The ideal Bayesian [can] never predict in which direction future information will alter his own estimates, and investors in an ideal stock market, [can] never predict in which direction prices will move
I suggest rewording this, it seems like you are making a different claim than the one you intended. An ideal Bayesian can predict in which direction future information will alter his own estimates.
I have been given a coin which I know is either fair or biased (comes up heads 75% of the time). After a sequence of tosses I have arrived at, say, 95% probability that the coin is biased. The probability I assign to the next toss giving ‘heads’ is:
p(heads) = 0.95 0.75 + 0.05 0.5 ~= 0.74
There is a 74% chance that I will alter my estimate upwards after this coin toss.
I predict with 95% confidence that should I continue to toss the coin long enough future information will alter my estimates upwads until it reaches ~100% confidence that the coin is biased. Naturally, I predict a 5% chance that my estimates would be eventually altered downwards until they approximate 0%, a far greater change.
The same applies to some stocks in an ideal stock market. For example, some companies may have a limit on their growth potential and yet have some chance of going bankrupt. The chance that these stocks could completley lose value should suggest that they are more likely to go up than to go down for their value to be what it is now.
Can someone suggest a concise replacement for “in which direction” that applies here?
I suggest rewording this, it seems like you are making a different claim than the one you intended. An ideal Bayesian can predict in which direction future information will alter his own estimates.
I have been given a coin which I know is either fair or biased (comes up heads 75% of the time). After a sequence of tosses I have arrived at, say, 95% probability that the coin is biased. The probability I assign to the next toss giving ‘heads’ is:
p(heads) = 0.95 0.75 + 0.05 0.5 ~= 0.74
There is a 74% chance that I will alter my estimate upwards after this coin toss.
I predict with 95% confidence that should I continue to toss the coin long enough future information will alter my estimates upwads until it reaches ~100% confidence that the coin is biased. Naturally, I predict a 5% chance that my estimates would be eventually altered downwards until they approximate 0%, a far greater change.
The same applies to some stocks in an ideal stock market. For example, some companies may have a limit on their growth potential and yet have some chance of going bankrupt. The chance that these stocks could completley lose value should suggest that they are more likely to go up than to go down for their value to be what it is now.
Can someone suggest a concise replacement for “in which direction” that applies here?
Expected future expectation is always the same as the current expectation.
Thanks Jim!
You’re right. Edited to Jim’s version, although it sounds kind of convoluted. I’m going to keep an eye out for how real statisticians describe this.
Expected value?