The Parameters of Bayes' Theorem, Part 1
First off, Thomas Bayes (1701–1761) had a stroke of brilliance in creating his theorem! This is how we wish everyone should think when evaluating claims, events and promises. In a way, one cannot help but be in awe of it. Nothing I say is to indicate otherwise. My only beef is how it's been misused in cases where it shouldn't be used. The formula is below. Notice that the prior probability of event "B" cannot be zero. That sets the major limitation for how Bayes is used. Anything that is given a zero prior probability is not the subject for Bayes' Theorem. Got it? To use it in cases where there is a zero probability is to use it incorrectly. That's the point, not that every claim has a nonzero prior probability to it.
In the following simplified video explaining Bayes' Theorem, take a close look at the 45 second mark.
The narrator says something like this:
"Let's say you know one student in a class of twenty has the flu. Then the prior probability that a student in that class named Sally has the flu, is 1/20. That is your prior probability." Notice you have some factual information, that is, one student in a class of twenty has the flu. This is significant. First comes data, then comes prior probabilities. Bayes is dealing with factual data from the beginning. Without it there is nothing to compute. Compute?
We're told that every claim has a nonzero probability to it, so long as it's not logically contradictory. For a logical contradiction's prior probability is impossible. Something like, "John both exists and doesn't exist in the same sense and the same time when using the word 'exists.'" Okay, got it.
But questions arise. Are the following claims logically impossible?
Can I ride Disney's elephant Dumbo?
When watching a Batman movie can the Joker leap out and attack me?
Who cares? The sane thing to say, unless we live in a cartoon world, is this ain't happening. Does any reasonable person wish to say this is possible? Some will respond by saying things like these can happen in a dream. Well, we're not dreaming. We know when we're awake even if our dreams can deceive us. This is a case I made in my book The Outsider Test for Faith. Nonetheess, we can revise the statements to ask:
When I'm awake, can I ride Disney's elephant Dumbo?
When I'm awake watching a Batman movie, can the Joker leap out and attack me?
If someone wants to say these claims are even remotely possible then quantifying them with numbers is the wrong thing to do with them. Someone may say they're one in a million, another will say one in a billion, still another might say one in a gazillion. How can we know, if no one can agree? Even if someone thinks otherwise, Borel's Law states that in some specific physical examples "phenomena with very low probabilities do not occur," odds below 1 in 10 to the 50 power. So it's impossible that I can ride Disney's elephant Dumbo, or that the Joker can pop out of a movie and attack me. I am closed-minded to these possibilities.
There are other impossibilities.
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Can I run a one second mile today, unaided by any technology and discounting the fact that I'm running on a moving earth traveling many miles per second around the sun?
Can I build a time machine before sundown today and travel back in time to prevent my grandparents from ever meeting, then come back prior to the time I built the time machine?
Can I build a spaceship and travel to the moon and back before sundown today? Hell, I don't have the money to do this, nor the know-how, nor access to the fuel needed without being arrested. I don't think even NASA could do this if the head received a directive to do this completely from scratch, even with the knowledge and materials. NASA wouldn't even attempt it due to safety concerns for the astronaut(s) inside.
Can pigs fly? Can they propel themselves through the sky without the aid of any technology? And for naysayers, a pig is a pig is a pig. They do not have the means to fly. And no, flying doesn't mean being thrown off a cliff, or riding in a plane.
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Those are mundane examples. Anyone who attempts to assign numbers to them is doing nothing more than guessing without any data at all. Data, that's what is lacking here. You need data to begin with, some evidence. There ought to be an example to help set the data, but in these cases and many many others none exists. I invite others to come up with other impossibilities. They abound.
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What does it mean then if we set the prior possibility to zero in these cases? I'm told it means I'm not open to the evidence to the contrary. Well, that's true. Pigs don't fly. How do I know pigs don't fly? They are too heavy and lack any appendage, such as wings, to create any air lift, even the minimal amount needed to glide if pushed off a cliff.
I think setting a zero prior probability means the available evidence I have seen so far, or could conceive of seeing, leaves me thinking something is impossible, not that I cannot be convinced if shown otherwise. Many people can say they couldn't be convinced of something only to later be convinced of it, if the evidence is there.
Now let's say someone can show me pigs do fly, and do so convincingly. Then what would I do? I would revise my calculated prior probability to account for it, and in so doing input a whole new set of numbers into the equation. I would be doing nothing more than what anti-evolutionists did before Darwin, who changed their minds due to the overwhelming evidence for evolution. That's because math does not dictate reality. Reality dictates the mathematical inputs. If reality convinced me against my reasonable close-mindedness about flying pigs, I would admit I was wrong and change my inputs. This is how progress is made. The evidence has a way of convincing closed-minded people. And with regard to pigs flying I need to be convinced. Because they cannot fly. Not today's pigs. Tomorrow's evolved pigs might be something else, but then there will be other actions those evolved pigs cannot do. Nor would they be pigs.
Now as to magical, mythical, superstitious, miraculous cases...later.
The narrator says something like this:
"Let's say you know one student in a class of twenty has the flu. Then the prior probability that a student in that class named Sally has the flu, is 1/20. That is your prior probability." Notice you have some factual information, that is, one student in a class of twenty has the flu. This is significant. First comes data, then comes prior probabilities. Bayes is dealing with factual data from the beginning. Without it there is nothing to compute. Compute?
We're told that every claim has a nonzero probability to it, so long as it's not logically contradictory. For a logical contradiction's prior probability is impossible. Something like, "John both exists and doesn't exist in the same sense and the same time when using the word 'exists.'" Okay, got it.
But questions arise. Are the following claims logically impossible?
Can I ride Disney's elephant Dumbo?
When watching a Batman movie can the Joker leap out and attack me?
Who cares? The sane thing to say, unless we live in a cartoon world, is this ain't happening. Does any reasonable person wish to say this is possible? Some will respond by saying things like these can happen in a dream. Well, we're not dreaming. We know when we're awake even if our dreams can deceive us. This is a case I made in my book The Outsider Test for Faith. Nonetheess, we can revise the statements to ask:
When I'm awake, can I ride Disney's elephant Dumbo?
When I'm awake watching a Batman movie, can the Joker leap out and attack me?
If someone wants to say these claims are even remotely possible then quantifying them with numbers is the wrong thing to do with them. Someone may say they're one in a million, another will say one in a billion, still another might say one in a gazillion. How can we know, if no one can agree? Even if someone thinks otherwise, Borel's Law states that in some specific physical examples "phenomena with very low probabilities do not occur," odds below 1 in 10 to the 50 power. So it's impossible that I can ride Disney's elephant Dumbo, or that the Joker can pop out of a movie and attack me. I am closed-minded to these possibilities.
There are other impossibilities.
-----------------------
Can I run a one second mile today, unaided by any technology and discounting the fact that I'm running on a moving earth traveling many miles per second around the sun?
Can I build a time machine before sundown today and travel back in time to prevent my grandparents from ever meeting, then come back prior to the time I built the time machine?
Can I build a spaceship and travel to the moon and back before sundown today? Hell, I don't have the money to do this, nor the know-how, nor access to the fuel needed without being arrested. I don't think even NASA could do this if the head received a directive to do this completely from scratch, even with the knowledge and materials. NASA wouldn't even attempt it due to safety concerns for the astronaut(s) inside.
Can pigs fly? Can they propel themselves through the sky without the aid of any technology? And for naysayers, a pig is a pig is a pig. They do not have the means to fly. And no, flying doesn't mean being thrown off a cliff, or riding in a plane.
------------------------
Those are mundane examples. Anyone who attempts to assign numbers to them is doing nothing more than guessing without any data at all. Data, that's what is lacking here. You need data to begin with, some evidence. There ought to be an example to help set the data, but in these cases and many many others none exists. I invite others to come up with other impossibilities. They abound.
------------------------
What does it mean then if we set the prior possibility to zero in these cases? I'm told it means I'm not open to the evidence to the contrary. Well, that's true. Pigs don't fly. How do I know pigs don't fly? They are too heavy and lack any appendage, such as wings, to create any air lift, even the minimal amount needed to glide if pushed off a cliff.
I think setting a zero prior probability means the available evidence I have seen so far, or could conceive of seeing, leaves me thinking something is impossible, not that I cannot be convinced if shown otherwise. Many people can say they couldn't be convinced of something only to later be convinced of it, if the evidence is there.
Now let's say someone can show me pigs do fly, and do so convincingly. Then what would I do? I would revise my calculated prior probability to account for it, and in so doing input a whole new set of numbers into the equation. I would be doing nothing more than what anti-evolutionists did before Darwin, who changed their minds due to the overwhelming evidence for evolution. That's because math does not dictate reality. Reality dictates the mathematical inputs. If reality convinced me against my reasonable close-mindedness about flying pigs, I would admit I was wrong and change my inputs. This is how progress is made. The evidence has a way of convincing closed-minded people. And with regard to pigs flying I need to be convinced. Because they cannot fly. Not today's pigs. Tomorrow's evolved pigs might be something else, but then there will be other actions those evolved pigs cannot do. Nor would they be pigs.
Now as to magical, mythical, superstitious, miraculous cases...later.
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