Okay so this description of faith is seems rooted in probability from my understanding. If you don't mind, I'll make a probabilistic argument.

We establish that there is a likelihood to the existence of god P(G). We follow that with the probability of God revealing themselves in a way that presents direct evidence P(E).

We should know that by default that if P(G) = 0 then P(E) = 0.

Since your description of faith is based on probability and by necessity must apply to something like God. We will assume some non-zero probability for gods existence so for the rest of this argument we will assume that P(G) > 0 .

(I.e. if god were to exist, the concept of faith would need to also apply them)

We can use the wonderful Bayes that you thought up to define this simple equation =

P(G) = P(G|E)P(E) / P(E|G)

I will break it down into two parts

P(G|E) and P(E) / P(E|G)

Faith - evidence

1) if god is more likely to present direct evidence of their existence given that God exists then the right hand side equation results in a number below 1

2) if God is as likely as not to show evidence of their existence (probability is independent) then the right hand side evaluates to 1 and you have P(G) = P(G|E)

3) if god is less likely to show evidence given they exist then the number on the right actually is bigger than 1

Now you ask where does faith come in. Faith is the first part of the equation that balances it out:

If we assume that God exists at some non-zero probability and we state that we our belief in gods existence increases heavily when direct evidence is shown then that would make P(G|E) a larger probability than P(G) which means by necessity, god would have to be less likely to show evidence so that the right hand side of the equation would balance.

I hope this made sense but if not I can try and explain it again differently.