There is actually much easier and intuitive proof.
For simplicity, let’s assume H takes only two values T(true) and F(false).
Now, let’s assume that God know that H = T, but observer (me) doesn’t know it. If I now make measurement of some dependent variable D with value d_i, I’all either:
1. Update my probability of T upwards if d_i is more probable under T than in general.
2. Update my probability of T downwards if d_i is less probable under T than in general.
3. Don’t change my probability of T at all if d_is is same as in general.
(In general here means without the knowledge whether T or F happened, i.e. assuming prior probabilities of observer)
Law of conservation of expected evidence tells us that in general (assuming prior probabilities), expected change in assigned probability for T is 0. However, if H=T, than those events that update probability of T upwards are more likely under T than in general, and those which update probability of T downwards are less likely. Thus expected change in assigned probability for T > 0 if T is true.
Very simple. To prove it for arbitrary number of values, you just need to prove that h_i being true increases its expected “probability to be assigned” after measurement for each i.
If you define T as h_i and F as NOT h_i, you just reduced the problem to two values version.
There is actually much easier and intuitive proof.
For simplicity, let’s assume H takes only two values T(true) and F(false).
Now, let’s assume that God know that H = T, but observer (me) doesn’t know it. If I now make measurement of some dependent variable D with value d_i, I’all either:
1. Update my probability of T upwards if d_i is more probable under T than in general.
2. Update my probability of T downwards if d_i is less probable under T than in general.
3. Don’t change my probability of T at all if d_is is same as in general.
(In general here means without the knowledge whether T or F happened, i.e. assuming prior probabilities of observer)
Law of conservation of expected evidence tells us that in general (assuming prior probabilities), expected change in assigned probability for T is 0. However, if H=T, than those events that update probability of T upwards are more likely under T than in general, and those which update probability of T downwards are less likely. Thus expected change in assigned probability for T > 0 if T is true.
QED
I had already proved it for two values of H before I contracted Sellke. How easily does this proof generalize to multiple values of H?
Very simple. To prove it for arbitrary number of values, you just need to prove that h_i being true increases its expected “probability to be assigned” after measurement for each i.
If you define T as h_i and F as NOT h_i, you just reduced the problem to two values version.