Probability

# A case study in non-centering for data augmentation: by Neal P.

By Neal P.

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Extra info for A case study in non-centering for data augmentation: Stochastic epidemics

Example text

12), we often use their combination this criterion depends both on the transition process and on the terminal state of the system. 14) provides the best (in contrast with any other system) result only in the worst operating mode. 14) were studied in [16, 40, 92, 1481. Synthesis Problems for Control Systems 15 If a dynamicsystem described by Eqs. 5) is totally on troll able,^ then optimal control problems with fixed endpoints or with some fixed terminal set are often considered together with control problems with variable right endpoint of the trajectory.

2. Strictly discontinuous Markov processes. Suppose that the state x of the process [ ( t ) varies by jumps a t random instants of time. By analogy with the case of discrete processes, we assume that the moments of jumps form an ordinary sequence of events; then we denote the intensity of jumps by X(t, x ) provided that [ ( t ) = x . A jump a t time t transfers the process [ ( t ) to a random state y with probability density ~ ( xy,,t ) = p ( [ ( t 0 ) = y 1 [ ( t ) = x ) . , t ;y 1 r ) = 11 - (7 - t ) X ( x ,t ) I d ( y - 2 ) + (r - t ) X ( x ,t ) r ( x ,y, t ) + +o(r - t ) .

N, are densities relative to thee conditional distributions of the process £(ti) provided that the instant valuess £(ti) = x - L , £ ( t 2 ) = x2, . . , £(^-i) = z,--i are chosen. 37) in the case of continuous Markov processes. 52) that to write any multidimensional density function, one needs to know the unconditional density p(x, t ) and the conditional density p(y, r 1 x , t ) for any t and r > t. The function p(y, r 1 x,t ) , just as Pap(t, r) in the discrete case, is called the transition probability.