1
$\begingroup$

Dennis Lindley's paper The Philosophy of Statistics in 2001 includes the following 'simple' example of statistical coherence: "A set of uncertainty statements is said to be coherent if they satisfy the rules of the probability calculus. Thus, the pair of statements p(A|B) = 0.7 and p(A| ~B) = 0.4 do not cohere with the pair p(B|A)= 0.5 and p(B|~ A) = 0.3. (Here ~B denotes the complement of B.) Think of A as a statement about data x and B as a statement about parameter theta. The first pair refers to uncertainties in the data and coheres with the first parameter statement, p(B|A)= 0.5, for data A. (Take p(B) = 0.4/1.1 = 0.36.) But all three do not cohere with the second parameter statement for data A, that p(B| A) = 0.3. With p(B) = 0.36, the coherent value is 0.22."

How is the coherent value of 0.22 calculated?

$\endgroup$

1 Answer 1

1
$\begingroup$

$$p(B|A)=\frac{p(A|B)p(B)}{p(A|B)p(B))+p(A|\neg B)p(\neg B)} =\frac{p(A|B)}{p(A|B)+p(A|\neg B)p(\neg B)/p(B)}$$ leads to $$0.5=\frac{0.7}{0.7+0.4p(\neg B)/p(B)}$$ hence to $p(B)=0.4/1.1=0.36$. This means that $$P(B|\neg A)=\frac{p(\neg A|B)p(B)}{p(\neg A|B)p(B)+p(\neg A|\neg B)p(\neg B)}$$ is equal to $$\frac{0.3\cdot 0.36}{0.3\cdot 0.36+0.6\cdot 0.64}\approx 0.22$$ And using instead $$p(B|\neg A)=\frac{p(\neg A|B)p(B)}{p(\neg A|B)p(B))+p(\neg A|\neg B)p(\neg B)} =\frac{p(\neg A|B)}{p(A|B)+p(\neg A|\sim B)p(\neg B)/p(B)}$$ leads to $$0.3=\frac{0.3}{0.3+0.6p(\neg B)/p(B)}$$ hence to $p(B)=0.6/1.3\approx 0.46$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.