### Bayes probability model

Bayes probability model is an important probability model to predict prior stage probabilities of a multi-stage probability model. Typically it's applied to a 2 stage probability model where the 2nd stage is dependent on the events in the first stage.

Unlike 2 part probability model where events in either part are independent of events in the other part, 2 stage probability model is more complicated because 2nd stage probability is dependent on the 1st stage probability, where conditional probability applies. Bayes probability model quantifies the relationship between probabilities of different events happening in the model. It can be used to compute the 1st stage probability given the conditional probability of 2nd stage events.

A simple dependent probability for event E and F is: P(E and F) = P(E) P(F|E) (the multiplication law), it says the probability that both E and F happen is the product of E happens and the conditional probability of F happens given that E happens. It often is written as P(F|E) = P(E and F)/P(E).

The symmetry in the formulation can be explored: P(E and F) = P(F) P(E|F) = P(E) P(F|E). Thus,

given P(E) = P(F) P(E|F)/P(F|E). Given conditional probabilities of P(E|F), P(F|E), and P(F), one can compute P(E).

Let E and F represent simple event Ei and Fji (Fj happens given that Ei happens) in event sets, P(Ei and Fj) = pE_i * pF_j_i where P(Ei) = pE_i and P(Fj|Ei) = pF_j_i, therefore the chance Fj happens in a 2 stage experiment is sigma(P(Ei and Fj), i = 0..I) = sigma(pE_i * pF_j_i, i = 0..I)

Thus we can rewrote the symmetry formula as P(Fj) P(Eij|Fj) = P(Ei) P(Fji|Ei). This illustrates that given 2nd stage probability P(Fj), we can reliably compute first stage conditional probability P(Eij|Fj) by rewrite it again as:

P(Eij|Fj) = P(Ei) P(Fji|Ei)/P(Fj) = P(Ei) P(Fji|Ei) / [sigma(P(Ei) * P(Fji), i = 0..I)]

Unlike 2 part probability model where events in either part are independent of events in the other part, 2 stage probability model is more complicated because 2nd stage probability is dependent on the 1st stage probability, where conditional probability applies. Bayes probability model quantifies the relationship between probabilities of different events happening in the model. It can be used to compute the 1st stage probability given the conditional probability of 2nd stage events.

A simple dependent probability for event E and F is: P(E and F) = P(E) P(F|E) (the multiplication law), it says the probability that both E and F happen is the product of E happens and the conditional probability of F happens given that E happens. It often is written as P(F|E) = P(E and F)/P(E).

The symmetry in the formulation can be explored: P(E and F) = P(F) P(E|F) = P(E) P(F|E). Thus,

given P(E) = P(F) P(E|F)/P(F|E). Given conditional probabilities of P(E|F), P(F|E), and P(F), one can compute P(E).

Let E and F represent simple event Ei and Fji (Fj happens given that Ei happens) in event sets, P(Ei and Fj) = pE_i * pF_j_i where P(Ei) = pE_i and P(Fj|Ei) = pF_j_i, therefore the chance Fj happens in a 2 stage experiment is sigma(P(Ei and Fj), i = 0..I) = sigma(pE_i * pF_j_i, i = 0..I)

Thus we can rewrote the symmetry formula as P(Fj) P(Eij|Fj) = P(Ei) P(Fji|Ei). This illustrates that given 2nd stage probability P(Fj), we can reliably compute first stage conditional probability P(Eij|Fj) by rewrite it again as:

P(Eij|Fj) = P(Ei) P(Fji|Ei)/P(Fj) = P(Ei) P(Fji|Ei) / [sigma(P(Ei) * P(Fji), i = 0..I)]

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