AFAIK, in mathematical literature there is met no definition of “guessed probability” like “I (or my computer) think rain probability is 34%” or more fundamentally “In my opinion, the probability of Euler-Stocks hypothesis being true is 67%”. In this short note I give a formal definition of guessed probability.
Let’s fix a (probabilistic or not) game (in the sense of game theory) or more generally a probabilistic distribution of games (being in some way “averaged” to be considered as a single game), a hypothesis (a logical statement) a “doubtful” player (a function that makes probabilistic decisions for the game) dependent on boolean variable (called trueness of our hypothesis) for a “side” (e.g. for white or black in chess) of the game. Let’s “factor” the game by restricting the side to only these decisions that the doubtful player can make. The guessed probability is defined as the probability that given (fixed!) player (a probabilistic function that makes decisions for our side of the game) will choose the variant in which our hypothesis is true.
A variation of this is when the doubtful player’s input is mediated by a function that takes on input logical statements instead of a boolean value. (The above is easy to rewrite in this case.)
More generally guessed probability can be generalized to guessing real number (or any measure space) variables by replacing the doubtful player by a real number function (or a function from our measure space) and defining the guessed value as the probability distribution of player’s decisions in the “factored” by restricting to guessed player decisions game.
A special case of the above are perceived or guessed value of an economical asset in an economical game (a game about becoming richer).
The guessed value of assets of a person is a scientific way to measure success. (It’s a well known fact, however denied by many pseudoscientists, that measuring success by money is often irrelevant.)
In economical games it’s often relevant to consider the entire market (a set of players) as our relevant player. So we obtain a kind of market value.
Further research is highly perspective. Particularly it defines scientific (not necessarily monetary) values of scientific hypotheses and is useful in research planning.