Another thing I don’t fully understand is the process of “updating” a prior. I’ve seen different flavors of Bayesian reasoning described. In some, we start with a prior, get some information and update the probabilities. This new probability distribution now serves as our prior for interpreting the next incoming piece of information, which then causes us to further update the prior. In other interpretations, the priors never change; they are always considered the initial probability distribution. We then use those prior probabilities plus our sequence of observations since then to make new interpretations and predictions. I gather that these can be considered mathematically identical, but do you think one or the other is a more useful or helpful way to think of it?
In this example, you start off with uncertainty about which process put in the balls, so we give 1⁄3 probability to each. But then as we observe balls coming out, we can update this prior. Once we see 6 red balls for example, we can completely eliminate Case 1 which put in 5 red and 5 white. We can think of our prior as our information about the ball-filling process plus the current state of the urn, and this can be updated after each ball is drawn.
Another thing I don’t fully understand is the process of “updating” a prior. I’ve seen different flavors of Bayesian reasoning described. In some, we start with a prior, get some information and update the probabilities. This new probability distribution now serves as our prior for interpreting the next incoming piece of information, which then causes us to further update the prior. In other interpretations, the priors never change; they are always considered the initial probability distribution. We then use those prior probabilities plus our sequence of observations since then to make new interpretations and predictions. I gather that these can be considered mathematically identical, but do you think one or the other is a more useful or helpful way to think of it?
In this example, you start off with uncertainty about which process put in the balls, so we give 1⁄3 probability to each. But then as we observe balls coming out, we can update this prior. Once we see 6 red balls for example, we can completely eliminate Case 1 which put in 5 red and 5 white. We can think of our prior as our information about the ball-filling process plus the current state of the urn, and this can be updated after each ball is drawn.