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felipec  unbelief  1yr ago (felipec.substack.com)   Edited 1yr ago   2908 thread views

not-guilty is not the same as innocent

felipec.substack.com

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.

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But the coin bias in reality is not true or false.

I'm not asking if the coin is biased, I'm asking if the next coin flip will land heads. It's a yes-or-no question that Bayesians would use a single number to answer.

So, when there's a binary event, like the Russia nuke question, a Bayesian says 6% probability, but a "burden-of-proofer" may say "I think the people that claim Russia will throw a nuke have the burden of proof"

No, I say "I don't know" (uncertain), which cannot be represented with a single probability number.

I'm not asking if the coin is biased, I'm asking if the next coin flip will land heads. It's a yes-or-no question that Bayesians would use a single number to answer.

Yeah

But at first it seems to me you were talking about the bias and what you can learn about it from repeated tosses (and were confused in thinking Bayesians wouldn't learn).

If I throw the coin 100 times and 50 of those it lands head the final percentage is 50%. If I throw the coin 1,000,000 times and 500,000 of those times it land heads it's still 50%, so I have gained zero information.

So like we've talked, they'd use many numbers to compute the probability of the yes-no question, they just give the final answer as one number. Bayesians do consider uncertainty, to all levels they feel they need. What they don't do is give uncertainties about uncertainties in their answers. And they see the probability of next toss heads as equivalent to "how certain am I that it's going to be heads?" (to a Bayesian, probabilities are also uncertainties in their minds, not just facts about the world). Iiuc, you would be happy saying you believe the next toss has 50%±20 chances of being heads. Why not add uncertainty to the 20% too since you are not sure it should be exactly 20%, as in 50%±(20±5)%? If that feels redundand in some sense, that's how a Bayesian feels about saying "coin will come up heads, I'm 50% sure, but I'm only 30% sure of how sure I am.". If it doesn't feel redundant, add another layer until it does :P

No, I say "I don't know" (uncertain), which cannot be represented with a single probability number.

Still, I think I see your point in part. There is clearly some relevant information that's not being given in the answer if the answer to "will this fair coin land heads?", 50%, is the same as the answer given to "plc ashetn ðßh sst?" (well-posed question in a language I just invented), now a lame 50% meaning "the whaat huuhhh?".

If it doesn't feel redundant, add another layer until it does :P

But they don't do that, they give a single number. Whatever uncertainty they had at the beginning is encoded in the number 0.5.

Later on when their decision turns out to be wrong, they claim it wasn't wrong, because they arrived at that number rationally, nobody would have arrived to a better number.

Still, I think I see your point in part. There is clearly some relevant information that's not being given in the answer if the answer to "will this fair coin land heads?

It's not just about the answer is given, it's about how the answer is encoded in your brain.

If the answer to some question is "blue", it may not be entirely incorrect, but later on when you are asked to recall a color you might very well pick any blue color. On the other hand if your answer was "sky blue", well then you might pick a more accurate color.

I claim the correct answer should be 50%±50%, but Bayesians give a single answer: 50%, in which case my answer uncertain is way better.

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.

In Bayesianese, "50%+-50%"

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.

But that's not Bayesian

Right. Which is what that very sentence you half quoted explains.

Me: They do learn. They use many numbers to compute.

They don't. The probability that the next coin flip is going to land heads is the same: 0/0, 50/50, 5000/5000 is 0.5. It does not get updated.

Me: They do, the probability is the uncertainty of the event.

No. It's not. p=0.5 is not the uncertainty.

You: 50%+-20% is analogous to saying "blue" whereas saying 50% is analogous to saying "sky blue".

I didn't say that.

Me: Not if probability means uncertainty.

Which is not the case.

Me: It depends on the question.

There is no other question. I am asking a specific question, and the answer is p=0.5, there's no "it depends".

p=0.5 is the probability the next coin flip is going to land heads. Period.

I'm going to attempt to calculate the values for n number of heads and tails with 95% confidence so there's no confusion about "the question":

  • 0/0: 0.5±∞

  • 5/5: 0.5±0.034

  • 50/50: 0.5±0.003

  • 5000/5000: 0.5±0.000

It 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.

the probability the next coin flip is going to land heads

0/0: 0.5±∞

5/5: 0.5±0.034

50/50: 0.5±0.003

5000/5000: 0.5±0.000

  1. If I ask for the probability that Putin is dead tomorrow, I'd say that fixes the date. You don't move "tomorrow" along with you so it never arrives. After the next coin flip happened, it either was heads or it wasn't, there's nothing left.

  2. There is that word "probability" in the question, so of course how one interprets that word changes the question. If you disagree, give an argument. Instead, you are just repeating that your way of interpreting the word is the only way. I'd ask you to rephrase the question without using the words "the probability/chances/odds" or any such synonym. Then ask how a Bayesian would answer that version of the question, and see if the disagreement persists.

Then ask how a Bayesian would answer that version of the question, and see if the disagreement persists.

I know the definitions of probability, I know what probability is according to a Bayesian, I know what a likelihood function is, and I know what the actual probability of this example is, because I wrote a computer simulation with the actual probability embedded in it.

You are just avoiding the facts.

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They don't. The probability that the next coin flip is going to land heads is the same: 0/0, 50/50, 5000/5000 is 0.5. It does not get updated.

Uff, I even told you how it's done. It's like I just pressed "new chat" on ChatGPT. Re-read or go Google "Bayesian inference coin flipping". It doesn't get more basic that that. I'm moving on, there's no progress to be made.

Uff, I even told you how it's done.

Show me step by step, I'll show you where you are wrong.

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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".)

See the first Google result for "Bayes theorem biased coin" for an example.

Did you actually read that? It clearly says:

P(“Heads”): The evidence term is the overall probability of getting heads and is the sum of all 101 (prior * likelihood) products.

It's a single value.

I looked at the code of the simulation:

evidence = sum(likelihood .* prior);

It's a single value.

I printed the variable at the end of the simulation:

p=0.498334

It's a single value.

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