I'm going to try to apply some Bayesian math to the question of whether it makes sense to believe "if you aren't sad about my bad situation then that means you don't care about me"

In this example, Person X is in a bad situation, and wants to know if Person Y cares about them.

To use Bayes' theorem, we are interested in the following probabilities:

P(A) is 'Person Y cares about Person X'

P(B) is 'Person Y feels sad about Person X's situation'

P(C) is 'Person Y expresses sadness about Person X's situation'

Let's use P(B) as an abbreviation for either P(B given C) or P(B given not C). Because we're doing these calculations after Person X already knows whether or not Person Y expressed sadness. In other words, I'm assuming that P(B) has already been updated on C.

Bayes' theorem says that P(A given B) is P(B given A) times P(A) over P(B).

P(B given A) and P(A) make it go up, P(B) makes it go down.

Bayes' theorem says that P(not A given not B) is P(not B given not A) times P(not A) over P(not B).

P(not B given not A) and P(not A) make it go up, P(not B) makes it go down.

So this tells us:

The more uncertain Person X is about whether Person Y cares about them, the more they'll worry about whether Person Y feels sad about any specific misfortune Person X is experiencing.

Different people probably have different beliefs about what P(A given B) is. Someone who thinks that this value is high will be more reassured by someone feeling sad about their situation, and someone who thinks this value is low will be less reassured. So this value will be different for a different person X, and also for a different person Y.

Different people probably have different beliefs about what P(not A given not B) is. Someone who thinks that this value is high will be more worried by someone not feeling sad about their situation, and someone who thinks this value is low will be less worried. So this value will be different for a different person X, and also for a different person Y.

If Person Y somehow feels equally sad about the misfortune of people e specifically cares about, and people e doesn't even know, then P(B given A) is equal to P(B), and whether they feel sad about any particular misfortune of Person X doesn't give any new information about whether Person Y cares about person X.

Similarly, if Person Y never feels sadness about anyone's misfortune, then P(B given A) is equal to P(B), and the fact that Person Y doesn't feel sad about any particular misfortune of Person X doesn't give any new information about whether Person Y cares about person X.

And if Person Y is somehow less likely to feel sad about the misfortunes of people e cares about, than people e doesn't care about... then all this would be reversed? This isn't really relevant anyway, so I won't bother checking the math.

I am very likely to have made a mistake somewhere in this comment. Halfway through writing this comment I started to get really fuzzyheaded.

Most of this was already obvious before doing the math, but I think there was at least some value to this exercise.

Also, I very strongly suspect that I'm completely missing the point of... something...

oh, right, I was trying to answer the question of whether it makes sense to believe "if you aren't sad about my bad situation then that means you don't care about me"

and the answer is... sometimes. It depends on the variables described in the math above.