STOCHASTIC AND STATE SPACE MODELS OF CARCINOGENESIS: SOME NEW MODELING APPROACHES - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 79

Also, since k1 (i) is very small, it can be shown 63 that with k1 (i) =
10 6 k1 (i),

t j1 1

X

E[P T (i; j)] = expf 1
10 6

EI k1 (t; i) k1 (i)

t=0

+ E log(1e 1

10 6 R(i;j) k1 (i) )g
B i (j) k1 (i)expfA i (j) k1 (i)g;

P
t j 1
t=0

1

EI k1 (t; i) 2 ER(i; j)g and

where A i (j) =

10 6 f

B i (j) =

1

10 6 ER(i; j):
From the above distribution results, it is obvious that the joint density
offX; U ; Ygis

Y

m

Y

n

PfX; U ; Yg= Pfy ij jn i (j); I k1 (i; j)g

i=1 j=1

t j

Y

Pf X (t)j X (t1); U (t1)g

t=t j1 +1

Pf U (t1)j X (t1)g: (9)

Notice that in the above equation, the birth rates, death rates and
mutation ratesf 0 (t); b r (t); d r (t); r (t); r = 1; : : : ; k1gare functions of
the dose level s i .
The above distribution will be used to derive the conditional posterior
distribution of the unknown parameters givenfX; U ; Yg. Notice that
because the number of parameters is very large, the classical sampling the-
ory approach by using the likelihood function PfYjX; Ugis not possible
without making assumptions about the parameters; however, this problem
can easily be avoided by new information from the stochastic system model
and the prior distribution of the parameters.

5.3. The Posterior Distribution of the Unknown
Parameters and State Variables

In many practical situations, one may assume that the birth rates, death
rates and mutation rates are time homogeneous. For the i-th dose level,
denote these rates byf i ; ji ; b ji ; d ji ; j = 1; : : : ; k1g. Then the set of un-
known parameters are =f i ; ji ; b ji ; d ji ; j = 1; : : : ; k1; i = 1; : : : ; mg.
To derive the posterior distribution of givenfX; U ; Yg, let Pfgbe the
prior distribution of and for the ith dose level, denote thef X (t); U (t)g
by X

(i) (t) =fI (i j (t); j = 1; : : : ; k1gand U (i) (t) =fB (i j (t); D (i j (t); j =