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Minimax still requires a probability distribution, which means you need a prior. Edit: Based on the downvotes, I see my audience is not convinced. I'll repeat

Minimax still requires a probability distribution, which means you need a prior. Edit: Based on the downvotes, I see my audience is not convinced. I'll repeat an explanation I posted a while ago. Probably should make this a blog post because I see this claim quite often. I'd love to know what book you read it in. -- In minimax regret, you have a set of available decisions D, and a set of possible states of nature N, and a utility U(D,N). Each state of nature also has a probability P(N) (which can be influenced by the decision too in some problems). States of nature include "interest rates rise 1%", "interest rates fall 1%", and "interest rates stay the same". Decisions include "invest in stocks" and "invest in bonds". Minimax regret proposes to ignore the probabilities P(N), instead suggesting a way to make a decision purely based on the utilities of the outcomes. But that is actually an illusion. Outside of math class word problems, we don't have N or U(D,N) handed to us on a silver platter. There is always an infinite range of possible states of nature, many of which have a probability approaching but never reaching zero, including states such as "win the lottery", "communist revolution", and "unexpected intergalactic nuclear war". In commonsense decision-making we don't include those states of nature in our decision matrix, because our common sense rules them out as being implausible before we even think about our options. You wouldn't choose to invest in bonds just because stocks have the most regret in the event of a communist takeover. So what actually happens is we intuitively apply some probability threshold that rules out states of nature falling below it from our consideration. Then we minimize max regret on the remaining "plausibly realistic" states of nature. Humans are so good at doing probability mentally that this step happens before we even realize it. But if you are writing code that makes decisions, you'll need to do it, and so you'll need to have at least a rough stab at the probability distributions.

deleted 7y ago
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perl4ever 7y ago
"There is always an infinite range of possible states of nature" Well, I think that is definitely and unambiguously false. The universe is not infinite, nor infinitely divisible, as far as we know, and the number of future states of any particular person (or humanity) are even smaller than those of the universe. Limits in time mean limits in space, and limits in space mean limits in particles and possibilities. I'm not sure I can make a case that it matters, but if it doesn't matter, why say infinite?
- jbay808 7y ago
  I guess this is a tangent. First of all my point really doesn't hinge on the infinity; it can be finite (but really big) but regardless, whenever you apply minmax you must first crop your decision space to a probability threshold, or else you'll make nonsensical decisions based on what gives the best outcome if the sun should happen to explode. But secondly, I think (although I'd happily concede if convinced otherwise) that the space of possible scenarios really is infinite, even if the observable universe is not. The space I'm talking about is not the actual state space of the universe, which in some interpretations of physics might be finite or even unitary. It spans the space of hypothetical universes that are all consistent with your information with nonzero probability, which I think is probably infinite, but again, if it's not infinite that's a technicality. If you include the states that have with zero probability (because why not? GGP was advocating that the probability is irrelevant to minmax decisions) then the space is definitely infinite, because even physically impossible states of nature will impact our decision making.
  - brownbat 7y ago
    Another way to conceptualize the "cropping" is to get rid of all future states where planning would have been meaningless anyway. We are momentary Boltzmann brains? We'll assume not, because if so, nothing really matters. Trivial difference, but that avoids potentially difficult threshold problems and cousins of the St. Petersburg paradox or even Pascal's mugger, at the risk of being slightly more hand wavy. Arguably an aesthetic distinction at this point, I generally think your description and approach are right.
- 6gvONxR4sf7o 7y ago
  >Well, I think that is definitely and unambiguously false. The universe is not infinite, nor infinitely divisible, as far as we know... This seems wrong. We don't know whether the universe is infinitely divisible. See this for a nice discussion: https://physics.stackexchange.com/questions/33273/is-spacetime-discrete-or-continuous https://physics.stackexchange.com/questions/33273/is-spaceti...
  - 7y ago
madhadron 7y ago
Conceptually you are right: all mathematical models have assumptions, including assumptions about their scope of applicability. But you are redefining "prior" to refer to all the assumptions of the model, and not its usual meaning as the prior distribution used in Bayes calculations.
- jbay808 7y ago
  Prior distribution is P(N|I), where I is the background information you have such as "historical interest rates in the USA looked like this", and "communist revolutions occurred 6 times in the 20th century" (made-up number). I is not itself the prior. For this investing example, it's also the only information we have, unless we're trying to update on something like a central bank announcement. So our probability distribution over N is just the prior distribution. When you're actually trying to make a decision, and not just solving a problem handed to you in math class, you can't avoid using P(N). You can either say "The minimax procedure requires knowing P(N) as an input, so that it isn't dominated by extremely improbable N", or you can say equivalently that "Minimax doesn't require P(N), but as an assumption of my model I'm ignoring all states of nature N with P(N) < y, then applying minimax regret over the remaining N".
  - madhadron 7y ago
    I think we must be coming from two different communities of practice where the words don't quite line up. All the operational things you are saying I agree with. I just put them under different verbal categories.

deleted 7y ago

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perl4ever 7y ago

"There is always an infinite range of possible states of nature" Well, I think that is definitely and unambiguously false. The universe is not infinite, nor infinitely divisible, as far as we know, and the number of future states of any particular person (or humanity) are even smaller than those of the universe. Limits in time mean limits in space, and limits in space mean limits in particles and possibilities. I'm not sure I can make a case that it matters, but if it doesn't matter, why say infinite?

jbay808 7y ago
I guess this is a tangent. First of all my point really doesn't hinge on the infinity; it can be finite (but really big) but regardless, whenever you apply minmax you must first crop your decision space to a probability threshold, or else you'll make nonsensical decisions based on what gives the best outcome if the sun should happen to explode. But secondly, I think (although I'd happily concede if convinced otherwise) that the space of possible scenarios really is infinite, even if the observable universe is not. The space I'm talking about is not the actual state space of the universe, which in some interpretations of physics might be finite or even unitary. It spans the space of hypothetical universes that are all consistent with your information with nonzero probability, which I think is probably infinite, but again, if it's not infinite that's a technicality. If you include the states that have with zero probability (because why not? GGP was advocating that the probability is irrelevant to minmax decisions) then the space is definitely infinite, because even physically impossible states of nature will impact our decision making.
- brownbat 7y ago
  Another way to conceptualize the "cropping" is to get rid of all future states where planning would have been meaningless anyway. We are momentary Boltzmann brains? We'll assume not, because if so, nothing really matters. Trivial difference, but that avoids potentially difficult threshold problems and cousins of the St. Petersburg paradox or even Pascal's mugger, at the risk of being slightly more hand wavy. Arguably an aesthetic distinction at this point, I generally think your description and approach are right.
6gvONxR4sf7o 7y ago
>Well, I think that is definitely and unambiguously false. The universe is not infinite, nor infinitely divisible, as far as we know... This seems wrong. We don't know whether the universe is infinitely divisible. See this for a nice discussion: https://physics.stackexchange.com/questions/33273/is-spacetime-discrete-or-continuous https://physics.stackexchange.com/questions/33273/is-spaceti...
- 7y ago

madhadron 7y ago

Conceptually you are right: all mathematical models have assumptions, including assumptions about their scope of applicability. But you are redefining "prior" to refer to all the assumptions of the model, and not its usual meaning as the prior distribution used in Bayes calculations.

jbay808 7y ago
Prior distribution is P(N|I), where I is the background information you have such as "historical interest rates in the USA looked like this", and "communist revolutions occurred 6 times in the 20th century" (made-up number). I is not itself the prior. For this investing example, it's also the only information we have, unless we're trying to update on something like a central bank announcement. So our probability distribution over N is just the prior distribution. When you're actually trying to make a decision, and not just solving a problem handed to you in math class, you can't avoid using P(N). You can either say "The minimax procedure requires knowing P(N) as an input, so that it isn't dominated by extremely improbable N", or you can say equivalently that "Minimax doesn't require P(N), but as an assumption of my model I'm ignoring all states of nature N with P(N) < y, then applying minimax regret over the remaining N".
- madhadron 7y ago
  I think we must be coming from two different communities of practice where the words don't quite line up. All the operational things you are saying I agree with. I just put them under different verbal categories.