et al.
Fig. 5. (left) Mode probabilities of the exclusive switch bridging problem over time for
the first lumped approximation (dashed lines) and the final approximation (solid lines)
with constraints X0= (32,0,0,1,0) and X10 = (0,32,0,0,1). (right) The expected
occupation time (excluding initial and terminal states) for the switching problem of
the toggle switch using Hill-type functions. The bridging problem is from initial (0,120)
to a first passage of (120,0) in t[0,10].
for a macro-state ¯x={a, . . . , b}.
We analyze the switching scenario from (0,120) to the first visit of state
(120,0) up to time T= 10. The initial lumping scheme covers up to 352 molecules
of Aand Band macro-states have size 32 ×32. The truncation threshold is
δ= 1e-4. The resulting truncation is shown in Figure 5 (right). It also illustrates
the kind of insights that can be obtained from the bridging distributions. For
an overview of the switching dynamics, we look at the expected occupation
time under the terminal constraint of having entered state (120,0). Letting the
corresponding hitting time be τ= inf {t0|Xt= (120,0)}, the expected
occupation time for some state xis ERτ
01=x(Xt)dt |τ10. We observe that
in this example the switching behavior seems to be asymmetrical. The main mass
seems to pass through an area where initially a small number of Amolecules is
produced followed by a total decay of Bmolecules.
5.3 Recursive Bayesian Estimation
We now turn to the method’s application in recursive Bayesian estimation. This
is the problem of estimating the system’s past, present, and future behavior un-
der given observations. Thus, the MJP becomes a hidden Markov model (HMM).
The observations in such models are usually noisy, meaning that we cannot infer
the system state with certainty.
This estimation problem entails more general distributional constraints on
terminal β(·, T ) and initial π(·,0) distributions than the point mass distributions
considered up until now. We can easily extend the forward and backward proba-
bilities to more general initial distributions and terminal distributions β(T). For
Analysis of Markov Jump Processes under Terminal Constraints 15
the forward probabilities we get
π(xi, t) = X
j
Pr(Xt=xi|X0=xj)π(xj,0),(15)
and similarly the backward probabilities are given by
β(xi, t) = X
j
Pr(XT=xj|Xt=xi)βT(xj).(16)
We apply our method to an SEIR (susceptible-exposed-infected-removed) model.
This is widely used to describe the spreading of an epidemic such as the current
COVID-19 outbreak [23,20]. Temporal snapshots of the epidemic spread are
mostly only available for a subset of the population and suffer from inaccuracies
of diagnostic tests. Bayesian estimation can then be used to infer the spreading
dynamics given uncertain temporal snapshots.
Model 5 (Epidemics Model). A population of susceptible individuals can
contract a disease from infected agents. In this case, they are exposed, mean-
ing they will become infected but cannot yet infect others. After being infected,
individuals change to the removed state. The mass-action reactions are as fol-
lows.
S+Iλ
E+I E µ
I I ρ
R
The parameter values are λ= 0.5,µ= 3,ρ= 3. Due to the stoichiometric
invariant X(S)
t+X(E)
t+X(I)
t+X(R)
t= const., we can eliminate Rfrom the
system.
We consider the following scenario: We know that initially (t= 0) one in-
dividual is infected and the rest is susceptible. At time t= 0.3 all individuals
are tested for the disease. The test, however, only identifies infected individuals
with probability 0.99. Moreover, the probability of a false positive is 0.05. We
like to identify the distribution given both the initial state and the measurement
at time t= 0.3. In particular, we want to infer the distribution over the latent
counts of Sand Eby recursive Bayesian estimation.
The posterior for nIinfected individuals at time t, given measurement Yt=
ˆnIcan be computed using Bayes’ rule
Pr(X(I)
t=nI|Yt= ˆnI)Pr(Yt= ˆnI|X(I)
t=nI) Pr(X(I)
t=nI).(17)
This problem is an extension of the bridging problem discussed up until now.
The difference is that the terminal posterior is estimated it using the result of the
lumped forward equation and the measurement distribution using (17). Based
on this estimated terminal posterior, we compute the bridging probabilities and
refine the truncation tailored to the location of the posterior distribution. In Fig-
ure 6 (left), we illustrate the bridging distribution between the terminal posterior
and initial distribution. In the context of filtering problems this is commonly re-
ferred to as smoothing. Using the learned truncation, we can obtain the posterior
distribution for the number of infected individuals at t= 0.3 (Figure 6 (middle)).
Moreover, can we infer a distribution over the unknown number of susceptible
and exposed individuals (Figure 6 (right)).
16 M. Backenk¨ohler et al.
Fig. 6. (left) A comparison of the prior dynamics and the posterior smoothing (bridg-
ing) dynamics. (middle) The prior, likelihood, and posterior of the number of infected
individuals nIat time t= 0.3 given the measurement ˆnI= 30. (right) The prior and
posterior distribution over the latent types Eand S.
6 Conclusion
The analysis of Markov Jump processes with constraints on the initial and ter-
minal behavior is an important part of many probabilistic inference tasks such
as parameter estimation using Bayesian or maximum likelihood estimation, in-
ference of latent system behavior, the estimation of rare event probabilities, and
reachability analysis for the verification of temporal properties. If endpoint con-
straints correspond to atypical system behaviors, standard analysis methods fail
as they have no strategy to identify those parts of the state-space relevant for
meeting the terminal constraint.
Here, we proposed a method that is not based on stochastic sampling and
statistical estimation but provides a direct numerical approach. It starts with an
abstract lumped model, which is iteratively refined such that only those parts of
the model are considered that contribute to the probabilities of interest. In the
final step of the iteration, we operate at the granularity of the original model
and compute lower bounds for these bridging probabilities that are rigorous up
to the error of the numerical integration scheme.
Our method exploits the population structure of the model, which is present
in many important application fields of MJPs. Based on experience with other
work based on truncation, the approach can be expected to scale up to at least
a few million states [33]. Compared to previous work, our method neither relies
on approximations of unknown accuracy nor additional information such as a
suitable change of measure in the case of importance sampling. It only requires
a truncation threshold and an initial choice for the macro-state sizes.
In future work, we plan to extend our method to hybrid approaches, in which
a moment representation is employed for large populations while discrete counts
are maintained for small populations. Moreover, we will apply our method to
model checking where constraints are described by some temporal logic [21].
Analysis of Markov Jump Processes under Terminal Constraints 17
Acknowledgements This project was supported by the DFG project MULTI-
MODE and Italian PRIN project SEDUCE.
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