5 ms·

> Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind). Isn't this only beca

by mhb 5y ago

> Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind). Isn't this only because you have strong priors for the coin and weak ones for Mars life? Suppose you didn't have your example's information about the coin. Then the situations are the same and a new piece of evidence in either case (e.g., discovering whether the coin is round, has two heads, or where its center of gravity is).

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lisper 5y ago
There is no such thing as a "strong" prior or a "weak" prior. A prior is jut a number between 0 and 1. But the coin example is really not very apt because there is potential information that will change your belief about the coin. For example, you might discover that the coin is magnetic, and there is an external field being generated somewhere in just the right way to make the coin land heads every time. Or something like that. Whatever the case, if you get (say) 100 heads in a row you should start to have some serious doubts about your initial analysis of the situation.
- unholiness 5y ago
  But ultimately, it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? So, the prior which is strong/weak is the probability distribution of probability distributions. There is a <<1% chance that I will ever believe there is a >90% chance of flipping heads. But there is a much greater probability that I will one day believe there is a >90% chance of life on Mars. What terminology would you use to describe that, if not strong/weak priors?
  - lisper 5y ago
    > it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? I would say it's the exact opposite. Encountering an unfair coin would not surprise me a bit because I know, for example, that double-headed coins exist. But encountering life on Mars would surprise me a lot because I've accumulated a lot of evidence already that there isn't any. (There might have been in the past, but there isn't any now.) The only thing that I consider "surprising" is encountering evidence that moves one of my priors from being close to 0 to being close to 1 or vice versa. That happens less and less as time goes by because (I hypothesize) my beliefs converge to reality, and reality is stable in the sense that there is something out there towards which it is possible to converge. But at no stage in this process are my beliefs about propositions modeled by anything other than numbers between 0 and 1, with 0 meaning, "I am absolutely certain this proposition is false", 1 meaning, "I am absolutely certain this proposition is true", and 0.5 meaning, "I have absolutely no idea whether or not this proposition is true."
    - goatlover 5y ago
      How does this work for the reasoning you just gave? Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1? Apply that to any philosophical or mathematical reasoning. Let's take an interpretation of QM. How would you assign a probability to Many Worlds Interpretation? Do you have some way of assigning your degree of belief between 0 and 1? Or are there just arguments you find more or less convincing for it? There was a thread on here the other day about black hole information paradox and how we may never be able to test any mathematical reasoning regarding what happens to the information inside a black hole. Can you assign a Bayesian probability to that?
      - lisper 5y ago
        > Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1? I think you misinterpreted what I said. My belief in Bayesian reasoning was very nearly 1 from the first time I encountered Bayes's theorem. Nothing I've seen since has changed this, i.e. nothing about Bayes's theorem has surprised me. The last time I can remember being surprised (in a Bayesian sense) about anything was when Donald Trump won the 2016 election. My prior on his winning was very close to zero when he announced his candidacy (and frankly I think his was too), and obviously my posterior after the election was very close to 1. (Contrast this with the reasoning currently being deployed by proponents of the Big Lie.) That event really rocked my world, but I can't think of anything that has surprised me in the same way since then. > How would you assign a probability to Many Worlds Interpretation? The same way you assign a probability to any proposition: you look at the evidence and decide whether the MWI is a good explanation that accounts for all the evidence. I did a very deep dive into this exact question about three years ago: https://blog.rongarret.info/2019/07/the-trouble-with-many-worlds.html https://blog.rongarret.info/2019/07/the-trouble-with-many-wo... > Can you assign a Bayesian probability to that? It depends on what you mean by "that". You can assign a Bayesian probability to any proposition, but you have to be careful because not all statements are propositions. For example: "Penguins are beautiful" is not a proposition. It makes no sense to say that it is "true" or "false" because it is ambiguous (do you mean all penguins or just some penguins?) and it involves a value judgement. The MWI is particularly thorny because it is really easy to hide ambiguity and value judgements in seemingly innocuous language. For example, you will often hear proponents of the MWI say something like, "The universe splits every time a quantum measurement is made." But that is ambiguous because it doesn't tell you what a quantum measurement is. It also has a tacit "value judgement" hidden in the phrase "the universe". The whole point of the MWI is that there is not one universe, there are unfathomably many of them. So which universe is "the" universe? Well, to me, "the" universe is the universe that "I" live in. But what does the word "I" mean in the context of the MWI? Because according to the MWI, every time the universe splits, I split along with it and become multiple me's in multiple universes. So which "me" is the real me? The MWI is very strongly analogous to the beauty of penguins.
- jwarden 5y ago
  Priors are not just numbers between 0 and 1. They are probability distributions. If they were not, it would be impossible to update beliefs based on evidence. My prior beliefs about the coin include the possibility that the coin has a bias other than 50%. This distribution may be centered at 50%, but I may assign more or less probability to extreme biases, and this determines how I update my belief when I observe several heads in a row.
  - lisper 5y ago
    Of course it is possible to update beliefs based on evidence if priors are numbers. If I flip a coin but don't show you the outcome, you will rationally assign a prior of 0.5 to the proposition "The coin landed heads-up". If I then show you that the coin did indeed land heads-up (or tails-up) you will update your assessment of the probability of "The coin landed heads up" to something close to 1 (or 0). Now, it is possible to do Bayesian analysis on probability distributions rather than "bare" priors, but that is just a way of assigning priors to a family of propositions rather than single propositions. For example, "This is a fair coin" is not a proposition to which one can assign a prior because it isn't a well-defined proposition. The intuition behind "fair coin" is something like "the ratio of heads to tails will approach 1 as the number of flips goes to infinity". But no coin can be flipped an infinite number of times, so this is a meaningless definition. Furthermore, consider a coin that always landed on the face opposite that of the previous flip. The ratio of heads to tails for such a bizarre coin would indeed approach 1 as the number of flips approached infinity, but that is not what is generally meant by a "fair coin". So the usual way of dealing with coins is assigning priors to the family of propositions, "If I flip this coin N times the probability that I will get M heads is X" for all possible values of N and M. But each of the propositions in that family gets a prior which is a number.
    - cf 5y ago
      That framing actually stops holding for continuous distributions. Like in a distribution on heights of people there isn't a probability or prior associated with a particular height. Probabilities are assigned to subsets of this distribution (sometimes called sigma algebras). Priors stands for prior distribution and it really is best to see Bayesian analysis as a machine that consumes probability distributions as input and spits out probability distributions as output.
- deleted 5y ago
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by mhb 5y ago

deleted 5y ago

[deleted]

lisper 5y ago

There is no such thing as a "strong" prior or a "weak" prior. A prior is jut a number between 0 and 1. But the coin example is really not very apt because there is potential information that will change your belief about the coin. For example, you might discover that the coin is magnetic, and there is an external field being generated somewhere in just the right way to make the coin land heads every time. Or something like that. Whatever the case, if you get (say) 100 heads in a row you should start to have some serious doubts about your initial analysis of the situation.

unholiness 5y ago
But ultimately, it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? So, the prior which is strong/weak is the probability distribution of probability distributions. There is a <<1% chance that I will ever believe there is a >90% chance of flipping heads. But there is a much greater probability that I will one day believe there is a >90% chance of life on Mars. What terminology would you use to describe that, if not strong/weak priors?
- lisper 5y ago
  > it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? I would say it's the exact opposite. Encountering an unfair coin would not surprise me a bit because I know, for example, that double-headed coins exist. But encountering life on Mars would surprise me a lot because I've accumulated a lot of evidence already that there isn't any. (There might have been in the past, but there isn't any now.) The only thing that I consider "surprising" is encountering evidence that moves one of my priors from being close to 0 to being close to 1 or vice versa. That happens less and less as time goes by because (I hypothesize) my beliefs converge to reality, and reality is stable in the sense that there is something out there towards which it is possible to converge. But at no stage in this process are my beliefs about propositions modeled by anything other than numbers between 0 and 1, with 0 meaning, "I am absolutely certain this proposition is false", 1 meaning, "I am absolutely certain this proposition is true", and 0.5 meaning, "I have absolutely no idea whether or not this proposition is true."
  - goatlover 5y ago
    How does this work for the reasoning you just gave? Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1? Apply that to any philosophical or mathematical reasoning. Let's take an interpretation of QM. How would you assign a probability to Many Worlds Interpretation? Do you have some way of assigning your degree of belief between 0 and 1? Or are there just arguments you find more or less convincing for it? There was a thread on here the other day about black hole information paradox and how we may never be able to test any mathematical reasoning regarding what happens to the information inside a black hole. Can you assign a Bayesian probability to that?
    - lisper 5y ago
      > Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1? I think you misinterpreted what I said. My belief in Bayesian reasoning was very nearly 1 from the first time I encountered Bayes's theorem. Nothing I've seen since has changed this, i.e. nothing about Bayes's theorem has surprised me. The last time I can remember being surprised (in a Bayesian sense) about anything was when Donald Trump won the 2016 election. My prior on his winning was very close to zero when he announced his candidacy (and frankly I think his was too), and obviously my posterior after the election was very close to 1. (Contrast this with the reasoning currently being deployed by proponents of the Big Lie.) That event really rocked my world, but I can't think of anything that has surprised me in the same way since then. > How would you assign a probability to Many Worlds Interpretation? The same way you assign a probability to any proposition: you look at the evidence and decide whether the MWI is a good explanation that accounts for all the evidence. I did a very deep dive into this exact question about three years ago: https://blog.rongarret.info/2019/07/the-trouble-with-many-worlds.html https://blog.rongarret.info/2019/07/the-trouble-with-many-wo... > Can you assign a Bayesian probability to that? It depends on what you mean by "that". You can assign a Bayesian probability to any proposition, but you have to be careful because not all statements are propositions. For example: "Penguins are beautiful" is not a proposition. It makes no sense to say that it is "true" or "false" because it is ambiguous (do you mean all penguins or just some penguins?) and it involves a value judgement. The MWI is particularly thorny because it is really easy to hide ambiguity and value judgements in seemingly innocuous language. For example, you will often hear proponents of the MWI say something like, "The universe splits every time a quantum measurement is made." But that is ambiguous because it doesn't tell you what a quantum measurement is. It also has a tacit "value judgement" hidden in the phrase "the universe". The whole point of the MWI is that there is not one universe, there are unfathomably many of them. So which universe is "the" universe? Well, to me, "the" universe is the universe that "I" live in. But what does the word "I" mean in the context of the MWI? Because according to the MWI, every time the universe splits, I split along with it and become multiple me's in multiple universes. So which "me" is the real me? The MWI is very strongly analogous to the beauty of penguins.
jwarden 5y ago
Priors are not just numbers between 0 and 1. They are probability distributions. If they were not, it would be impossible to update beliefs based on evidence. My prior beliefs about the coin include the possibility that the coin has a bias other than 50%. This distribution may be centered at 50%, but I may assign more or less probability to extreme biases, and this determines how I update my belief when I observe several heads in a row.
- lisper 5y ago
  Of course it is possible to update beliefs based on evidence if priors are numbers. If I flip a coin but don't show you the outcome, you will rationally assign a prior of 0.5 to the proposition "The coin landed heads-up". If I then show you that the coin did indeed land heads-up (or tails-up) you will update your assessment of the probability of "The coin landed heads up" to something close to 1 (or 0). Now, it is possible to do Bayesian analysis on probability distributions rather than "bare" priors, but that is just a way of assigning priors to a family of propositions rather than single propositions. For example, "This is a fair coin" is not a proposition to which one can assign a prior because it isn't a well-defined proposition. The intuition behind "fair coin" is something like "the ratio of heads to tails will approach 1 as the number of flips goes to infinity". But no coin can be flipped an infinite number of times, so this is a meaningless definition. Furthermore, consider a coin that always landed on the face opposite that of the previous flip. The ratio of heads to tails for such a bizarre coin would indeed approach 1 as the number of flips approached infinity, but that is not what is generally meant by a "fair coin". So the usual way of dealing with coins is assigning priors to the family of propositions, "If I flip this coin N times the probability that I will get M heads is X" for all possible values of N and M. But each of the propositions in that family gets a prior which is a number.
  - cf 5y ago
    That framing actually stops holding for continuous distributions. Like in a distribution on heights of people there isn't a probability or prior associated with a particular height. Probabilities are assigned to subsets of this distribution (sometimes called sigma algebras). Priors stands for prior distribution and it really is best to see Bayesian analysis as a machine that consumes probability distributions as input and spits out probability distributions as output.
deleted 5y ago
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