debate 1:
on the value of bayesian analysis
Inquirer doesn't understand Bayesian analysis :o
Ad hom, Inq wins!
As a side note, I just bought a book on Bayes analysis and have been going through it.
As a side note, I just bought a book on Bayes analysis and have been going through it.
Oh really? I'm excited for you. Tell me how you like it :D
Is it Sivia's book? That one is fantastic.
As a side note, I just bought a book on Bayes analysis and have been going through it.Oh really? I'm excited for you. Tell me how you like it :D
Is it Sivia's book? That one is fantastic.
No, it's a book by a Statistician from University of Vermont, it's way more beginners than I would have liked but I should have it done by the end of the week. I'll pick up a more advanced text after that.
Any other recommendations other than Sivia, if not I guess I'll just go with that.
As a side note, I just bought a book on Bayes analysis and have been going through it.Oh really? I'm excited for you. Tell me how you like it :D
Is it Sivia's book? That one is fantastic.
No, it's a book by a Statistician from University of Vermont, it's way more beginners than I would have liked but I should have it done by the end of the week. I'll pick up a more advanced text after that.
Any other recommendations other than Sivia, if not I guess I'll just go with that.
Nah, the way I see Bayesian analysis is that you master the fundamentals and then you just apply it to different scenarios. The trick is to be smart about your assumptions.
Sivia is just the best book for mastering the fundamentals.
He's an Indian professor, from what what I understand. He was on trial for murdering his colleagues, but it turned out his colleague had attacked him and so he was found not guilty.
The book is written by him and Skilling. But IMHO, all the good parts are Sivia.
last edit on
11/18/2020 11:37:11 PM
I think I understand and I can see the utility of this form of analysis.
let there be events A and B and conditional probabilities P(A|B) and P(B|A).
P(A|B) = P(A∩B)/P(B) → P(A∩B) = P(A|B)*P(B)
P(B|A) = P(A∩B)/P(A) → P(A∩B) = P(B|A)*P(A)
Hence, P(A|B)*P(B) = P(B|A)*P(A)
→ P(A|B) = (P(B|A)*P(A))/P(B) and P(B|A) = (P(A|B)*P(B))/P(A)
Bayes theorem is then a statement of the relationship between two inverse conditional probabilities. The relationship can then be explored my assumption is that this is what Bayesian Analysis is used for, exploring this relationship and using it to make inferences.
That analysis would begin with replacing the marginal probabilities P(A) and P(B) with their joint equivalents.
That is,
P(A) = P(A∩B) + P(A∩~B) and P(B) = P(A∩B) + P(~A∩B)
So,
P(A|B) = (P(B|A)*P(A))/(P(A∩B) + P(A∩~B))
P(B|A) = (P(A|B)*P(B))/(P(A∩B) + P(~A∩B))
Now we sub the joint probabilities with there conditional probabilities,
P(A∩~B) = P(A|~B) * P(~B)
P(~A∩B) = P(~A|B) * P(~A)
So,
P(A|B) = (P(B|A)*P(A))/(P(B|A)*P(A) + P(B|~A)*P(~A))
P(B|A) = (P(A|B)*P(B))/(P(A|B)*P(B) + P(A|~B)*P(~B))
The expressions convey that P(A|B) and P(B|A) are therefore ratios between the product of some conditional and prior probability divided by the sum of that product and the sum of some of its conditional in relation to its negation.
In statistical terms, specifically the use case of analyzing Hypothesis as you've stated, where A and ~A are hypothesis and B is a observation,
P(B|A) is the probability of observation B given hypothesis A
P(A) is the probability of hypothesis A before observation B
P(B|~A) is the probability of observation B given hypothesis ~A
P(~A) is the probability of hypothesis ~A before observation B
Finally,
P(A|B) is the probability of A given observation B
As such we can compare hypothesis A and ~A given some set of observations B (We can account for a set of observations by summating the denominator) and verify the validity of a given hypothesis through the ratio of these hypothesis when observations B are given.
Is this correct and do you have anything to add?
last edit on
11/20/2020 3:16:40 AM