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shminux comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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The homoeopathy claims interaction of water molecules with each other leads to water memory... it can even claim that all the physics is the same (hypothesis size per Solomonoff induction is same) yet the physics works out to the stuff happening. And we haven't really ran sufficiently detailed simulation to rule it out, just arguments of uncertain validity. There's no way to estimate it's probability.
We do something different than estimating probability of homoeopathy being true. It's actually very beautiful method, very elegant solution. We say, well, let's strategically take the risk of one-in-a-million that we discard a true curing method homoeopathy with such and such clinical effect. Then we run the clinical trials, and find evidence that there's less than one-in-a-million chance that those trials happened as they were, if the homoeopathy was true. We still do not know the probability of homoeopathy, but we strategically discard it, and we know the probability of wrongfully having discarded it, we can even find upper bound on how much that strategy lost in terms of expected utility, by discarding. That's how science works. The cut off strategy can be set from utility maximization considerations (with great care to avoid pascal's wager).
So, suppose that Sabadil cured your allergies 10 times out of 10, you will not take again unless forced to, because "There's no way to estimate it's (sic) probability."? Maybe you need to reread chapter 1 of HPMOR, and brush up on how to actually change your mind.
If it did cure allergies 10 times out of 10, and that ALL other possible cure-causes had been eliminated as causal beforehand (including the placebo effect which is inherent to most acts of taking a homoeopathic pill, even when the patient doesn't believe it'll work, simply out of subconscious memory of being cured by taking a pill), then yes, the posterior belief in its effectiveness would shoot up.
However, "the body curing itself by wanting to and being willing to even try things we know probably won't work based on what-ifs alone" is itself a major factor, one that has also been documented.
Par contre, if it did work 10 times out of 10, then I almost definitely would take it again, since it has now been shown to be, at worst, statistically correlated with whatever actually does cure me of my symptoms, whether that's the homoeopathic treatment or not. While doing that, I would keep attempting to rationally identify the proper causal links between events.
The point is that there is a decision method that allows me to decide without anyone having to make a prior.
Say, the cost of trial is a, the cost (utility loss) of missing valid cure to strategy failure is b, you do the N trials , N such that a * N < (the probability of trials given assumption of validity of cure) * b , then you proclaim cure not working. Then you can do more trials if the cost of trial falls. You don't know the probability and you still decide in an utility-maximizing manner (on choice of strategy), because you have the estimate on the utility loss that the strategy will incur in general.
edit: clearer. Also I am not claiming it is the best possible method, it isn't, but it's a practical solution that works. You can know the probability that you will end up going uncured if the cure actually works.