General state transition network for a 3 states system. |

Here you see we have the option of going from any state to any state with a probability. To not get confused we show the probabilities with 'a' instead of 'p' so for any node of the graph the sum of the outgoing arrows should be 1, like the following:

*a11 + a21 + a31 = 1*

*a12 + a22 + a32 = 1*

*a13 + a23 + a33 = 1*

As we talked about the superposition of states in previous considering the following is our transition matrix:

Transition matrix for a 3 state system |

And the following will give us the next state of the system:

New state of the system can be derived by multiplying transition matrix to the current state of the system |

Here we remind you again that state of our system at any time is a superposition of possible states, this is why the S(t) itself is a vector of possible states in which any possible state is a vector or matrix itself. It is like in t+1 your system can be in any of these 3 states each with a probability. So the S(t+1) will be something like the following:

New state of a system based on the previous state |

And again like the previous post we always can go forward and calculate the future states:

*S(t)*

*S(t+1) = T*x

*S(t)*

*S(t+2) = T*x

*S(t+1) =*

*T*x

*(*

*T*x

*S(t)*

*)*

*.*

*.*

*.*

*S(t+n) = T*x

*S(t+n-1) =*

*T*x

*( T*x (

*T*x ...

*(*

*T*x

*S(t)*

*)*)

*)*

What basically this says is that you can predict the state probabilities of the system in future, regardless of how far the selected future is and how complicate your system is. The more complicated the system the more number of states it has, so the transition matrix grows, and this is why predicting the future of a system like a single human life or oil price is so much difficult. Because first of all we don't have access to all elements and their values which defines every single state of a human life or oil price or may have some effect on the system and second of all the transition matrix will have millions if not billions of dimensions which practically makes it impossible to get transition calculated with the technologies we have today.

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