The Hypotheses Reevaluation Math

 

From the logic and math point of view, there are two alternatives and probabilities to consider P(H), P(~H),  P(H)+P(~H)=1, namely: whether macroscopic life on Mars is present (H), or it is not (~H).

 

According to the Bayes’ theorem, every relevant observation A changes the probabilities ratio R=P(H)/P(~H)  of these two hypotheses:

              

                                     R|A = R*K(A), where K(A)=P(A|H)/P(A|~H)

 

For any two independent observations A, B, holds P(AB|H)=P(A|H)*P(B|H), so we have

 

                                    K(AB)=K(A)*K(B)

 

Consequently the probabilities ratio after taking into account observations - is a product of initial a priory probabilities ratio estimate, and of the correction coefficients for all observations.

 

If we have enough observations, the initial bias of the hypothesis probability estimate does not really matter /if it is not zero/, since the corrections will eventually overcome it.

 

We will need more general rule based on the Bayes theorem:

 

with   R(H)=P(H)/P(H0),  R(H|A)=P(H|A)/P(H0|A),   K(H|A)=P(A|H)/P(A|H0):

 

R(H|A)=R(H)* K(H|A),   K(H|AB)=K(H|A)*K(H|B) 

 

I(H|A)= -log(K(H|A)),  I(H|AB)=I(H|A)+I(H|B)