I must admit to some amount of silliness – the first thought I had upon stumbling onto LessWrong, some time ago, was: “wait, if probability does not exist in the territory, and we want to optimize the map to fit the territory, then shouldn’t we construct non-probabilistic maps?” Indeed, if we actually wanted our map to fit the territory, then we would not allow it to contain uncertainty – better some small chance of having the right map, then no chance, right?
Of course, in actuality, we don’t believe that (p with x probability) with probability 1. We do not distribute our probability-mass over actual states of reality, but rather, over models of reality; over maps, if you will! I find it helpful to visualize two levels of belief: on the first level, we have an infinite number of non-probabilistic maps, one of which is entirely correct and approximates the territory as well as a map possibly can. On the second level, we have a meta-map, which is the one we update; it consists of probability distributions over the level-one maps. What are we actually optimizing the level-two map for, though? I find it misleading to talk of “fitting the territory”; after all, our goal is to keep a meta-map that best reflects the state of the data we have access to. We alter our beliefs based (hopefully!) on evidence, knowing full well that this will not lead us to a perfect picture of reality, and that a probabilistic map can never reflect the territory.
I think a concrete example is good for explaining this concept. Imagine you flip a coin and then put your hand over it before looking. The state of the coin is already fixed on one value. There is no probability or randomness involved in the real world now. The uncertainty of it’s value is entirely in your head.
I rather like this way of thinking. Clever intuition pump.
What are we actually optimizing the level-two map for, though?
Hmmm, I guess we’re optimizing out meta-map to produce accurate maps. It’s mental cartography, I guess. I like that name for it.
So, Occam’s Razor and formal logic are great tools of philosophical cartographers. Scientists sometimes need a sharper instrument, so they crafted Solomonoff induction and Bayes’ theorem.
Formal logic being a special case of Bayesian updating, where only p=0 and p=1 values are allowed. There are third alternatives, though. Instead of binary Boolean logic, where everything most be true or false, it might be useful to use a 3rd value for “undefined”. This is three-value logic, or more informally, Logical Positivism. You can add more and more values, and assign them to whatever you like. At the extreme is Fuzzy Logic, where statements can have any truth value between 0 and 1. Apparently there’s also something which Bayes is just a special case of, but I can’t recall the name.
Of all these possible mental cartography tools though, Bayes seems to be the most versatile. I’m only dimly aware of the ones I mentioned, and probably explained them a little wrong. Anyone care to share thoughts on these, or share others they may know? Has anyone tried to build a complete ontology out of them the way Eliezer did with Bayes? Are there other strong metaphysical theories from philosophy which don’t have a formal mathematical corollary (yet)?
I must admit to some amount of silliness – the first thought I had upon stumbling onto LessWrong, some time ago, was: “wait, if probability does not exist in the territory, and we want to optimize the map to fit the territory, then shouldn’t we construct non-probabilistic maps?” Indeed, if we actually wanted our map to fit the territory, then we would not allow it to contain uncertainty – better some small chance of having the right map, then no chance, right? Of course, in actuality, we don’t believe that (p with x probability) with probability 1. We do not distribute our probability-mass over actual states of reality, but rather, over models of reality; over maps, if you will! I find it helpful to visualize two levels of belief: on the first level, we have an infinite number of non-probabilistic maps, one of which is entirely correct and approximates the territory as well as a map possibly can. On the second level, we have a meta-map, which is the one we update; it consists of probability distributions over the level-one maps. What are we actually optimizing the level-two map for, though? I find it misleading to talk of “fitting the territory”; after all, our goal is to keep a meta-map that best reflects the state of the data we have access to. We alter our beliefs based (hopefully!) on evidence, knowing full well that this will not lead us to a perfect picture of reality, and that a probabilistic map can never reflect the territory.
I think a concrete example is good for explaining this concept. Imagine you flip a coin and then put your hand over it before looking. The state of the coin is already fixed on one value. There is no probability or randomness involved in the real world now. The uncertainty of it’s value is entirely in your head.
Sure; including probability in the map means admitting that it is a map (or a meta-map as you called it).
I rather like this way of thinking. Clever intuition pump.
Hmmm, I guess we’re optimizing out meta-map to produce accurate maps. It’s mental cartography, I guess. I like that name for it.
So, Occam’s Razor and formal logic are great tools of philosophical cartographers. Scientists sometimes need a sharper instrument, so they crafted Solomonoff induction and Bayes’ theorem.
Formal logic being a special case of Bayesian updating, where only p=0 and p=1 values are allowed. There are third alternatives, though. Instead of binary Boolean logic, where everything most be true or false, it might be useful to use a 3rd value for “undefined”. This is three-value logic, or more informally, Logical Positivism. You can add more and more values, and assign them to whatever you like. At the extreme is Fuzzy Logic, where statements can have any truth value between 0 and 1. Apparently there’s also something which Bayes is just a special case of, but I can’t recall the name.
Of all these possible mental cartography tools though, Bayes seems to be the most versatile. I’m only dimly aware of the ones I mentioned, and probably explained them a little wrong. Anyone care to share thoughts on these, or share others they may know? Has anyone tried to build a complete ontology out of them the way Eliezer did with Bayes? Are there other strong metaphysical theories from philosophy which don’t have a formal mathematical corollary (yet)?