It may be useful to actually type out how you use the above thought experiment to explain Bayes. That would make it more useful for those of us still confused or unsure about what Bayes means (hey, I’m a newbie, be nice), and it would help people critique the example in how it teaches the theorem.
For example, why is it better to ask, “If a 3 is pulled, is it more likely to be an 8-sided dice or not?” than to ask, “If a random dice is rolled, is it more likely to be a 3 or not?”
For example, why is it better to ask, “If a 3 is pulled, is it more likely to be an 8-sided dice or not?” than to ask, “If a random dice is rolled, is it more likely to be a 3 or not?”
Good question. The second question is “just a probability” question. The first question asks you to condition on evidence (“If the randomly chosen die is rolled and comes up 3″) and infer “backward” to what this tells you about the die. That’s why Bayesian reasoning applies.
The reasoning goes like this: before I roll the die, the two kinds of dice are equally likely.
P(die is 8 sided)=P(die is only 3′s)=1/2
Then I rolled the die and saw a three. The conditional probability of this if the die is eight sided is 1⁄8. The conditional probability of this if the die is only 3′s is 1.
P(3|die is 8 sided)=1/8
P(3|die is only 3′s)=1
The Bayesian update is to multiply out the probability of observing the evidence in the two cases:
P(3 and die is 8 sided)=1/2*1/8=1/16
P(3 and die is only 3′s)=1/2*1=1/2
And then renormalize:
P(die is 8 sided | 3)=(1/16)/(1/2+1/16)=1/9=0.111
P(die is only 3′s | 3)=(1/2)/(1/2+1/16)=8/9=0.888
JQuinton mentioned that he uses this to argue about falsifiability. I’d like to hear that explained more. I think the example is meant to show that a hypothesis that “can explain anything” (the 8 sided die), should lose probability if we obtain evidence that is “better explained” by the more specific hypothesis (the 3′s only die).
JQuinton mentioned that he uses this to argue about falsifiability. I’d like to hear that explained more. I think the >example is meant to show that a hypothesis that “can explain anything” (the 8 sided die), should lose probability if >we obtain evidence that is “better explained” by the more specific hypothesis (the 3′s only die).
Yes, that’s correct. The thing I was trying to illustrate is that some hypotheses are more falsifiable than others. A hypothesis that can explain too much data (e.g. a 1,000 sided die) would lose probability to a more restricted hypothesis like a 6 sided die if the numbers 1 − 6 are rolled. The compliment to that is if the numbers 7 − 1,000 are rolled this refutes the idea that the 6 sided die was rolled. Accounting for too much data and falsifiability are two sides of the same coin; explaining too much data tends towards unfalsfiability.
It may be useful to actually type out how you use the above thought experiment to explain Bayes. That would make it more useful for those of us still confused or unsure about what Bayes means (hey, I’m a newbie, be nice), and it would help people critique the example in how it teaches the theorem.
For example, why is it better to ask, “If a 3 is pulled, is it more likely to be an 8-sided dice or not?” than to ask, “If a random dice is rolled, is it more likely to be a 3 or not?”
Good question. The second question is “just a probability” question. The first question asks you to condition on evidence (“If the randomly chosen die is rolled and comes up 3″) and infer “backward” to what this tells you about the die. That’s why Bayesian reasoning applies.
The reasoning goes like this: before I roll the die, the two kinds of dice are equally likely.
P(die is 8 sided)=P(die is only 3′s)=1/2
Then I rolled the die and saw a three. The conditional probability of this if the die is eight sided is 1⁄8. The conditional probability of this if the die is only 3′s is 1.
P(3|die is 8 sided)=1/8
P(3|die is only 3′s)=1
The Bayesian update is to multiply out the probability of observing the evidence in the two cases:
P(3 and die is 8 sided)=1/2*1/8=1/16
P(3 and die is only 3′s)=1/2*1=1/2
And then renormalize:
P(die is 8 sided | 3)=(1/16)/(1/2+1/16)=1/9=0.111
P(die is only 3′s | 3)=(1/2)/(1/2+1/16)=8/9=0.888
JQuinton mentioned that he uses this to argue about falsifiability. I’d like to hear that explained more. I think the example is meant to show that a hypothesis that “can explain anything” (the 8 sided die), should lose probability if we obtain evidence that is “better explained” by the more specific hypothesis (the 3′s only die).
Yes, that’s correct. The thing I was trying to illustrate is that some hypotheses are more falsifiable than others. A hypothesis that can explain too much data (e.g. a 1,000 sided die) would lose probability to a more restricted hypothesis like a 6 sided die if the numbers 1 − 6 are rolled. The compliment to that is if the numbers 7 − 1,000 are rolled this refutes the idea that the 6 sided die was rolled. Accounting for too much data and falsifiability are two sides of the same coin; explaining too much data tends towards unfalsfiability.