Proof of Immortality, the IV league

[caveman1917]

What? Of course you can, that is the whole point of Bayesian reasoning. My priors can be anything such as "Zeus has it out for me" or "Chewing gum prevents lightning strikes".

That's not a prior, that's a hypothesis, a prior is a probability of a hypothesis. Bayesian inference does not update a hypothesis but its probability.

While a Bayes' formula can include equal and opposite hypotheses, it certainly isn't required to.

Nobody said anything about equal and opposite hypotheses.

 

[jsfisher]

Let's call these two purported uses of "exists" exists_jabba and exists_caveman. The issue with exists_jabba is that it has the property that Jabba can only observe exists_jabba if exists_jabba is indeed true. This property is shared by exists_caveman, hence even if there were further differences between exists_jabba and exists_caveman they are not relevant for this argument.

Not quite. Exists_caveman always included a property (was alive, was the offspring of Alice and Bob, was human) that was entirely relevant.

 

[Loss Leader]

Dave,
- I'd say that we're all talking about the same concept, just that Susan and others see it as an illusion or process rather than a "thing."
- What I'm trying to show is that the likelihood of a particular sense of self (like YOU, or ME -- and whether illusory or real) to be currently existing is infinitesimal -- there being no preexisting pool of potential (even illusory) selves, there is also no limit on their number. And, the likelihood of your current existence -- given OOFLam -- is no more than 7 billion over infinity.

Dave -

Please allow me to give lip service to the word 'illusion' without ever demonstrating any hint of an understanding as to what it means to my argument.

I'll be back.

 

[jsfisher]

Priors are not something which can be taken into account, evidence can be taken into account. If you were consistently using Bayesian inference to arrive at your current beliefs regarding getting hit by lightning (ie posterior becomes new prior as new piece of evidence comes in and so on) then we can iterate backwards to what your priors were before you took the first piece of evidence into account. You can assert no prior distribution for this which has non-maximal entropy without acquiring a burden of proof.

You seem to have a difference understanding of Bayesian inference than I. I was taught that the prior probability was an estimate based on all the available background information. The posterior probability was calculated from the prior adjusted by the impact of new information.

It was not a start from nothing effort of adding data one item at a time.

 

[Hokulele]

That's not a prior, that's a hypothesis, a prior is a probability of a hypothesis. Bayesian inference does not update a hypothesis but its probability.

They would be hypotheses if that is what we are testing. Since we are testing the hypothesis of "I will be struck by lightning when I walk out the door", they are priors. If my prior is "Chewing gum prevents lightning strikes", I will put greater weight on the odds I will not be struck by lightning.

Edit: If I were approaching this is a non-silly manner, my prior would be the rarity of lightning strikes where I live.

 

[caveman1917]

Not quite. Exists_caveman always included a property (was alive, was the offspring of Alice and Bob, was human) that was entirely relevant.

It also always included the required property, namely that a person can only observe "I exist_caveman" if exist_caveman is indeed true, thereby showing that having this property does not make an argument incorrect. If you find fault in Jabba's argument based on lack of relevancy then attack that instead of willy-nilly inserting symbols into equations (changing "P(E | H)" to "P(E | H, E)" etc) because it suits your argument and passing it off as math.

 

[jsfisher]

It also always included the required property, namely that a person can only observe "I exist_caveman" if exist_caveman is indeed true, thereby showing that having this property does not make an argument incorrect.

That isn't the point, though. Jabba selected an existing something (himself) then expressed shock and surprise that the thing selected existed. Your examples didn't feed on themselves that way. It's that other characteristic (e.g. not dead) that was important.

If you find fault in Jabba's argument based on lack of relevancy then attack that instead of willy-nilly inserting symbols into equations (changing "P(E | H)" to "P(E | H, E)" etc) because it suits your argument and passing it off as math.

His notation is only one of his shortcomings. I'd put the inability to define things much higher on the list.

 

[caveman1917]

You seem to have a difference understanding of Bayesian inference than I. I was taught that the prior probability was an estimate based on all the available background information. The posterior probability was calculated from the prior adjusted by the impact of new information.

It was not a start from nothing effort of adding data one item at a time.

Here's how I'm understanding it:

Let F be a sigma-algebra in a probability space.
Let H be a subset of F. (our "hypotheses")
Let I_n be a sequence of subsets of F such that for each i: |I_i| = |I_{i - 1}| + 1 and I_i strict superset of I_{i - 1}. (our "information sets")
Let P_n be a sequence of probability distributions over F, such that P_i is the posterior of the prior P_{i - 1} by applying Bayes' theorem using I_{i} \ I_{i - 1}

I_1 is a chosen set of propositions considered to be true as "background information" and P_1 is chosen to maximize entropy under the constraints of I_1.

Then setting I_1 to anything other than the empty set acquires a burden of proof, since any other possible I_1 must contain at least one proposition asserted to be true. The mere fact that people can decide to agree on a non-empty I_1 and implicitly consider the burden of proof to have been met does not change this.

But this is not the only problem with that reasoning. For one, it uses edge-case behaviour (Bayesian inference when probabilities are exactly 1 or 0) to hide certain properties.

 

[caveman1917]

That isn't the point, though. Jabba selected an existing something (himself) then expressed shock and surprise that the thing selected existed.

And? That does also not make an argument incorrect.

Your examples didn't feed on themselves that way.

They can if you want them to. I didn't specify the likelihoods other than which one is larger than the other one, so you can play around with it as much as you like to get something as shocking and surprising as you want. That's not the point, I might be extremely surprised that the wire wasn't live and I still exist, that doesn't negate the validity of me concluding that wire likely wasn't live.

It's that other characteristic (e.g. not dead) that was important.

What was apparently important was the characteristic of only being observable if true.

His notation is only one of his shortcomings. I'd put the inability to define things much higher on the list.

Be that as it may, this then primarily concerns jt512's notation and ability to define things given that I disputed his claim and not Jabba's.

 

[Hokulele]

Then setting I_1 to anything other than the empty set acquires a burden of proof, since any other possible I_1 must contain at least one proposition asserted to be true. The mere fact that people can decide to agree on a non-empty I_1 and implicitly consider the burden of proof to have been met does not change this.

Incorrect. This is the same mistake Nonpareil was making earlier in the thread. Priors do not have a burden of proof, and depending on the hypothesis under review, can have different values depending on perspective. That is one thing that makes this type of analysis so powerful is that it can be used to calculate posterior probabilities, or backtrack to determine prior probabilities.

Edit: As I mentioned earlier in the thread, this type of analysis was never meant to determine which out of all possible hypotheses is the one that is correct!

 

[marplots]

When I walk out the door, I can either be hit by lightning, or not be hit by lightning. That means that there is a 50% chance I will be hit by lightning when I walk out the door.

Your magnetic personality increases it to 54% in favor of lightning.

 

[Hokulele]

Your magnetic personality increases it to 54% in favor of lightning.

A rather striking observation.

 

[caveman1917]

Incorrect. This is the same mistake Nonpareil was making earlier in the thread. Priors do not have a burden of proof, and depending on the hypothesis under review, can have different values depending on perspective.

False. Using earlier notation, for each possible P_1 there exists an I_1 such that P_1 is the result of applying Bayes' theorem with I_1 to a prior P_0 which is the maximal entropy distribution when I_0 is the empty set. Since every possible I_1 is a subset of F the intersection of the elements of I_1 (that which we are conditioning on) is a subset of the sample space, and hence equivalent to asserting a certain proposition regarding the sample space to be true.

It is generally accepted that the burden of proof is on the person making the claim (asserting a certain proposition to be true), and it is definitely clear that between two persons asserting two different sets of propositions that if one person's set of propositions is a strict superset of the other's that the former person has the burden of proof.

Let's apply your reasoning:
Let H be "God exists".
I choose my prior to be P(H) = 1, since people seem to love abusing this particular edge-case behaviour.

Now for each piece of evidence I can claim that it isn't evidence because it doesn't change my prior, unless that piece of evidence entails that it is absolutely impossible for God to exist, ie P(H | E) = 0. But then that's what you get setting your priors to either 1 or 0, it's not that no Bayesian evidence can exist, it's that you've made yourself effectively impervious to evidence in a Bayesian sense.

 

[jt512]

But that's the same population I used to construct my model to match.

It wasn't even a single population. It compared a sample from the population of one state with a national sample.

I'll try one more example to see if we can agree.
Suppose I sample (using good random technique) the natural numbers. I take a sufficiently large sample (say a few million) and find that about half are evenly divisible by two. I now suggest that the entire population of natural numbers (if infinity bothers us here, we can set some very high limit instead) has this property.

I sample again and get the same results (within some acceptable margin of error).

Now, you seem to be saying the second sampling confirms my hypothesis in some way.

Of course it does.

I'm saying that it adds no new information, since one of the characteristics of my first sampling was that it represents the entire population fairly.

Of course it's new information. It's an independent sample.

The best I might say is I have demonstrated good sampling technique, not that my hypothesis is anything more than a restatement of the results of my sampling.

Huh?

If that's misleading or wrong, perhaps you could give an example which illustrates your point - because I suspect we are talking at cross purposes here.

I did give you an example that illustrates my point. Furthermore, you just came up with your own perfectly good example.

I have no idea how you could possibly think more well-sampled data from the ideal population doesn't give you more evidence. It's a bizarre misconception. You might want to spend less time studying philosophy and more time studying mathematical statistics.

 

[jt512]

How does this qualify as a response?

I think I misunderstood what you wrote. But I did reply more fully to your argument the first time you made it. You can dig up my post if you're interested.

 

[marplots]

I have no idea how you could possibly think more well-sampled data from the ideal population doesn't give you more evidence. It's a bizarre misconception. You might want to spend less time studying philosophy and more time studying mathematical statistics.

Is that the disconnect here? Are you committed to evidence being a mathematical construct, the proper purview of statisticians as if we lived in a world of givens?

This makes me sad.

 

[jt512]

Is that the disconnect here? Are you committed to evidence being a mathematical construct, the proper purview of statisticians as if we lived in a world of givens?

I have no idea what that means (especially what a "world of givens" is and what it means for a statistician to live in one). Maybe there is some sort of evidence that is outside the realm of statistics. My point is that there is evidence that is in the realm of statistics that your preoccupation with philosophy has confused you about.

There are theorems in statistics that show that the larger the sample size, the more information the sample provides about the parameter it is estimating. If you doubt this, then you're confused, and I think we can trace your confusion to your preoccupation with philosophy.

 

[marplots]

I have no idea what that means (especially what a "world of givens" is and what it means for a statistician to live in one). Maybe there is some sort of evidence that is outside the realm of statistics. My point is that there is evidence that is in the realm of statistics that your preoccupation with philosophy has confused you about.

In this case, philosophy has a better grasp on reality than the math. At least in the abstract way you seem keen on using the math.

There are theorems in statistics that show that the larger the sample size, the more information the sample provides about the parameter it is estimating. If you doubt this, then you're confused, and I think we can trace your confusion to your preoccupation with philosophy.

Oh, let's try another story. Why not? They are ever so much fun.

Two ancient scientists are asked by the King to investigate the properties of water. Specifically, the King suggests that water is cold - for some understood meaning of cold, perhaps "causes numbness when a finger is placed in it."

Scientist One walks a half mile to a mountain fed stream and tests the water. It is cold. He tells the King about the evidence he has found.

Scientist Two travels several days away to an oasis he has heard about and finds the water there to be tepid, if not actually warm. He returns and reports to the King as well.

"Ah," says the King, "We have evidence both for and against my hypothesis. What are we to do?"

"Simple," says Scientist One, "I shall return to the stream and draw another sample. In fact, I shall return every day for a week and have seven tests, one each day. For, I have been told by a statistician of merit that resampling counts as new data and provides further evidence."

"Wait," says Scientist Two. "I could do the same, but I have to travel too far. I can never keep up with the number of samples Scientist One can accumulate."

"Tough bananas," says the King. "I don't need you to keep checking anyhow, I can just keep accumulating new evidence from the local stream until we are all convinced by the mounting evidence in favor of my hypothesis. Water is cold, and that's that."

"But," says Scientist Two, "If you are just resampling the same data set, you are going to keep getting the same information over and over again."

Turning to the guard, the King says, "Put him in the dungeon. I suspect he is no scientist at all, but a dreaded philosopher!"

 

[jt512]

In this case, philosophy has a better grasp on reality than the math. At least in the abstract way you seem keen on using the math.

I have no idea how you can think that. Statistics says that the information in a sample is proportional to the sample size. You say it doesn't add any information at all. Go take a sample of size 1 form the population of US voters and use it to predict the outcome of the election. Next, repeat the sample 1 million times, and do the same. Let us know which gave you more information about the preferences of the electorate: 1 sample of size 1 or 1 million samples of size 1.

Oh, let's try another story. Why not? They are ever so much fun.

Two ancient scientists are asked by the King to investigate the properties of water. Specifically, the King suggests that water is cold - for some understood meaning of cold, perhaps "causes numbness when a finger is placed in it."

"Ah," says the King, "We have evidence both for and against my hypothesis. What are we to do?"

The first thing we are to do is to define what water we are talking about.

To be less glib: the problem is ill defined.

 

[marplots]

I have no idea how you can think that. Statistics says that the information in a sample is proportional to the sample size. You say it doesn't add any information at all. Go take a sample of size 1 form the population of US voters and use it to predict the outcome of the election. Next, repeat the sample 1 million times, and do the same. Let us know which gave you more information about the preferences of the electorate: 1 sample of size 1 or 1 million samples of size 1.

One million. Except we were talking about getting new evidence from old data. What sample size should I select from the old data after the election is over? It seems to me that, in that condition, neither one nor a million adds any new evidence at all.

The first thing we are to do is to define what water we are talking about.

All water, everywhere.

To be less glib: the problem is ill defined.

Just so, and why philosophy is a better tool. If I am allowed to abstract reality in a certain manner which fits my formulas, then the math works fine. But if I am not allowed that luxury, it doesn't. The statistics are only as useful as the available input and the way the problem is constructed is part of the input.