Thread by @Light_Cynapse

Zuboff’s empiricism draws upon Bayesian inference, in which Bayes’ theorem has an objective a priori status. One issue with this theorem is that it requires prior probabilities to update the probability of a hypothesis being true in light of new evidence.

This has led some philosophers to adopt a subjectivist stance, arguing that ultimate prior probabilities cannot be known objectively and must instead be assigned based on subjective credences. Zuboff contends that these philosophers conflate subjectivity with perspectivity.

While probability is perspectival, that is a separate issue for another time.
Zuboff defends objective Bayesianism, which is possible due to his solution to the problem of induction.

Max Albert showed that the issue of a priori priors in Bayesianism is equivalent to Hume’s problem of induction. Thus, if the problem of induction can be solved, then the problem of priors can be solved as well.

A thought experiment illustrating Zuboff's solution involves two urns, each containing a trillion beads. In one urn (A), all trillion beads are blue, whereas in the other (B), only one of the trillion beads is blue.

"This second urn has been well stirred so that the single blue bead has nestled into a random location among the other beads. First, let us say, a toss of a fair coin decides which of the two urns is pushed forward for sampling.

Then a single bead that is randomly drawn from that urn is shown to an observer who has no other basis for judging what it contains and who understands all the circumstances I have described. If the bead that is shown is blue, the observer should infer

that it is a trillion times more probable that the urn being sampled is the urn with beads that were all of them blue. If it were instead the urn with only one blue bead, then this random drawing of a bead that was blue would have had to be something overwhelmingly improbable.

But it is overwhelmingly improbable that something overwhelmingly improbable is what has occurred. Hence that hypothesis, combined with this evidence, is in itself overwhelmingly improbable and we must infer that the other hypothesis, of the urn being that with all blue beads,

is overwhelmingly more probable to be true. We should expect this inference to give us the wrong answer in something like once in every trillion times this is tried. But it is overwhelmingly improbable that this is such a time.“

In this case, the prior probabilities were 50%, since the fair coin toss determined which urn was chosen. However, in many real-world situations, prior probabilities are unknown.

This uncertainty seemingly prevents us from making any probabilistic claims about which urn was selected based on the evidence of drawing a blue bead. For instance, if it were a trillion times more probable that urn B would be selected,

this could counterbalance the one-in-a-trillion chance of drawing the single blue bead from it. However, according to Zuboff, it was overwhelmingly improbable that the prior probabilities were like this. He writes:

"Can we not still, as in our earlier uncontroversial case, confidently say that it is overwhelmingly more probable that the urn pushed forward was the one with all blue beads? Because if it had been the other urn, something overwhelmingly improbable must have occurred—

and it is overwhelmingly improbable that something overwhelmingly improbable occurred."

If the prior probability of selecting urn A (which contains only blue beads) were low, then drawing a blue bead would be improbable regardless of which urn had been selected.

Consider all possible scenarios:

1) If we suspect that urn B was chosen but believe the prior probability of selecting it was low, then drawing a blue bead would have been improbable.

2) If we suspect that urn B was chosen but believe the prior probability of selecting it was high, then drawing a blue bead would still have been improbable.

3) If we suspect that urn A was chosen but believe the prior probability of selecting it was low, then drawing a blue bead would have been improbable.

4) If we suspect that urn A was chosen and believe the prior probability of selecting it was high, then drawing a blue bead would have been certain.

Drawing a blue bead is improbable in all cases except the last one.
Zuboff concludes:

"The right view of the mathematics, I think, is that weighing the hypotheses simply in terms of their favorability to the evidence already gives you their objective probabilities when combined with that evidence.

Then, the objective prior probabilities would, if not directly known, be merely missing further information—information that one should anyway expect to favor the hypothesis most favorable to the evidence.

Otherwise, a less probable event would have to be what actually occurred, which is itself less probable.
When the difference in how much the hypotheses favor the evidence is small, the unknown prior probabilities might be decisive, making their absence significant.

However, in cases like the trillion-bead urn scenario, where the difference in evidential favorability is vast, the unknown prior probabilities do not undermine the inference."

@ThreadReaderApp unroll