Sociopath Community

AliceInWonderland said: 

I think I understand and I can see the utility of this form of analysis. 

Great! I'm very excited for you and see you got pretty far already. That's pretty impressive :)

The main advantage comes about from the fact that we often are in a situation where we can derive an easy expression from p(A|B) but not for P(B|A). Or the conditional probabilities are simpler ( p(A,B) is a hard expression to write out but p(A|B) and p(B) are easy ). Or we have access to some additional information we usually don't have access to.

It's all about learning how to re-write the conditional probabilities in a way that simplifies the problem, basically.

For example,  what is the probability that a person A will buy a stock at a price x?

p(buy a stock at price x)

Without some appropriate information, it's really difficult to evaluate that probability. However, let's say that we break the problem into smaller parts: The person has to be both willing to buy the stock at price x, and the stock also needs to be available at price x. Let's also say that the person thinks that the fair price of the stock is f, which means he'll buy it in 100% of the cases if x<f.

Then the above is

p(buy at x|f)

=p(willing to buy the stock at price x and the stock is available at price x|f)      (Eq. 1)

However, this is still very tricky to evaluate. We know that *if* the stock is available, he'll buy it if the price is right (x<f), but the above requires also that the stock is available. So what we do know is:

p(willing to buy the stock at price x|the stock is available at price x, f)=Heaviside(x<f)

But we don't know the expression of (Eq. 1). However, we can expand Eq. 1 into a more convenient form:

p(buy at x|f)

=p(willing to buy the stock at price x|the stock is available at price x, f) p(the stock is available at price x|f)

=Heaviside(x<f) * p(the stock is available at price x|f)

=Heaviside(x<f) * p(the stock is available at price x)

Where the stock availability obviously depends only on what is available on the stock market, and not on person A's subjective notion of what a fair price is (which is why I dropped the f). So to evaluate that you can look at the stock market. So now we have an expression we can evaluate easily.

Then you can inspect a population of people (instead of one person) by basically marginalizing over f. So assume some fair price distribution p(f), which however will be largely unknown. Then you can let go of the assumption that you magically know the person's view on a fair price. Or you can use the Bayes' theorem to get the fair price distribution, given that you have observed people's buying habits or something:

p(f|buy at x) = p(buy at x|f) p(f) / p(buy at x).

Edit: Actually, we'd need to include (market data) into that expression to really derive it properly, but you get the idea.

let there be events A and B and conditional probabilities P(A|B) and P(B|A). 

P(A|B) = P(A∩B)/P(B) → P(A∩B) = P(A|B)*P(B) 

P(B|A) = P(A∩B)/P(A) → P(A∩B) = P(B|A)*P(A) 

Hence, P(A|B)*P(B) = P(B|A)*P(A) 

→ P(A|B) = (P(B|A)*P(A))/P(B) and P(B|A) = (P(A|B)*P(B))/P(A) 

Bayes theorem is then a statement of the relationship between two inverse conditional probabilities. The relationship can then be explored my assumption is that this is what Bayesian Analysis is used for, exploring this relationship and using it to make inferences. 

Yes. It's Bayes' theorem. I use Bayesian analysis as a catch-all word to deal with conditional probabilities, where one of the probabilities involve data and we make use of some prior knowledge (so I don't only use it in the context of Bayes' theorem).

That analysis would begin with replacing the marginal probabilities P(A) and P(B) with their joint equivalents. 

That is, 

P(A) = P(A∩B) + P(A∩~B) and P(B) = P(A∩B) + P(~A∩B)

So, 

P(A|B) = (P(B|A)*P(A))/(P(A∩B) + P(A∩~B))

P(B|A) = (P(A|B)*P(B))/(P(A∩B) + P(~A∩B))

Now we sub the joint probabilities with there conditional probabilities, 

P(A∩~B) = P(A|~B) * P(~B)

P(~A∩B) = P(~A|B) * P(~A)

So, 

P(A|B) = (P(B|A)*P(A))/(P(B|A)*P(A) + P(B|~A)*P(~A))

P(B|A) = (P(A|B)*P(B))/(P(A|B)*P(B) + P(A|~B)*P(~B))

The expressions convey that P(A|B) and P(B|A) are therefore ratios between the product of some conditional and prior probability divided by the sum of that product and the sum of some of its conditional in relation to its negation. 

In statistical terms, specifically the use case of analyzing Hypothesis as you've stated, where A and ~A are hypothesis and B is a observation, 

P(B|A) is the probability of observation B given hypothesis A

P(A) is the probability of hypothesis A before observation B 

P(B|~A) is the probability of observation B given hypothesis ~A 

P(~A) is the probability of hypothesis ~A before observation B

Finally, 

P(A|B) is the probability of A given observation B

As such we can compare hypothesis A and ~A given some set of observations B (We can account for a set of observations by summating the denominator) and verify the validity of a given hypothesis through the ratio of these hypothesis when observations B are given. 

Yes, we can compare two hypotheses (it can be A and (not A) or it can be A and B) , and determine which of them is more likely to be correct. It's amazing once you get used to it. Sivia's book is great at illustrating practical applications that are both easy to get started with and interesting.

You've already gotten pretty far. The fundamentals are quite simple, but extremely powerful once you get comfortable with them.

Is this correct and do you have anything to add?

Only notation-wise. I use P(A,B) to state "probability of A and B", and I_Alice to mark prior knowledge accessible to Alice.

For example, you probably now understand the issue with one property of the Inquirer evidence, which used to state, among other things, that everyone has access to the same prior knowledge (before it was re-defined), such that

p(A|I_Alice)=p(A|I_Legga),

which is generally not true.

What can be said is that if we're willing to share data, and I trust you, then the probability evaluation from both our perspectives will be more similar after sharing the data than before sharing the data (Aumann's agreement theorem).

AppleGenius said: 

AliceInWonderland said: 

I think I understand and I can see the utility of this form of analysis. 

Great! I'm very excited for you and see you got pretty far already. That's pretty impressive :)

The main advantage comes about from the fact that we often are in a situation where we can derive an easy expression from p(A|B) but not for P(B|A). Or the conditional probabilities are simpler ( p(A,B) is a hard expression to write out but p(A|B) and p(B) are easy ). Or we have access to some additional information we usually don't have access to. So it's all about identifying the right conditional probabilities (the ones you can write down for the problem).

It's all about learning how to re-write the conditional probabilities in a way that simplifies the problem, basically.

For example,  what is the probability that a person A will buy a stock at a price x?

p(buy a stock at price x)

Without some appropriate information, it's really difficult to evaluate that probability. However, let's say that we break the problem into smaller parts: The person has to be both willing to buy the stock at price x, and the stock also needs to be available at price x. Let's also say that the person thinks that the fair price of the stock is f, which means he'll buy it in 100% of the cases if x<f.

Then the above is

p(willing to buy the stock at price x and the stock is available at price x|f)      (Eq. 1)

However, this is very tricky to evaluate. We know that *if* the stock is available, he'll buy it if the price is right (x<f), but the above requires also that the stock is available. So what we do know is:

p(willing to buy the stock at price x|the stock is available at price x, f)=Heaviside(x<f)

However, we can expand Eq. 1 into a more convenient form:

=p(willing to buy the stock at price x|the stock is available at price x, f) p(the stock is available at price x|f)

=Heaviside(x<f) * p(the stock is available at price x|f)

=Heaviside(x<f) * p(the stock is available at price x)

Where the stock availability obviously depends only on what is available on the stock market, and not on person A's subjective notion of what a fair price is (which is why I dropped the f). So to evaluate that you can look at the stock market.

Then you can let go of the one person assumption by basically marginalizing over a population of people (with some fair price distribution p(f), which however will be largely unknown)

Okay, yes. 

That is essentially what I noticed as the power behind this, given some set of information that you do have access to you can not only use that information to discern the other probabilities through their basic relations but it's also a very efficient means of doing so given the relations through the ratio with is Bayes theorem are made so explicitly. 

That is to say that a specific probability may be extremely difficult quantify do to a lack of information but even if that's so it may be possible to quantify it in a round about way by using information that allows us to quantify another probability that is related to that probability of interest. And, if not, we can through these relationships still say a lot about the system itself. 

let there be events A and B and conditional probabilities P(A|B) and P(B|A). 

P(A|B) = P(A∩B)/P(B) → P(A∩B) = P(A|B)*P(B) 

P(B|A) = P(A∩B)/P(A) → P(A∩B) = P(B|A)*P(A) 

Hence, P(A|B)*P(B) = P(B|A)*P(A) 

→ P(A|B) = (P(B|A)*P(A))/P(B) and P(B|A) = (P(A|B)*P(B))/P(A) 

Bayes theorem is then a statement of the relationship between two inverse conditional probabilities. The relationship can then be explored my assumption is that this is what Bayesian Analysis is used for, exploring this relationship and using it to make inferences. 

Yes. It's Bayes' theorem. I use Bayesian analysis as a catch-all word to deal with conditional probabilities, where one of the probabilities involve data and we make use of some prior knowledge (so I don't only use it in the context of Bayes' theorem).

That analysis would begin with replacing the marginal probabilities P(A) and P(B) with their joint equivalents. 

That is, 

P(A) = P(A∩B) + P(A∩~B) and P(B) = P(A∩B) + P(~A∩B)

So, 

P(A|B) = (P(B|A)*P(A))/(P(A∩B) + P(A∩~B))

P(B|A) = (P(A|B)*P(B))/(P(A∩B) + P(~A∩B))

Now we sub the joint probabilities with there conditional probabilities, 

P(A∩~B) = P(A|~B) * P(~B)

P(~A∩B) = P(~A|B) * P(~A)

So, 

P(A|B) = (P(B|A)*P(A))/(P(B|A)*P(A) + P(B|~A)*P(~A))

P(B|A) = (P(A|B)*P(B))/(P(A|B)*P(B) + P(A|~B)*P(~B))

The expressions convey that P(A|B) and P(B|A) are therefore ratios between the product of some conditional and prior probability divided by the sum of that product and the sum of some of its conditional in relation to its negation. 

In statistical terms, specifically the use case of analyzing Hypothesis as you've stated, where A and ~A are hypothesis and B is a observation, 

P(B|A) is the probability of observation B given hypothesis A

P(A) is the probability of hypothesis A before observation B 

P(B|~A) is the probability of observation B given hypothesis ~A 

P(~A) is the probability of hypothesis ~A before observation B

Finally, 

P(A|B) is the probability of A given observation B

As such we can compare hypothesis A and ~A given some set of observations B (We can account for a set of observations by summating the denominator) and verify the validity of a given hypothesis through the ratio of these hypothesis when observations B are given. 

Yes, we can compare two hypotheses (it can be A and (not A) or it can be A and B) , and determine which of them is more likely to be correct. It's amazing once you get used to it. Sivia's book is great at illustrating practical applications that are both easy to get started with and interesting.

You've already gotten pretty far. The fundamentals are quite simple, but extremely powerful once you get comfortable with them.

Is this correct and do you have anything to add?

Only notation-wise. I use P(A,B) to state "probability of A and B", and I_Alice to mark prior knowledge of Alice.

For example, you probably now understand the issue with one property of the Inquirer evidence, which used to state that everyone has access to the same prior knowledge (before it was re-defined), such that

p(A|I_Alice)=p(A|I_Legga).

What can be said is that if we're willing to share data, and I trust you, then the probability evaluation from both our perspectives will be more similar after sharing the data than before sharing the data (Aumann's agreement theorem).

 Yes, in fact my exploration of this over the past few days has made that specific interaction on SC all the more funnier and I   understand your frustration a bit more. 

https://en.wikipedia.org/wiki/Aumann%27s_agreement_theorem 

'Cannot agree to disagree' or 'reach agreement efficiently' I'll have to read more into this, maybe I'll go through Aumanns' paper. 

If I understand it correctly from this brief glance at the wiki page, the difficulty with the Inq debate is the continuous redefining of evidence which leads to the conclusion that he is not a Rational Bayesian. That is that if evidence is given instead of either party revaluating their hypothesis rationally Inq will instead change the definition of evidence and therefore make all priors useless? 

AliceInWonderland said: 

Okay, yes. 

That is essentially what I noticed as the power behind this, given some set of information that you do have access to you can not only use that information to discern the other probabilities through their basic relations but it's also a very efficient means of doing so given the relations through the ratio with is Bayes theorem are made so explicitly. 

That is to say that a specific probability may be extremely difficult quantify do to a lack of information but even if that's so it may be possible to quantify it in a round about way by using information that allows us to quantify another probability that is related to that probability of interest. And, if not, we can through these relationships still say a lot about the system itself. 

Yep, you're much smarter than me. It took me a long time. And yes, it's also a matter of being able to fold in additional information into your probabilities. For example, your Bayesian analysis of the stock market with and without access to all stock data should give different results.

For example, you probably now understand the issue with one property of the Inquirer evidence, which used to state that everyone has access to the same prior knowledge (before it was re-defined), such that

p(A|I_Alice)=p(A|I_Legga).

What can be said is that if we're willing to share data, and I trust you, then the probability evaluation from both our perspectives will be more similar after sharing the data than before sharing the data (Aumann's agreement theorem).

 Yes, in fact my exploration of this over the past few days has made that specific interaction on SC all the more funnier and I   understand your frustration a bit more. 

https://en.wikipedia.org/wiki/Aumann%27s_agreement_theorem

'Cannot agree to disagree' or 'reach agreement efficiently' I'll have to read more into this, maybe I'll go through Aumanns' paper. 

The issue with Aumann's agreement theory is that it builds upon four premises, none of which are correct:

1. You should trust what you are told as much as you should trust your own eyes.

2. You will be able to exchange all relevant information with one another with perfect accuracy.

3. You are fully rational actors

4. You know that the other person is a fully rational actor

However, if those four premises are true, then indeed Aumann's agreement theorem seems to guarantee convergence of the two evidences towards a singular point. However, the issue is that the third premise contradicts the first and the last one. No rational actor will trust everything they're told, otherwise atheism would be irrational.

So Aumann's agreement theorem states that even if

p(A|I_Alice)!=p(A|I_Legga)

we can share information and then both our priors become I=I_Legga+I_Alice

So that my assessment of A is now p(A|I_Legga, I_alice)

however, this is incorrect. The actual probability is

p(A|I_Legga, I_alice)p(I_alice|I_Legga)

where p(I_alice|I_Legga) now represents, colloquially, my trust in what you say.

If you've been on a space ship, then you might have very good evidence to believe aliens exist. However, if you tell me you've been on a space ship, I only have anecdotal evidence. It will certainly help me in supporting the existence of aliens, but to say that I should assign the same weight to your personal experiences as you should is nonsensical, even if now MissCommunication and Xadem also say they saw you on an alien space ship.

If I understand it correctly from this brief glance at the wiki page, the difficulty with the Inq debate is the continuous redefining of evidence which leads to the conclusion that he is not a Rational Bayesian. That is that if evidence is given instead of either party revaluating their hypothesis rationally Inq will instead change the definition of evidence and therefore make all priors useless? 

So the constant re-definitions were certainly an issue, and as you point out, that will essentially make the analysis useless because his framework doesn't have any predictive power and indeed the priors are useless then. One other problem I pointed out is that he is presuming that anecdotal evidence is equivalent to personal experience, which is simply not correct.

Had he accepted that something you've seen with your own eyes should count as much better evidence than hearsay, he would've needed to admit that John Johnson evidence is better than Inquirer evidence (because it was based on personal experiences/data accessible to Inquirer and the other one was not).

As a consequence, he would be forced to provide John Johnson evidence as per his own stance and promise:

He demanded that I provide Inquirer evidence because Inquirer evidence is more stringent than Legga evidence; applying his own logic, if John Johnson evidence is more stringent than Inquirer evidence, he needs to provide John Johnson evidence -- which he can't and won't.

AppleGenius said: 

AliceInWonderland said: 

Okay, yes. 

That is essentially what I noticed as the power behind this, given some set of information that you do have access to you can not only use that information to discern the other probabilities through their basic relations but it's also a very efficient means of doing so given the relations through the ratio with is Bayes theorem are made so explicitly. 

That is to say that a specific probability may be extremely difficult quantify do to a lack of information but even if that's so it may be possible to quantify it in a round about way by using information that allows us to quantify another probability that is related to that probability of interest. And, if not, we can through these relationships still say a lot about the system itself. 

Yep, you're much smarter than me. It took me a long time. And yes, it's also a matter of being able to fold in additional information into your probabilities. For example, your Bayesian analysis of the stock market with and without access to all stock data should give different results.

For example, you probably now understand the issue with one property of the Inquirer evidence, which used to state that everyone has access to the same prior knowledge (before it was re-defined), such that

p(A|I_Alice)=p(A|I_Legga).

What can be said is that if we're willing to share data, and I trust you, then the probability evaluation from both our perspectives will be more similar after sharing the data than before sharing the data (Aumann's agreement theorem).

 Yes, in fact my exploration of this over the past few days has made that specific interaction on SC all the more funnier and I   understand your frustration a bit more. 

https://en.wikipedia.org/wiki/Aumann%27s_agreement_theorem

'Cannot agree to disagree' or 'reach agreement efficiently' I'll have to read more into this, maybe I'll go through Aumanns' paper. 

The issue with Aumann's agreement theory is that it builds upon four premises, none of which are correct:

1. You should trust what you are told as much as you should trust your own eyes.

2. You will be able to exchange all relevant information with one another with perfect accuracy.

3. You are fully rational actors

4. You know that the other person is a fully rational actor

However, if those four premises are true, then indeed Aumann's agreement theorem seems to guarantee convergence of the two evidences towards a singular point. However, the issue is that the third premise contradicts the first and the last one. No rational actor will trust everything they're told, otherwise atheism would be irrational.

So Aumann's agreement theorem states that even if

p(A|I_Alice)!=p(A|I_Legga)

we can share information and then both our priors become I=I_Legga+I_Alice

So that my assessment of A is now p(A|I_Legga, I_alice)

however, this is incorrect. The actual probability is

p(A|I_Legga, I_alice)p(I_alice|I_Legga)

where p(I_alice|I_Legga) now represents, colloquially, my trust in what you say.

If you've been on a space ship, then you might have very good evidence to believe aliens exist. However, if you tell me you've been on a space ship, I only have anecdotal evidence. It will certainly help me in supporting the existence of aliens, but to say that I should assign the same weight to your personal experiences as you should is nonsensical, even if now MissCommunication and Xadem also say they saw you on an alien space ship.

Interesting enough this theorem seems to have some conceptual overlap with the efficient-market hypothesis as it assumes all of those same premises. I've always had a problem with the idea of rational actors in any social construct as most people don't seem rational as they ill equipped to even process information in a rigorous way and as such are incapable of acting upon it.

Furthermore, there is an outright rejection of most information merely because that information, if were to act upon it rationally, would actually change our hypothesis and inform new ways of acting. Most people do not want to admit that they are wrong and as such would much rather choose to be irrational actor, this is especially true in the markets. 

If I understand it correctly from this brief glance at the wiki page, the difficulty with the Inq debate is the continuous redefining of evidence which leads to the conclusion that he is not a Rational Bayesian. That is that if evidence is given instead of either party revaluating their hypothesis rationally Inq will instead change the definition of evidence and therefore make all priors useless? 

So the constant re-definitions were certainly an issue, and as you point out, that will essentially make the analysis useless because his framework doesn't have any predictive power and indeed the priors are useless then. One other problem I pointed out is that he is presuming that anecdotal evidence is equivalent to personal experience, which is simply not correct.

Had he accepted that something you've seen with your own eyes should count as much better evidence than hearsay, he would've needed to admit that John Johnson evidence is better than Inquirer evidence (because it was based on personal experiences/data accessible to Inquirer and the other one was not).

As a consequence, he would be forced to provide John Johnson evidence as per his own stance and promise:

He demanded that I provide Inquirer evidence because Inquirer evidence is more stringent than Legga evidence; applying his own logic, if John Johnson evidence is more stringent than Inquirer evidence, he needs to provide John Johnson evidence -- which he can't and won't.

I see, he notes two types of evidence as equivalent specifically weighing anecdotal evidence and personal experience (which I assume is empirical) the same and is only willing to accept Inq evidence which is heresay. 

Because he is only willing to accept Inq evidence based on a certain premise that he is applying to you then if you were to provide the same kind of evidence but technically better, so meets all his criteria but is more stringent than his own, than he would have to accept that evidence given his own logic. He will not though and as such this would imply that he either has a irrational belief in himself, does not really understand his own premises, or is being disingenuous. 

Indeed, the Aumann's agreement theory is also related  to information flow, rational actors, and fail for much the same reasons as the efficient market hypothesis. I would argue that everything in Bayesian analysis is related to information flow, which is why I love it so much.

So I guess that the efficient market hypothesis basically proves that whatever money we make off of the stock market is through exploitation of irrational people lol.

You have it right mostly regarding Inquirer evidence, with the exception that he was not rejecting the John Johnson evidence. Instead, he contradicted his own stance by demanding I give him Inquirer evidence because Inquirer evidence is more stringent, but then flipped his stance when I said that he needs to give John Johnson evidence because John Johnson evidence is more stringent than Inquirer evidence.

I only made the demonstration because I wanted to point out an obvious flaw in his logic. Even after a year, he hasn't admitted he made a mistake. In fact, every time Turncoat or I demonstrated how he contradicted himself, he just moved the goalpost and claimed he meant something else. It became kind of funny after the 20th time, when `evidence` meant `reasoning`, `more stringent evidence` meant `more reasonable evidence`, `reasonable` meant `stringent but not too stringent`, and `point me to where Ed said he's Swedish` meant `hmm, I wonder in what context Ed could've said he is Swedish` (these re-definitions occurred every time only it was shown that he contradicted himself).

Indeed, it would either imply that he is disingenuine or doesn't understand the implications of his own stance.

AppleGenius said: 

Indeed, the Aumann's agreement theory is also related  to information flow, rational actors, and fail for much the same reasons as the efficient market hypothesis. I would argue that everything in Bayesian analysis is related to information flow, which is why I love it so much.

The informational paradigm is powerful and becoming all the more dominant in a multitude of fields, and Bayesian Analysis is a really simple and efficient means to not only explore that paradigm but also use it for decision making. 

I do like it. 

So I guess that the efficient market hypothesis basically proves that whatever money we make off of the stock market is through exploitation of irrational people lol.

I don't mind this at all. 

As the old joke goes, if you're playing poker and you can't spot the sucker at the table then you're the sucker. 

You could also poke at Marxs' with this idea, his statement "The history of all hitherto existing society is the history of class struggles" may actually be "The history of all hitherto existing society is the history of irrational struggles". 

You have it right mostly regarding Inquirer evidence, with the exception that he was not rejecting the John Johnson evidence. Instead, he contradicted his own stance by demanding I give him Inquirer evidence because Inquirer evidence is more stringent, but then flipped his stance when I said that he needs to give John Johnson evidence because John Johnson evidence is more stringent than Inquirer evidence.

I only made the demonstration because I wanted to point out an obvious flaw in his logic. Even after a year, he hasn't admitted he made a mistake. In fact, every time Turncoat or I demonstrated how he contradicted himself, he just moved the goalpost and claimed he meant something else. It became kind of funny after the 20th time, when `evidence` meant `reasoning`, `more stringent evidence` meant `more reasonable evidence`, `reasonable` meant `stringent but not too stringent`, and `point me to where Ed said he's Swedish` meant `hmm, I wonder in what context Ed could've said he is Swedish` (these re-definitions occurred every time only it was shown that he contradicted himself).

Indeed, it would either imply that he is disingenuine or doesn't understand the implications of his own stance.

 Okay, I understand you now. 

AliceInWonderland said: 

I don't mind this at all. 

As the old joke goes, if you're playing poker and you can't spot the sucker at the table then you're the sucker. 

You could also poke at Marxs' with this idea, his statement "The history of all hitherto existing society is the history of class struggles" may actually be "The history of all hitherto existing society is the history of irrational struggles".

Haha, I would mind exploiting people. Or maybe not.

I like how you flipped Marx's statement

Bayesianism is fundamentally an epistemic study of belief.

As such you can come to the completely wrong conclusion and then say my belief in the wrong conclusion is justified.

Furthermore it seems to make a lot of assumptions that transcendental realists and empirical idealists make, so it could be said all considering that it's epistemologically flawed but that is a very complicated subject I am not ready to flesh out.

bayesian theory is useless if i cant still have a legal loli bot gf

I thought i was ok at stats. at uni, but I think now I really wasn't as good as i thought. oh well my days of using stats are long behind me but maybe another thing to look into when im bored