Humanities Data Analysis “125 85018 Karsdrop Humanities ch01 3p” — 2020/8/19 — 11 04 — page 227 — #27 Introduction to Probability • 227 evidence in favor of Madison being the author or as evidence in[.]
“125-85018_Karsdrop_Humanities_ch01_3p” — 2020/8/19 — 11:04 — page 227 — #27 Introduction to Probability evidence in favor of Madison being the author or as evidence in favor of Hamilton being the author? 6.5 Appendix 6.5.1 Bayes’s rule In order to derive Bayes’s rule we first begin with the third axiom of probability: Pr(A and B|C) = Pr(B|C)Pr(A|BC) If we let C be an event which encompasses all possible events (A and C = A) we have a simpler statement about a conditional probability: Pr(A|B) = Pr(A and B) Pr(B) Replacing A and B with Hj and E, respectively, we arrive at an initial form of Bayes’s rule: Pr(E and Hj ) Pr(E) Pr(Hj |E)Pr(E) = Pr(E) Pr(Hj |E) = The denominator, Pr(E) may be unpacked by using the rule of marginal probability:11 Pr(E) = K Pr(E and Hk ) k=1 = K Pr(E|Hk )Pr(Hk ) k=1 Replacing Pr(E) in the initial statement we have the final, familiar form of Bayes’s rule: Pr(E|Hj )Pr(Hj ) Pr(Hj |E) = K k=1 Pr(E|Hk )Pr(Hk ) 11 The rule of marginal probability for discrete random variables is the following: Pr(X = x) = y∈Y Pr(X = x, Y = y) = y∈Y Pr(X = x|Y = y)Pr(Y = y), where Y is the set of values which Y may take In prose, the rule for marginal probability tells us how to calculate the marginal distribution, Pr(X = x), if we know the joint distribution Pr(X = x, Y = y) (Appealing to the third axiom of probability, we can observe that the joint distribution of X and Y is equal to the product of the conditional distribution of X given Y times the marginal distribution of Y, Pr(X = x|Y = y)Pr(Y = y).) • 227