where I[...] is an indicator function. In terms of interpretation: this is the frequency at which I will see B take on value b, if I sample A from the distribution P[A] and then compute B via B = f(A).
What do you want to do which is not that, and why do you want to do it?
Most of the time, the data you gather about the world is that you have a bunch of facts about the world and probabilities about the individual data points and you would want as an outcome also probabilities over individual datapoints.
As far as my own background goes, I have not studied logic or the math behind the AI algorithm that David Chapman wrote. I did study bioinformatics in that that study we did talk about probabilities calculations that are done in bioinformatics, so I have some intuitions from that domain, so I take a bioinformatics example even if I don’t know exactly how to productively apply predicate calculus to the example.
If you for example get input data from gene sequencing and billions of probabilities (a_1, a_2, …, a_n) and want output data about whether or not individual genetic mutations exist (b_1, b_2, …, b_m) and not just P(B) = P(b_1) * P(b_2) * … * P(b_m).
If you have m = 100,000 in the case of possible genetic mutations, P(B) is a very small number with little robustness to error. A single bad b_x will propagate to make your total P(B) unreliable. You might have an application where getting a b_234, b_9538 and b _33889 wrong is an acceptable error because most of the values where good.
The standard push forward here would be:
P[B=b]=∑aI[f(a)=b]P[A=a]
where I[...] is an indicator function. In terms of interpretation: this is the frequency at which I will see B take on value b, if I sample A from the distribution P[A] and then compute B via B = f(A).
What do you want to do which is not that, and why do you want to do it?
Most of the time, the data you gather about the world is that you have a bunch of facts about the world and probabilities about the individual data points and you would want as an outcome also probabilities over individual datapoints.
As far as my own background goes, I have not studied logic or the math behind the AI algorithm that David Chapman wrote. I did study bioinformatics in that that study we did talk about probabilities calculations that are done in bioinformatics, so I have some intuitions from that domain, so I take a bioinformatics example even if I don’t know exactly how to productively apply predicate calculus to the example.
If you for example get input data from gene sequencing and billions of probabilities (a_1, a_2, …, a_n) and want output data about whether or not individual genetic mutations exist (b_1, b_2, …, b_m) and not just P(B) = P(b_1) * P(b_2) * … * P(b_m).
If you have m = 100,000 in the case of possible genetic mutations, P(B) is a very small number with little robustness to error. A single bad b_x will propagate to make your total P(B) unreliable. You might have an application where getting a b_234, b_9538 and b _33889 wrong is an acceptable error because most of the values where good.