6 ms·

Each additional unchecked assumption that is added to a theory is another opportunity to be wrong. These probabilities of failure, however small, are compounded

by tbabb 7y ago

Each additional unchecked assumption that is added to a theory is another opportunity to be wrong. These probabilities of failure, however small, are compounded with multiplication, so the probability of failure rises exponentially with the number of unchecked assumptions. (10 assumptions each with 95% chance of correctness have only a collective 59% chance of being right— .95^10 = .59). To maximize the chance of choosing a model which corresponds to reality, we must involve as few unchecked assumptions as possible when deciding between theories that agree with the data; and this matters more than any particular assumption having a relatively high chance of being right, because of this exponential sensitivity. So if we want theories that are correct, yes, we should value simplicity.

klyrs 7y ago
> So if we want theories that are correct, yes, we should value simplicity. I'm reminded of a problem I was given on a math exam: a hand-drawn polynomial on a 5x5 grid, where the curve intersected the grid in 5 points. "Find the equation of this cubic polynomial." The given integral points weren't possible for any cubic; if the function was a polynomial, it was a quartic at a minimum. Sadly, my teacher was not trying to trick us. Simplicity, itself, was an unchecked assumption which rendered correctness impossible.
vladf 7y ago
I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics. Under the frequentist interpretation of probability, 95% of having a "correct assumption" means that as you observe iid experiments of this assumption, the long-run average of them will have 95/100 of such experiments exhibit the assumption as true. But "experiment" here isn't a physics experiment, it's a _universe_ in which this assumption could be true. We're in some _fixed_ universe. This assumption holds in ours with probability 0 or 1 (we don't know which). So, I think the 95% you're referring to here is a Bayesian belief level. This is coherent in that it doesn't require some ambient set of multiple universes to meaningfully describe what is meant by "probability", but the prior required for this interpretation is a bit of weird beast. That is, we have some prior over the space of all possible assumptions for models of physics and update our beliefs based on evidence (we can collect evidence from our single universe multiple times and update our beliefs in a coherent way). In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question. Within specific settings for statistical learning theory, we find that simple models generalize well. But that's an implication: _if_ you have a class of simple models and it fits the data well, _then_ you'll generalize well. When it comes to answering the question, "Which theories generalize well?", such analysis is incomplete.
- tbabb 7y ago
  > I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics. A sketch: Suppose we have two theories, one ("A") with assumptions (a, b); the other ("B") with assumptions (a, p, q, r, s, t); and our evidence 'e'. Pr(A) ~= Pr(a|e) * Pr(b|e) Pr(B) ~= Pr(a|e) * Pr(p|e) * Pr(q|e) * Pr(r|e) * Pr(s|e) * Pr(t|e) That is, the probability of each theory being right is the probability that each of its assumptions are simultaneously true. It might be difficult to come up with a specific number for each of those component terms, but we'd do well to estimate the total probability of each theory simply by counting the terms, since in the limit that will matter more than the probability of each term (assuming we think none of them are obviously low). Also note that when counting, we can trivially factor out the common assumptions "a" (a.k.a. the things we don't wish to doubt or vary between theories, e.g. QM in "ordinary" regimes, GR, the Newtonian approximation to GR, etc.) Yes, there is a "ground truth" theory which is absolutely true or false, but we don't have access to it. And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different? Example stress tests of this idea: e = the motion of the planets; A := "G = m1*m2/r^2"; B := [long list of epicycle parameters] --> pick "A". e = the varied appearance/adaptations of animals/species; A := [reproduction, inheritance, variation, selection]; B := foreach animal x {"God zotted $x into existence like that because just_so_story($x)" --> pick "A". e = my empty garage; A := "there's nothing in it"; B := "there's a dragon in it; the dragon is invisible; the dragon dodges your touch; the dragon has no heat signature; the dragon floats and leaves no footprints; ..." --> pick "A". e = the behavior of the universe; A := [the standard model]; B := [the standard model; also it's a simulation; there are intelligent beings who set up the simulation; there is an external universe in which the simulation is occurring; the number of simulations happening in this universe is large; ...] --> pick "A".
  - vladf 7y ago
    I think that you may have missed my point. First off, why are assumptions independent? Why are you allowed to factor p(a|e) * p(b|e) = p(a,b|e)? Assumptions aren't just independent binary variables -- it's not necessarily true that you can have some product measure over the set of assumptions (a, b, p, q, r, s, t) simultaneously. More importantly, you dived write into some notation (Pr) without telling me what it _means_, which is what my OP was about. > And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different? Here's a crucial difference. We can roll a die 100 times, 1000 times, 10K times, and the frequency of a six landing as we increase the number of trials will tend to 1/6. That's what we mean (if we're frequentists) when we say the probability of a six landing is 1/6. We can't "roll" universes with physics models.
whatshisface 7y ago
Assumption 1: Whathisface's universal star force lightning theory is true. Assumption 2: No other assumptions. General relativity has a couple postulates, my theory has one. I guess I'm more likely to be right.
- tbabb 7y ago
  I suspect you're being willfully dense. The theory you paint would be composite of other assumptions (involving lighting, forces, or stars, e.g.?) which in turn are either in contradiction with observation (implying Pr=0%), or rely on further (very low probability) assumptions in order to avoid contradiction. If that's not true, then you've just given a strange name to the empty set of assumptions, which is the null hypothesis; and the null hypothesis DOES get epistemological privilege.
  - deleted 7y ago
    [deleted]
glial 7y ago
>> So if we want theories that are correct, yes, we should value simplicity. There is no a priori reason why simple theories should be correct. The suggestion that any model can be correct is presumptuous - you might call it an assumption ;-) To borrow from George Box, all models are wrong, but some are useful...
- deepnotderp 7y ago
  Sure there is! The Universal Distribution is the ultimate prior, see for example, Solomonoff induction: https://en.m.wikipedia.org/wiki/Solomonoff%27s_theory_of_inductive_inference https://en.m.wikipedia.org/wiki/Solomonoff%27s_theory_of_ind...

by tbabb 7y ago

klyrs 7y ago

> So if we want theories that are correct, yes, we should value simplicity. I'm reminded of a problem I was given on a math exam: a hand-drawn polynomial on a 5x5 grid, where the curve intersected the grid in 5 points. "Find the equation of this cubic polynomial." The given integral points weren't possible for any cubic; if the function was a polynomial, it was a quartic at a minimum. Sadly, my teacher was not trying to trick us. Simplicity, itself, was an unchecked assumption which rendered correctness impossible.

vladf 7y ago

I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics. Under the frequentist interpretation of probability, 95% of having a "correct assumption" means that as you observe iid experiments of this assumption, the long-run average of them will have 95/100 of such experiments exhibit the assumption as true. But "experiment" here isn't a physics experiment, it's a _universe_ in which this assumption could be true. We're in some _fixed_ universe. This assumption holds in ours with probability 0 or 1 (we don't know which). So, I think the 95% you're referring to here is a Bayesian belief level. This is coherent in that it doesn't require some ambient set of multiple universes to meaningfully describe what is meant by "probability", but the prior required for this interpretation is a bit of weird beast. That is, we have some prior over the space of all possible assumptions for models of physics and update our beliefs based on evidence (we can collect evidence from our single universe multiple times and update our beliefs in a coherent way). In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question. Within specific settings for statistical learning theory, we find that simple models generalize well. But that's an implication: _if_ you have a class of simple models and it fits the data well, _then_ you'll generalize well. When it comes to answering the question, "Which theories generalize well?", such analysis is incomplete.

tbabb 7y ago
> I don't understand how we can reason with "probabilities of failure" about such fundamental things such as laws of physics. A sketch: Suppose we have two theories, one ("A") with assumptions (a, b); the other ("B") with assumptions (a, p, q, r, s, t); and our evidence 'e'. Pr(A) ~= Pr(a|e) * Pr(b|e) Pr(B) ~= Pr(a|e) * Pr(p|e) * Pr(q|e) * Pr(r|e) * Pr(s|e) * Pr(t|e) That is, the probability of each theory being right is the probability that each of its assumptions are simultaneously true. It might be difficult to come up with a specific number for each of those component terms, but we'd do well to estimate the total probability of each theory simply by counting the terms, since in the limit that will matter more than the probability of each term (assuming we think none of them are obviously low). Also note that when counting, we can trivially factor out the common assumptions "a" (a.k.a. the things we don't wish to doubt or vary between theories, e.g. QM in "ordinary" regimes, GR, the Newtonian approximation to GR, etc.) Yes, there is a "ground truth" theory which is absolutely true or false, but we don't have access to it. And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different? Example stress tests of this idea: e = the motion of the planets; A := "G = m1*m2/r^2"; B := [long list of epicycle parameters] --> pick "A". e = the varied appearance/adaptations of animals/species; A := [reproduction, inheritance, variation, selection]; B := foreach animal x {"God zotted $x into existence like that because just_so_story($x)" --> pick "A". e = my empty garage; A := "there's nothing in it"; B := "there's a dragon in it; the dragon is invisible; the dragon dodges your touch; the dragon has no heat signature; the dragon floats and leaves no footprints; ..." --> pick "A". e = the behavior of the universe; A := [the standard model]; B := [the standard model; also it's a simulation; there are intelligent beings who set up the simulation; there is an external universe in which the simulation is occurring; the number of simulations happening in this universe is large; ...] --> pick "A".
- vladf 7y ago
  I think that you may have missed my point. First off, why are assumptions independent? Why are you allowed to factor p(a|e) * p(b|e) = p(a,b|e)? Assumptions aren't just independent binary variables -- it's not necessarily true that you can have some product measure over the set of assumptions (a, b, p, q, r, s, t) simultaneously. More importantly, you dived write into some notation (Pr) without telling me what it _means_, which is what my OP was about. > And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different? Here's a crucial difference. We can roll a die 100 times, 1000 times, 10K times, and the frequency of a six landing as we increase the number of trials will tend to 1/6. That's what we mean (if we're frequentists) when we say the probability of a six landing is 1/6. We can't "roll" universes with physics models.

whatshisface 7y ago

Assumption 1: Whathisface's universal star force lightning theory is true. Assumption 2: No other assumptions. General relativity has a couple postulates, my theory has one. I guess I'm more likely to be right.

tbabb 7y ago
I suspect you're being willfully dense. The theory you paint would be composite of other assumptions (involving lighting, forces, or stars, e.g.?) which in turn are either in contradiction with observation (implying Pr=0%), or rely on further (very low probability) assumptions in order to avoid contradiction. If that's not true, then you've just given a strange name to the empty set of assumptions, which is the null hypothesis; and the null hypothesis DOES get epistemological privilege.
- deleted 7y ago
  [deleted]

glial 7y ago

>> So if we want theories that are correct, yes, we should value simplicity. There is no a priori reason why simple theories should be correct. The suggestion that any model can be correct is presumptuous - you might call it an assumption ;-) To borrow from George Box, all models are wrong, but some are useful...

deepnotderp 7y ago
Sure there is! The Universal Distribution is the ultimate prior, see for example, Solomonoff induction: https://en.m.wikipedia.org/wiki/Solomonoff%27s_theory_of_inductive_inference https://en.m.wikipedia.org/wiki/Solomonoff%27s_theory_of_ind...