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Judea Pearl wrote a whole book about proving causality (and when it can't be proven).

A rough summary is available as a YouTube video:

https://m.youtube.com/watch?v=HUti6vGctQM

They are very very closely related, and philosophers and Pearl cite each other on the topic of causation, but Hume and philosophers are usually concerned with a slightly different problem. Hume is concerned about the nature of the necessity between any cause and effect which seemed to be implied by our experience of events. Whereas Pearl (much less knowledgeable about Pearl) is interested in how we can model events to give us high confidence in the relation applying in a particular case while leaving the topic of necessity and the fundamental nature of causation to the side and assuming a much more vague /ad hoc notion of what causation is.

Hume's point is much more fundamental: you can't know if the world is indeed governed by 'natural laws'. Maybe the sun will just not rise tomorrow, ignoring all of the causal relationships we think we 'discovered'. There is no way to prove it will.

Or in more modern terms: we didn't discover anything inherent in nature, we invented models for prediction. Which are very useful, but don't tell anything about how the world 'really is'.

And, to be very explicit: this includes probability theory. You can prove that a series of observations from the past follows some pattern that matches the definition of probabilistic causality. But you can't prove that the events in the future will continue to fit that same pattern. So again you cannot prove anything about how the world 'really is'.

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