Bayesianism (probabilism, conditioning, priors as mathematical objects):
Let a possible world be a way the world could be. To say something about the world is to say that the actual world (our world) is one of a set of possible worlds. Like, to say that the sky is blue is to say that the actual world is one of the set of possible worlds in which the sky is blue. Some possible worlds might be ours for all we know (maybe they look like ours, at least so far). For others we are pretty sure that they aren’t ours (like all the possible worlds where the sky is pink). Let’s put a number on each possible world that stands for how much we believe that our world is that possible world: 1 means we’re sure it is, 0 means we’re sure it’s not, and everything between 0 and 1 means we’re not sure (higher numbers mean we’re more sure in some way).
Talking about each single possible world is hard and annoying. We want to have the same sort of numbers for sets of possible worlds too, to tell us how much we believe that our world is one of that set of possible worlds. Like, let’s say we’re sure that the sky is blue. This means that the numbers for all possible worlds where the sky is blue add up to 1. Let’s say we’re sure the sky is not pink. This means that the number for each possible world where the sky is pink is 0.
Each time we learn something with our senses, what we believe about the world changes. From what I see with my own eyes, I can definitely tell that the sky looks blue to me today. So I am now sure that the world is one of the set of possible worlds where the sky looks blue to me today. This means that I set my numbers for all the rest of the worlds, where the sky doesn’t look blue to me today, to 0 (I have to take care at this step, since I can’t take it back). Also, since I’m sure the sky looks blue, my new numbers for the possible worlds where the sky looks blue have add up to 1. What I do is I just add together all the old numbers that I just set to 0, and then give a little bit of what I get to each of the other possible worlds, so that worlds that had higher numbers before get more. (This last part is called normal-making. This way of changing our possible-world-numbers is called bringing them up to date.)
What’s interesting is that learning by seeing things and bringing our numbers up to date can also end up changing our numbers for something we can’t see. Let’s say that I start out not sure about whether there is an animal way under my bed where I couldn’t see it (that is, I’m not sure whether our world is one of the possible worlds where an animal is under my bed). I also believe some other things that bring together things I can and can’t see. Like, I believe that if there is an animal under my bed there will probably be noise under the bed (that is, my added-together number for the set of possible worlds where there is noise and an animal under my bed is almost as large as my number for the set of possible worlds where an animal is under my bed). I also believe that if there isn’t an animal under my bed, there will probably not be noise under the bed (my number for the set of worlds where there is noise and there isn’t an animal under my bed is much smaller than my number for the set of worlds where there isn’t an animal under my bed). These sorts of things I believe in part because it’s just the way I am, and in part because of what I’ve seen in the past. Anyway, if I actually listen and hear noise coming from under the bed, and if I bring my numbers up to date, I will end up more sure than I was before that there is an animal under the bed. You can see this if you draw it out like so (ignore the words).
Bayesianism (probabilism, conditioning, priors as mathematical objects):
Let a possible world be a way the world could be. To say something about the world is to say that the actual world (our world) is one of a set of possible worlds. Like, to say that the sky is blue is to say that the actual world is one of the set of possible worlds in which the sky is blue. Some possible worlds might be ours for all we know (maybe they look like ours, at least so far). For others we are pretty sure that they aren’t ours (like all the possible worlds where the sky is pink). Let’s put a number on each possible world that stands for how much we believe that our world is that possible world: 1 means we’re sure it is, 0 means we’re sure it’s not, and everything between 0 and 1 means we’re not sure (higher numbers mean we’re more sure in some way).
Talking about each single possible world is hard and annoying. We want to have the same sort of numbers for sets of possible worlds too, to tell us how much we believe that our world is one of that set of possible worlds. Like, let’s say we’re sure that the sky is blue. This means that the numbers for all possible worlds where the sky is blue add up to 1. Let’s say we’re sure the sky is not pink. This means that the number for each possible world where the sky is pink is 0.
Each time we learn something with our senses, what we believe about the world changes. From what I see with my own eyes, I can definitely tell that the sky looks blue to me today. So I am now sure that the world is one of the set of possible worlds where the sky looks blue to me today. This means that I set my numbers for all the rest of the worlds, where the sky doesn’t look blue to me today, to 0 (I have to take care at this step, since I can’t take it back). Also, since I’m sure the sky looks blue, my new numbers for the possible worlds where the sky looks blue have add up to 1. What I do is I just add together all the old numbers that I just set to 0, and then give a little bit of what I get to each of the other possible worlds, so that worlds that had higher numbers before get more. (This last part is called normal-making. This way of changing our possible-world-numbers is called bringing them up to date.)
What’s interesting is that learning by seeing things and bringing our numbers up to date can also end up changing our numbers for something we can’t see. Let’s say that I start out not sure about whether there is an animal way under my bed where I couldn’t see it (that is, I’m not sure whether our world is one of the possible worlds where an animal is under my bed). I also believe some other things that bring together things I can and can’t see. Like, I believe that if there is an animal under my bed there will probably be noise under the bed (that is, my added-together number for the set of possible worlds where there is noise and an animal under my bed is almost as large as my number for the set of possible worlds where an animal is under my bed). I also believe that if there isn’t an animal under my bed, there will probably not be noise under the bed (my number for the set of worlds where there is noise and there isn’t an animal under my bed is much smaller than my number for the set of worlds where there isn’t an animal under my bed). These sorts of things I believe in part because it’s just the way I am, and in part because of what I’ve seen in the past. Anyway, if I actually listen and hear noise coming from under the bed, and if I bring my numbers up to date, I will end up more sure than I was before that there is an animal under the bed. You can see this if you draw it out like so (ignore the words).