We do science using the data we collected through our senses and we use science to understand how our senses work. Although I lack any rigorous formulation of the problem, the following plan seems interesting and I want to share it with you.

"Human senses collect accurate data about reality."

$q =$ "Reality is governed by laws of physics."

Now we want to know $Pr (p)$ and $Pr (q)$ . I don't know any way to directly calculate these values. It seems to me that $Pr (p | q)$ and $Pr (q | p)$ are easier to get our hands dirty. The former seems to be a scientific question and I heard that there are research done already, and the latter could be addressed by something like Solomonoff's Induction. At this point, we should be able to calculate the ratio $Pr (p) / Pr (q)$ .

The last stage is to calculate $Pr (q | \neg p)$ . Here for simplicity I assume that the trust in our senses is something binary, we either believe all data we collect is about reality or none (I hope that the scheme can be extended later). Now the term $Pr (q | \neg p)$ is simply our prior belief about the laws of physics before we analyzed any data, and that again could be addressed by the universal prior probabilities from Solomonoff's Induction.

Let's name the quantities we have so far. $Pr (p | q) = a$ , $Pr (q | p) = b$ , and $Pr (q | \neg p) = c$ .

\frac{Pr (p)}{Pr (q)} = \frac{a}{b}

\frac{Pr (\neg p)}{Pr (q)} = \frac{Pr (\neg p | q)}{Pr (q | \neg p)} = \frac{1 - a}{c}

\frac{Pr (p)}{Pr (\neg p)} = \frac{a c}{b (1 - a)}

Pr (p) = \frac{a c}{a c - a b + b} and Pr (q) = \frac{b c}{a c - a b + b}

Now I'm aware that we still need to assume the statement "Reality is governed by some algorithm, some fixed set of rules.", because Solomonoff's Induction needs that assumption.

I'd be very happy to hear your thoughts and comments on this framework. Is it dumb in some obvious way, or does it remind you of some research you have already seen before?

Update: I think I changed my belief that there is a circularity here. I feel pretty confident accepting the statement "There are some data I receive" without needing any science. The interesting question seems to be how much of the reality should we expect to reach using our senses.