What does it mean for a probability not to be well defined in this context? I mean, I think I share the intuition, but I'm not really comfortable with it either. Doesn't it seem strange that a probability could be well defined until I start learning more about it and trying to change it? How little do I have to care about the probability before it becomes well defined again?

As soon as the oracle is trying to make predictions that are affected by what the oracle says, the problem she has to solve shifts from "estimate the probabilities" to "choose what information to give, so as to produce consistent results given how that information will affect what happens". In some cases there might not be anything she can say that yields consistent results. Exactly where (if at all) that becomes impossible depends on the details of the situation.

The Value of Uncertainty Reduction - references to academic literature appreciated

by MichaelBishop 1 min read29th Apr 201410 comments

2


My intuition (probably widely shared) suggests that uncertainty about the future is stress inducing and reducing uncertainty about the future is helpful because it allows us to plan. So I started trying to invent thought experiments that would begin to help me quantify how much I (and others) value uncertainty reduction... and then I began to get confused. Below I'll share two examples focused on knowledge about the next five years of one's career but similar psychological/philosophical issues would arise in many other contexts.

Thought Experiment #1: the risk of job loss
Imagine the true probability you involuntarily lose your job at some point in the next 5 years is either 0% or 50% and that your current best guess is that you have a 25% chance of losing your job. For a price, an oracle will tell you whether the truth is 0% of 50%. How much will you pay?

If you think you can answer this question, please do so. Part of my confusion is that knowing the probability I will lose my job seems certain to affect the probability that I lose my job. If you told me the probability was 50% then I'd do a combination of working overtime and looking for other jobs that should reduce that probability, and if the probability remains 50% then I'm in a much less pleasant situation than I would be in the case that the probability is 50% but only because I'm assuming its a more reasonable 25%.

Thought Experiment #2: uncertainty about future earnings
Imagine your estimate of your total income over the next 10 years is unbiased, and that the random error in your estimate is normally distributed. (Admittedly a normally distributed error term is unrealistic in this problem but bear with me for simplicity). What's a reasonable standard deviation? Let's say 3 years worth of income. How much would you pay to reduce that standard deviation to 1.5 years of income?

Once again, go ahead and to answer this if you can, but I've got myself confused here as well... I'm trying to get at the present value of reducing uncertainty about the future, but in this example it appears I getting an offer to reduce the actual risk of *experiencing* a much lower than expected income at the expense of reducing the chances that I make a much higher than expected income, not just reducing uncertainty.

Any insight into what's going on with my thought experiments would be greatly appreciated. I see some parallels between them and Newcomb's Paradox, but I'm not sure what to make of Newcomb's Paradox either. If people have relevant references to the philosophy literature that's great...
relevant references to judgment and decision-making or economics literature would be even better.

2