Krdytkyu

I'm a mathematics/probability novice, but interested in the use and abuse of these things in other fields (historical, legal, etc.). Sorry about the lack of technicality here.

Here's a stupid hypothetical to explain the question.

You are friends with Vicky. Vicky is married to David. You wonder whether David has had an affair, but Vicky has not said anything to you.

Part 1: Probability of Vicky remaining silent

Vicky has not said anything. If Vicky knows of an affair, there is a 60% chance she would remain silent. If Vicky is ignorant of any affair, there is (obviously) a 100% chance she would remain silent.

P(silent|know) = 0.6. 

P(silent|ignorant) = 1.

In either case ("know" or "ignorant"), it is more probable that Vicky would remain silent.

Is it valid to compare these two results to consider the relative probability of "know" and "silent"?

I.e.:

P(silent|know) < P(silent|ignorant)

therefore it is more probable that Vicky's silence comes from "ignorant", and less probable that it comes from "know".

I have seen people do this comparison, and conclude that Vicky more probably does not know of any affair. Is that a valid inference?

Part 2: Probability that Vicky is ignorant

Assuming that the inference of Part 1 is correct, we may run the same kind of problem again:

Vicky is ignorant of any affair. If David had an affair, there is a 30% chance Vicky would remain ignorant. If David did not have an affair, there is (obviously) a 100% chance Vicky would remain ignorant.

P(ignorant|affair) = 0.3

P(ignorant|no-affair) = 1

Is it valid to then compare these to determine the relative probability of "affair" and "no-affair"?

P(ignorant|affair) < P(ignorant|no-affair) 

Thus, it is more probable that Vicky remains ignorant on "no-affair" than on "affair" - so "no-affair" is more probable.

Again - is this a valid inference?

What's the point?

Leaving aside all the other theoretical/methodological questions, someone could take this as a probabilistic argument to show David probably did not have an affair:

"silent" is more probably derived from "ignorant" than "know".

"ignorant" is more probably derived from "no-affair" than "affair"

therefore, the fact of "silence" allows the conclusion "no affair" is more probable than "affair".

In other words, the fact that Vicky hasn't said anything means that on the balance of probability (excluding all other factors), David has probably not had an affair.

So - is all that valid, or is there some sort of illegitimate transfer or comparison happening here?

In a word, yes this is a valid line of reasoning and is the essence of hypothesis testing and Bayesian updating. Everything else being equal, Vicky staying silent is a data point that increases (or in the extreme case at least doesn't decrease) the probability that David didn't have an affair. But the crucial aspect is the "everything else being equal." You are implicitly assuming that David having and not having an affair was roughly 50/50 before gathering evidence from Vicky. If you saw with your own eyes that David was kissing Mary, then you are already 100% sure he did have an affair. Whether Vicky stays silent or not will not affect your assessment of the situation in this case.