[halpmeh]

Technically, updating priors wouldn't necessarily be warranted. Consider a statement X implies Y, e.g. The government is corrupt, which implies SBF won't go to jail. Just because X implies Y does not mean ~Y implies ~X. E.g. SBF going to jail does not imply the government is not corrupt.

[sebzim4500]

Sure but it is a simple result in Bayesian statistics that if event X increases your confidence in fact Y then ~X should decrease your confidence in Y.

For example, if SBF evading jail would increase your confidence in the statement "The US justice system is wholly corrupt" then SBF being sentenced should decrease your confidence in it.

[shkkmo]

We've seen enough examples of X to maintain confidence in Y. You need a lot more than a single ~X to significantly impact that assessment.

[electrondood]

You're conflating real life with logic.

In your example, according to logic, if X implies Y, then if you don't have Y, you necessarily don't have X. If this were a logic exercise, then not "SBF goes to jail" necessarily implies not "the government is not corrupt."

However, in real life there's no connection between the two.

[zopa]

https://en.m.wikipedia.org/wiki/Contraposition

I imagine you’re remembering that P -> Q does not mean ~ P -> ~Q. That’s right, but you can absolutely get to ~Q -> ~P.

[davidktr]

X -> Y litteraly means ¬Y -> ¬X because of (¬X ∨ Y) = (Y ∨ ¬X).

[posterboy]

¬Y=='Gov not corrupt' is not an option for those people who argue that the government is corrupt.

In conclusion, naysayers say he wont be convicted is imying and thus proves that the gov is corrupt. The top comment says he may get a sentence, meaning the government is not necessarily corrupt. Yaysayers say the gov is not corrupt and he will get a conviction iff he is guilty.

This is trivial, but difficult to formalize. Thanks for your correction.

[panda-giddiness]

> Just because X implies Y does not mean ~Y implies ~X.

As others have mentioned, X implies Y does in fact require ~Y implies ~X. I think your example is confusing because "the government is corrupt" means many different things, but you're using it in a rather specific way ("the government is protecting SBF"). The equivalence of `X implies Y` and `~Y implies ~X` is more manifest through the following example

   "The government is protecting SBF, so SBF won't go to jail"

and

   "SBF went to jail, so the government wasn't protecting him."

[AkshatM]

I get what you mean, but I think you've formulated this incorrectly. Let P(Y) be your prior about government corruption. Let X be the event that SBF is arrested. You want to compute P(Y|X) using the Bayesian update formula and then set P(Y) = P(Y|X). That is what is meant by re-evaluating your priors.

You're modelling X and Y as propositions and you're correct about the inference of ~X and ~Y, but Bayesian updating is about degree of belief in those propositions, which your inference is not a claim about.