In many discussions I'm pulled back to the distinction between not-guilty and innocent as a way to demonstrate how the burden of proof works and what the true default position should be in any given argument. A lot of people seem to not have any problem seeing the distinction, but many intelligent people for some reason don't see it.
In this article I explain why the distinction exists and why it matters, in particular why it matters in real-life scenarios, especially when people try to shift the burden of proof.
Essentially, in my view the universe we are talking about is {uncertain,guilty,innocent}, therefore not-guilty is guilty', which is {uncertain,innocent}. Therefore innocent ⇒ not-guilty, but not-guilty ⇏ innocent.
When O. J. Simpson was acquitted, that doesn’t mean he was found innocent, it means the prosecution could not prove his guilt beyond reasonable doubt. He was found not-guilty, which is not the same as innocent. It very well could be that the jury found the truth of the matter uncertain.
This notion has implications in many real-life scenarios when people want to shift the burden of proof if you reject a claim when it's not substantiated. They wrongly assume you claim their claim is false (equivalent to innocent), when in truth all you are doing is staying in the default position (uncertain).
Rejecting the claim that a god exists is not the same as claim a god doesn't exist: it doesn't require a burden of proof because it's the default position. Agnosticism is the default position. The burden of proof is on the people making the claim.
Jump in the discussion.
No email address required.
Notes -
The correct answer depends on what the question is.
If the question is "what's the color of that thing you last saw 5 days ago?", Bayesians would be just like you and answer "blue" and not "sky blue #020fe8".
When you ask "how will the next coin toss land?", an answer disregarding uncertainty would be "it will be heads". An answer that takes uncertainty into account could be "I'm almost sure it will be heads", or "I suspect it will be tails", or "I haven't got a clue". A Bayesian would phrase those as "95%" (almost sure heads), "40%" (suspect tails), or "50%" (no idea).
In Bayesianese, answering that specific question with "50%+-50%" would mean something like "I have no clue if I have a clue whether the next coin toss will be heads or tails", which sounds weird. So I am inferring that you mean "50%+-50%" as an answer the a slightly different question, such as "how frequently would this coin land heads over many tosses?". Which one may phrase as "what's the probability that this coin comes up heads if I toss it?"; but then with this phrasing, a (subjective) Bayesian during a nitpicky philosophical discussion might parse it as "how will the next coin toss land (please, answer in a way that conveys your level of uncertainty)?". That's why I suspect there was talking past each other in your discussions with other people.
But that's not Bayesian. That's the whole point. And you accepted they use a single number to answer.
You: They use a single number for probabilities. They should use 2 like 50%+-20%
Me: Yes, they use a single number. No they shouldn't use 2 when they interpret probability as meaning subjective uncertainty. They should if they interpret it to mean something obejctive.
You: They don't learn from multiple coin tosses, they would need more than one number for that.
Me: They do learn. They use many numbers to compute.
You: They don't take uncertainty into account.
Me: They do, the probability is the uncertainty of the event.
You: 50%+-20% is analogous to saying "blue" whereas saying 50% is analogous to saying "sky blue".
Me: Not if probability means uncertainty. Then 50% maps to "blue", and 50%+-20% maps to nonsense.
You: My answer is correct.
Me: It depends on the question.
I'm not sure what's left here to discuss. I didn't get this follow up.
Right. Which is what that very sentence you half quoted explains.
They don't. The probability that the next coin flip is going to land heads is the same:
0/0,50/50,5000/5000is0.5. It does not get updated.No. It's not.
p=0.5is not the uncertainty.I didn't say that.
Which is not the case.
There is no other question. I am asking a specific question, and the answer is
p=0.5, there's no "it depends".p=0.5is the probability the next coin flip is going to land heads. Period.I'm going to attempt to calculate the values for
nnumber of heads and tails with 95% confidence so there's no confusion about "the question":0/0:0.5±∞5/5:0.5±0.03450/50:0.5±0.0035000/5000:0.5±0.000It should be clear now that there's no "the question". The answer for Bayesians is
p=0.5, and they don't encode uncertainty at all.This is false. Bayesian calculations are quite capable of differentiating between epistemic and aleatory uncertainty. See the first Google result for "Bayes theorem biased coin" for an example.
(edit to add: not a mathematically perfect example; the real calculations here treat a bias as a continuous probability space, where a Bayesian update turns into an integral equation, and instead discretizing into 101 bins so you can use basic algebra is in the grey area between "numerical analysis" and "cheating".)
Did you actually read that? It clearly says:
It's a single value.
I looked at the code of the simulation:
It's a single value.
I printed the variable at the end of the simulation:
It's a single value.
I read, and I understood, and I also looked at the graphs discretized over hundreds of values, and I'm able to understand that when a probability distribution has a mean value in ℝ, that does not mean that the probability distribution itself lies in ℝ, no matter how many times you repeat "single value" after you finish downvoting me. You seem to believe that any space on which a functional can give a single value is a one-dimensional space? This is again false.
Let's try going through an example more slowly:
A uniform prior on an unknown coin bias θ∈[0,1] (p₀≡1) will marginalize to a probability of 1/2 (∫p₀(θ)θdθ) for the next coin flip being heads, for example, and that mean is the exact same "single value" as a delta function (pᴅ≡δ(θ-1/2)) at p=1/2, for example (∫δ(θ-1/2)θdθ), but the delta function will give p=1/4 for the next two flips in a row being heads (∫δ(x-1/2)θ²dθ) and the uniform prior will give p=1/3 (∫p₀(θ)θ²dθ).
Do a Bayesian update on the uniform prior after one flip of heads (pʜ(θ) = θp₀(θ)/∫φp₀(φ)dφ = 2θ), and it'll now say that p(heads next) is 2/3 (∫pʜ(θ)θdθ) and p(2 heads in the next 2 flips) is 1/2 (∫pʜ(θ)θ²dθ); do the same update on the delta function prior and it'll say that the next p(heads) is still 1/2 and the next p(2 heads) is still 1/4, because updating the delta function to θpᴅ(θ)/∫φpᴅ(φ)dφ just gives back the exact same delta function. They started with the exact same "single value", but they updated in different ways because that single value was just an integral of an entire probability distribution function, and Bayesian analysis saves the whole distribution to bring to an update.
Do another Bayesian update after a flip of tails following the flip of heads, and the originally-uniform prior will be back to p(heads next)=1/2 (for a third time, the exact same single value!)... but it won't be back to uniform (pʜᴛ(θ) = (1-θ)pʜ(θ)/∫(1-φ)pʜ(φ)dφ = 6θ(1-θ)); ask the posterior here for p(2 heads on the 3rd and 4th flip) and it'll be at 3/10; it's now squeezing probability mass much closer to fair than the uniform prior did, though it's not the same output at the delta function either.
The difference between a probability distribution and a marginalization of a probability distribution is really important. There's actually a kernel of truth to complaining here: doing Bayes on "the entire universe" is intractable, so in practice we always marginalize away "anything we don't expect to matter" (notice even when representing a uniform prior over coin biases I still haven't accounted for "the coin might land on its edge" or "someone might grab it in midair", or...) when updating, and we have to watch out for accidentally ignoring too much. But just because you have to marginalize away a lot doesn't mean you're required to marginalize away everything!
You're the guy from "2+2≠4" who was having trouble with equivalence classes and Shannon information, right? I was feeling bad about how many people were just downvoting you and moving on, and at that point I wasn't one of them, but now I kind of get it: downvoting falsehoods and moving on is fast, correcting falsehoods is slow, and I fear sometimes dissuading further falsehoods is impossible. I'd love to find out that you can now compute a posterior pʜᴛʜ, apologize to everyone you've been rude to, and start asking questions rather than making false assertions when there's something you don't understand, but if not, well, I guess that is what the downvote button is for after all.
I don't understand what would make you think I believe that.
Yes, so this confirms what I'm saying: the result is a single value. I understand what you are saying: the function that was used to generate this single value is not the same as the original function, but I don't think you are understanding what I am saying.
You are operating from a bottom-up approach: you claim you have a perfectly rational method to arrive to the value of
p=0.5. This method integrates all the observed evidence (e.g 0/0, or 1/1 head/tails), so any new evidence is going to adjust the value correctly, for example 1 head is going to adjust the value differently if there have been 1/1 head/tails versus 10/10 head/tails.I do not disagree that this method arrives to a mathematically correct
p=0.5.But I'm operating from the other side: the top-down approach. If I need to make a decision,
p=0.5does not tell me anything. I don't care if the process that was used to generatep=0.5is "mathematically correct", I want to know how confident should I be in concluding thatpis indeed close to0.5.If in order to make that decision I have to look at the function that was used to generate
p=0.5, then that shows thatp=0.5is not useful, it's the function that I should care about, not the result.My conclusion is the same:
p=0.5is useless.Now, you can say "that's what Bayesians do" (look at the function, not the final value), but as I believe I already explained: I debated Scott Alexander, and his defense for making a shitty decision was essentially: "I arrived to it in a Bayesian way". So, according to him if
p=0.9, that was a "good" decision, even if turned out to be wrong.So, maybe Scott Alexander is a bad Bayesian, and maybe Bayesians don't use this single value to make decisions (I hope so), but I've seen enough evidence otherwise, so I'm not convinced Bayesians completely disregard this final single value. Especially when even the Stanford Encyclopedia of Philosophy explicitly says that's what Bayesians do.
I was not the one having trouble, they were. Or do you disagree that in computing you can't put infinite information in a variable of a specific type?
You are making the assumption that they are falsehoods, when you have zero justification for believing so, especially if you misinterpret what is actually being said.
It's the straightforward interpretation of
If you wanted to say "they don't encode uncertainty-about-uncertainty in the number 0.5", and not falsely imply that they don't encode uncertainty at all (0.5 is aleatory uncertainty integrated over epistemic uncertainty!) or that they don't encode all their uncertainty anywhere, you should have made that true claim instead.
You said of "They use many numbers to compute",
This is flatly false. I just gave you two examples, still at the "toy problem" level even, the first discretizing an infinite-dimensional probability measure and using 101 numbers to compute, the second using polynomials from an infinite-dimensional subspace to compute!
You said,
Which is false, because it ignores that the uncertainty is retained for further updates. That uncertainty is also typically published; that posterior probability measure is found in all sorts of papers, not just those graphs you ignored in the link I gave you. I'm sorry if not everybody calling themselves "Bayesian" always does that (though since you just ignored a link which did do that, you'll have to forgive me for not taking your word about it in other cases).
You said,
This is false. p=0.5 is what you need to combine with utilities to decide how to make an optimal decision without further data. If you have one binary outcome (like the coin flip case) then a single scalar probability does it, you're done. If you have a finite set of outcomes then you need |S|-1 scalars, and if you have an infinite set of outcomes (and/or conditions, if you're allowed to affect the outcome) you need a probability measure, but these are not things that Bayesians never do, they're things that Bayesians invented centuries ago.
This is trivially true in the end with any decision-making system over a finite set of possible decisions. You eventually get to "Do X1" or "Do X2". If you couldn't come up with that index as your result then you didn't make a decision!
If maximizing expected utility, you get that value from plugging marginalized probabilities times utilities and finding a maximum, so you need those probabilities to be scalar values, so scalar values is usually what you publish for decision-makers, in the common case where you're only estimating uncertainties and you're expecting others to come up with their own estimates of utilities. If you expect to get further data and not just make one decision with the data you have, you incorporate that data via a Bayesian update, so you need to retain probabilities as values over a full measure space, and so what you publish for people doing further research is some representation of a probability distribution.
Your title was literally "2 + 2 is not what you think", and as an example you used [2]+[2]=[4] in ℤ/4ℤ (with lazier notation), except you didn't know that there [0]=[4] so you just assumed it was definitively "0", then you wasted a bunch of people's time arguing against undergrad group theory.
What I disagreed with was
And I disagreed with it because it was flatly false! The foundation of information theory is I = -log(P); this is only I=1 (bit) if P=1/2, i.e. equal probability in the case of 1 bit of data. I gave you a case where I=1/6, and a more complicated case where I=0.58 or I=1.58, and you flatly refuted the latter case with "it cannot be more". It can. -log₂(P) does exceed 1 for P<1/2. If I ask you "are you 26 years old", and it's a lucky guess so you say "yes", you've just given me one bit of data encoding about 6 bits of information. The expected information in 1 bit can't exceed 1 (you're probably going to say "no" and give me like .006 bits of information), but that's not the same claim; you can't even calculate the expected information without a weighted sum including the greater-than-1 term(s) in the potential information.
Distinctions are important! If you want to talk like a mathematician because you want to say accurate things then you need to say them accurately; if you want to be hand-wavy then just wave your hands and stop trying to rob mathematics for more credible-sounding phrasing. The credibility does not rub off, especially if instead of understanding the corrections you come up with a bunch of specious rationalizations for why they have "zero justification".
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link