AFAIK, in mathematical literature there is met no definition of “subjective 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%”. Moreover, encyclopedias say that there is no formula for subjective probability. In this short note I give a formal definition of subjective 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.
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Project summary
Subjective probability (like the opinion that the tomorrow rain is of 86%) is known for its “elusive” character defying formal mathematical consideration, unlike the “real” probability theory’s probabilities. I however found a way to define them mathematically.
I definitely have found a way to formalize what is subjective probability. Contrary to academic pride, subjective probability can be defined by following a red-neckers’ saying “it is how much you are ready to pay for it, bro”. This, having fixed a formal market gives rise of two numbers: the maximum price one is going to pay in a bet of for this probability and the minimum price one would not agree to pay. Through these numbers we can define subjective probability. This has some complexities like dealing with too small sums for one to consider and the case when risk outweights even a big sum of money, but I am confident I have ideas of all math (of infinitesimals and of infinitely big amounts) needed for this, because I have formal training in mathematical analysis specialization.
I want to spend a month thinking (and, of course, writing my thoughts down for publication in peer review) about subjective probabilities. I will try to answer such questions, as how to unify two above subjective probabilities (one minimum and one maximum), do subjective probabilities match the definition of probability?, what may be an analogue(s) of Bayesian inference for subjective probabilities, why it is considered that the probability should be set to 1/2, if we know nothing about the outcome, etc.
Among subjective probabilities, I also consider other subjective quantities, such as subjective market value predictions. Also the entire market can be considered as one player, producing interesting results for such applications as legal theory (what is to blame the market?) and of course economy.
What are this project’s goals and how will you achieve them?
The goal is to describe most important properties of subjective probabilities. An additional goal is me to learn advanced game theory, while thinking how to apply it to subjective probabilities or apply subjective probabilities to game theory.
Following the John Nash’s game theory, there were several random people’s decisions that prevented nuclear war, not founded in any science. My theory aims to (partly) formalize such people’s decisions to prevent nuclear war and it may lead to prevention of a nuclear war in the future.