There is an odd paper by Jeanne Peijnenburg in the latest Mind. (It’s subscription only, so no link.) There’s a formal point and a philosophical point.
The formal point concerns the following question. Are there values of a1, b1, a2, b2, … such that given that P(Ei|Ei+1) = ai, and P(Ei|~Ei+1) = bi for all i, we can compute the value of P(E1)? This is answered in the affirmative, in some complicated cases where we have to compute some tricky infinite sequences.
The philosophical point is that this is meant to be a defence of infinitism, a la Peter Klein. The idea, if I’ve understood it, is that we can (contra Klein’s critics) say that we can deduce unconditional probabilities from an infinite string of conditional probabiilties. So probabilities don’t have to be ‘grounded’ in unconditional probabilities, as Klein suggests.
But there’s a much simpler way to prove the formal point. If a1 = b1 = x, the Pr(E1) = x, whatever the other values are. Here is a way to get from conditional probabilities to unconditional probabilities. And we don’t even need an infinite chain. So I don’t see how this is meant to give any support to infinitism. Maybe I’m just missing something here. At the very least, I’m certainly missing how these computations of particular probabilities support the idea that infinite chains can justify old-fashioned, non-probabilistic, belief.