So what should you conclude from this?

• VOI is higher when the experiment shifts your beliefs a lot, lower when the expected change in belief is small. For example, praying is sufficiently unlikely to work that it's not worth my time to test it. There are other cases where my uncertainty is high, but I can't think of sufficiently good cheap experiments.

• VOI is higher when you would gain a lot if it told you to change your plans. For example, if you would have taken Adderall without an experiment, and Adderall is expensive, then finding out it doesn't work saves you a lot of money. This is less true for melatonin.

UPDATE: Math corrected. thanks!

Put simply, VOI is the difference between your expected value with and without the information.

So with Melatonin, let's simplify to 2 possibilities:

A) Melatonin has no effect, costs $10 per year, for a value of -1

B) Saves you 15 minutes per day (+5 utilons), costs $10 per year (-1 utilon), for a net value of +4 utilons.

Now, let's say you think that A and B are equally likely. Then the expected value of not taking Melatonin is 0, and the expected value of taking it is 0.5 * -1 + 0.5 * 4 = 1.5. With only this information available, you will always take Melatonin, so your expected value is 1.5.

Then let's say you are considering a definitive experiment (so you will know with p=1 whether A or B is true).

If A is true then you will not take Melatonin, so the value of that outcome is 0 utilons.

If B is true, then you will take Melatonin, for a value of 4 utilons.

And by conservation of expected evidence, it is equally likely that the experiment will decide for A or B.*

Then the expected value of your decision with perfect info is 0.5 * 0 + 0.5 * 4 = 2 > 1.5, so the VOI is 0.5 utilons..

*Equally likely only because of how I set up the problem. Conservation of expected evidence would also be satisfied if the experiment would probably favor one side weakly, but improbably favor the other side strongly.

This makes me wonder... What "for dummies" books should I be using as checklists right now? Time to set a 5-minute timer and think about it.

Note that this explanation does not require Magic to simulate or calculate anything aside from creating a copy of a past state of the universe.

Much less than half the time. Remember, if Harry1 uses his time turner, he creates Universe2 with Harry2=Harry1_6_hours_ago. But in 6 hours, Harry2 will use his time turner, creating Universe3 with Harry3...

So IF there is a stable time loop of any kind, most universes will have that loop.

This raises the interesting prospect of stable sets of universes, with 6-hour histories A, B, and C. If a Harry that experiences A uses his time turner and does B, and a Harry experiencing B does C, and Harry experiencing C does A, then most copies of Harry will experience an inconsistent time loop, and it will seem like he actually went back and changed time.

If time loops are generally observed to be consistent, then this is evidence that single-state equilibria are much more probable than multiple-state equilibria.

Some things I decided to like (when I was young) in order to "be more grown up." (Liquor, coffee, classical music, opera)

Or because cool people or people I admired were doing it (smoking a tobacco pipe, philosophy, math).

Some things to add variety to my life, just like MixedNuts. For example, learning to appreciate and distinguish between different types of wine, teas, cheeses, classical music.

Some because I thought they were good for me, so I might as well like them. (Yogurt, sushi)

Actually, it is pretty useful in timescales up to about a week. After that I can't really imagine myself or predict the future very well.

For some reason I compartmentalized and failed to notice that this is the same technique applied to much longer timespans.

OTOH if true it does provide some evidence for Dumbledore's belief that souls are real things distinct from the body they work on.

It seems like that's a questionable assumption that Harry would be eager to test, once he found out about Horcruxes. For example, can you cast a Horcrux on the power of, say, Avada Kedavra-ing a nonmagical nonhuman creature? If not, how about a magical creature?

What if you could create a low-quality backup that way? Wouldn't it still be better than nothing?

It actually made me sit and think for a minute (though not the full five - oops) about whether there was any way I could contribute to improving conditions in prisons, that was comparatively low-cost, that I had overlooked.

I didn't think of one, but it's worth thinking about some more, probably.