Transition probability function

Figure 10.11 shows a smooth sigmoidal threshold function that is often used in practice. It has the same form as the transition probability function used for stochastic nets ... 

TRANSITION DIPOLE MOMENT TRANSITION POLARIZATION TRANSITION PROBABILITY FUNCTIONS TRANSITION STATE... 

If in a simple Pgl system A -B more than one potential curves V +,(R) lie energetically below the potential V (R)—corresponding to the condition that the excitation energy of A is larger than the ionization energies Et(B) of the target—then the total transition probability function T(R)/h branches into the individual transition probabilities Ti(R)/ti. The electronic branching ratios, defined by... 

DeHnition. The one-step transition probability function pjk for a Markov chain is a function that gives the probability of going from state j to state k in one step (one time interval) for each j and k. It will be denoted by ... 

In general, the one-step transition probability function is given by ... 

In other words, pjk(n), the n-step transition probability function, is the conditional probability of occupying Sk at the nth step, given that the system initially occupied Sj. pjk(n), termed also higher transition probability, extends the one-step transition probability pjk(l) = Pjk and gives an answer to question 2 in 2.1-2. Note also that the function given by Eq.(2-26) is independent of t, since we are concerned in homogeneous transition probabilities. 

P(y, T, X, t) transition probability function defined by Eq.(2-185). p, q constant one-step transition probabilities,... 

In a hybrid method, molecules are displaced in time according to conventional molecular dynamics (MD) algorithms, specifically, by integrating Newton s equations of motion for the system of interest. Once the initial coordinates and momenta of the particles are specified, motion is deterministic (i.e., one can determine with machine precision where the system will be in the near future). In the context of Eq. (2.1), the probability of proposing a transition from a state 0 to a state 1 is determined by the probability with which the initial velocities of the particles are assigned from that point on, motion is deterministic (it occurs with probability one). If initial velocities are sampled at random from a Maxwellian distribution at the temperature of interest, then the transition probability function required by Eq. 

Using assumption (4) and adopting the condition that the process is homogeneous in time (i.e transition probability function depends only on t - s and not specifically on / and s) the Kolmogorov equations (5.9) and (5.10) can be reduced to (linear) ordinary differential-difference equation ( differential in time, and difference in state and in accordance with (5.27) we get, for a deterministic initial condition... 

Sometimes the state space of a Markov chain consists of all possible values in an interval. In that case, we would say the Markov chain has a one-dimensional continuous state space. In other Markov chains, the state space consists of all possible values in a rectangular region of dimension p and we would say the Markov chain has a p-dimensional continuous state space. In both of these cases, there are an uncountably infinite number of possible values in the state space. This is far too many to have a transition probability function associated with each pair of values. The probability of a transition between all but a countable number of pairs of possible values must be zero. Otherwise the sum of the transition probabilities would be infinite. A state to state transition probability function won t work for all pairs of states. Instead we define the transition probabilities from each possible state X to each possible measurable set of states A We call... 

More generally, the transition CDF may be a combination of an absolutely continuous part and a discrete part. Then we would have the transition density given in Equation S.16 for the absolute continuous part and a transition probability function for the discrete part. The number of points in the discrete part can be either finite, or countably infinite, otherwise the sum of the discrete probabilities would be infinite. Random variables having this type distributions are discussed in Mood et al. (1974). In practice, we will be dealing with either absolutely continuous transition CDF, or a mixed absolutely continuous and discrete transition CDF, where the only discrete part is pi,i, the probability of remaining at the same state, which may be needed for the Metropolis algorithm. 

For Markov chains with a continuous state space, there are too many states for us to use a transition probability function. Instead we define a transition kernel which measures the probability of going from each individual state to every measurable set of states. 

In order to avoid these shortcomings, we propose using the heuristic distance information probability function, introduced in (Fan et al., 2003), to calculate the probabilities of transition instead of the using the traditional transition probability function. 

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