advanced-algorithmic-trading

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P (B)P (A|B) = P (A ∩ B) (2.2)
But, we can simply make the same statement about P (B|A), which is akin to asking "What
is the probability of seeing clouds, given that it is raining?":
P (B|A) =
P (B ∩ A)
P (A)
(2.3)
Note that P (A ∩ B) = P (B ∩ A) and so by substituting the above and multiplying by P (A),
we get:
P (A)P (B|A) = P (A ∩ B) (2.4)
We are now able to set the two expressions for P (A ∩ B) equal to each other:
P (B)P (A|B) = P (A)P (B|A) (2.5)
If we now divide both sides by P (B) we arrive at the celebrated Bayes’ rule:
P (A|B) =
P (B|A)P (A)
P (B)
(2.6)
However, it will be helpful for later usage of Bayes’ rule to modify the denominator, P (B)
on the right hand side of the above relation to be written in terms of P (B|A). We can actually
write:
P (B) = ∑ a∈A
P (B ∩ A) (2.7)
This is possible because the events A are an exhaustive partition of the sample space.
So that by substituting the defintion of conditional probability we get:
P (B) = ∑ a∈A
P (B ∩ A) = ∑ a∈A
P (B|A)P (A) (2.8)
Finally, we can substitute this into Bayes’ rule from above to obtain an alternative version of
Bayes’ rule, which is used heavily in Bayesian inference:
P (A|B) =
P (B|A)P (A)
∑a∈A P (B|A)P (A) (2.9)
Now that we have derived Bayes’ rule we are able to apply it to statistical inference.