Don’t think of probability as being mutable, as getting updated. Instead, consider a fixed comprehensive state space, that has a place on it for every possible future behavior, including the possible questions asked, possible pieces of evidence presented, possible actions you make. Assign a fixed probability measure to this state space.
Now, when you do observe something, this is information, an event, a subset on the global state space. This event selects an area on it, and encompasses some of the probability mass. The statements, or beliefs (such as “the ball #2 will be red”), that you update on this info, are probabilistic variables. A probabilistic variable is a function that maps the state space on a simpler domain, for example a binary discrete probabilistic variable is basically an event, a subset of the state space (that is, in some states, the ball #2 is indeed defined to be red, these states belong to the event of ball #2 being red).
Your info about the world retains only the part of the state space, and within that part of the state space, some portion of the probability mass goes to the event defining your statement, and some portion remains outside of it. The “updating” only happens when you focus on this info, as opposed to the whole state space.
If that picture is clear, you can try to step back to consider what kind of probability measure you’d assign to your state space, when its structure already encodes all possible future observations. If you are indifferent to a model, the assignment is going to be some kind of division into equal parts, according to the structure of state space.
IAWYC, but as pedagogy it’s about on the level of “How should you imagine a 7-dimensional torus? Just imagine an n-dimensional torus and let n go to 7.”
Don’t think of probability as being mutable, as getting updated. Instead, consider a fixed comprehensive state space, that has a place on it for every possible future behavior, including the possible questions asked, possible pieces of evidence presented, possible actions you make. Assign a fixed probability measure to this state space.
Now, when you do observe something, this is information, an event, a subset on the global state space. This event selects an area on it, and encompasses some of the probability mass. The statements, or beliefs (such as “the ball #2 will be red”), that you update on this info, are probabilistic variables. A probabilistic variable is a function that maps the state space on a simpler domain, for example a binary discrete probabilistic variable is basically an event, a subset of the state space (that is, in some states, the ball #2 is indeed defined to be red, these states belong to the event of ball #2 being red).
Your info about the world retains only the part of the state space, and within that part of the state space, some portion of the probability mass goes to the event defining your statement, and some portion remains outside of it. The “updating” only happens when you focus on this info, as opposed to the whole state space.
If that picture is clear, you can try to step back to consider what kind of probability measure you’d assign to your state space, when its structure already encodes all possible future observations. If you are indifferent to a model, the assignment is going to be some kind of division into equal parts, according to the structure of state space.
IAWYC, but as pedagogy it’s about on the level of “How should you imagine a 7-dimensional torus? Just imagine an n-dimensional torus and let n go to 7.”
Eliezer’s post on priors explains the same idea more accessibly.
EDIT: Sorry, I didn’t notice you already linked it below.