probability space

arbitrary measurable space (for each *t*∈[*t*
_{0},*∞*) with
)

measurable random variable

continuous-time Markov process with general state space

state space of

the pushforward measure of P under *X*
_{
t
}, i.e.
for all

the pushforward measure of P under
(with the corresponding product algebra)

the state space restricted to the interval [*t*
_{0},*t*]

the corresponding pushforward measure

*N* number of particles

state samples at time *t*

*K*
_{
s,t
}(*x*
_{
s
}, d*x*
_{
t
}) the Markov kernel of the process
from time *s* to time *t*

*a*(*x*,*t*) drift vector

*B*(*x*,*t*) diffusion matrix

multidimensional standard Wiener process

measurable space

*Y*
_{1:M
} observation random variable with values in measurable spaces

conditional probability density with respect to a reference measure

reference measure
on

*t*
_{
j
} (*j*=1,…,*M*) observation times

*T*
_{
j
} random variables modelling the uncertainty about exact observation times

Lebesgue measure on the interval [*t*
_{0},*∞*)

*γ*
_{
j
}(*t*
_{
j
}) probability density of *T*
_{
j
} with respect to

,
, and Markov chain, pushforward

measure, and kernel for importance sampling

Radon-Nikodym derivative

Radon-Nikodym derivative at start time *t*
_{0}

unnormalized weight of particle *i* at time *t*

normalized weight of particle *i* at time *t*

*s*
_{
ℓ
}
*ℓ*-th resampling time

(unnormalized) selection weight

normalized selection weight (probability that particle *i* will be selected during resampling)

*ι*
_{
ℓ
}:*I*→*I* selection function on the index set *I*: ={1,…,*N*} for the *ℓ*-th resampling step

data likelihood

*f*
_{
t
} full density at time *t*

filter density at time *t*, i.e. only those observations *y*
_{
j
} are included for which *t*
_{
j
}≤*t*

conditional probability density
if *T*
_{
j
}≤*t*, 1 otherwise

stochastic process given by d*W*
_{
j,t
}(*ω*)=(*g*
_{
j
}(*y*
_{
j
} | *X*
_{
t
}(*ω*),*t*) -1)*γ*
_{
j
}(*t*) d*t*

product process of the *W*
_{
j,t
}, where *j*=1,…,*M*

expectation of *h*(*X*
_{
t
}) given *Y*
_{1:M
}=*y*
_{1:M
} with respect to the filtered state *X*
_{
t
}

partial weight at time *t* (w.r.t. the *j*-th observation)

weight at time *t*

cumulative distribution function

*t*
_{0}=*τ*
_{0}<*τ*
_{1}…<*τ*
_{
D
} time discretization

cumulative product of the selection weights for particle *i*

*Δ*
*τ*
_{
d
}=*τ*
_{
d
}-*τ*
_{
d-1} stepsize

correction factor for the data likelihood at time *τ*
_{
d
}

*Q*
_{
i
} mass of the tracee in compartment *i*

*q*
_{
i
} mass of the tracer in compartment *i*

*U*
_{
i
} input for the tracee in compartment *i* (e.g. *U*
_{1} denotes the influx into compartment 1)

*u*
_{
i
} input for the tracer in compartment *i*

*k*
_{
j,i
} transfer coefficient of the tracers from compartment *i* to compartment *j*

*σ*
_{
i
} diffusion parameter

*ξ*
_{
t
} log-normal noise (
)

,
degradation rate of plasma leucine for people in the diabetes group and the control group, respectively

*ζ*
_{
p
}, *η*
_{
p
} patient-dependent random factors modelling the parametric uncertainties among individuals (*ζ*
_{
p
}= exp(*η*
_{
p
}) with
)