# Correspondence between beliefs and predictions

It’s obvious that a set of beliefs can imply certain predictions
(“make your beliefs pay rent”, etc.), but what about the reverse? Given
an arbitrary statement *X*, is it possible to encode whether I
believe *X* by writing down a series of predictions?

Another way to phrase this is to consider the following scenario. I
want to find out whether Alice believes *X*. Alice doesn’t want
to tell me whether she believes *X*. However, she has agreed to
make a bunch of predictions that I ask for, and to make those
predictions truthfully. Can I say whether she believes *X* based
on the predictions?

To take a trivial example, any prediction like “The sun will rise tomorrow” can just be passed along to Alice, and she will have to answer truthfully.

Another example of a statement is “This is a fair coin”. The
statement itself is *not a prediction*, so I’m not allowed to ask
it directly. The way to find out whether Alice believes the statement is
to instead ask “If I toss this coin 1,000,000 times, it will come up
heads roughly half the times” or something similar.

One strategy is to “go meta” by asking for Alice’s prediction for
“Alice will believe *X* one minute from now”, but I think that’s
kind of “boring”, in the sense that I think it goes against the spirit
of my original query. We might stipulate that we are not allowed to ask
Alice to predict her own future belief states.