Probability Theory and CTs

[defaultdotxbe]

In other words, regarding 9/11, a 'small-ish' group of people with a very strong disincentive to talk would be quite consistent with a situation fitting the the 9/11 CT ( LIHOP or MIHOP )

how small is "small-ish" and how would such a group go about pulling off 9/11 in the manner you think they did it?

 

[Mel Odious]

Dave Rogers, your response to my post and explanation of the problem was very helpful. Thank you.

 

[Nick Terry]

This is nitpicking, and doesn't challenge the overall premise of the thread, but I don't think that is correct. If your two witnesses give testimony independent of the other, the odds are 75% that at least one of them tells the truth; the probability that both tell the truth is 25%. If you enforce the condition that both must tell the same story, they are either both telling the truth or both lying: probability 50% for each.

The condition is that both have told the same story, and one is starting with a default 50/50 position of neutrality, neither wholly credulous nor wholly sceptical. In this default position of .5, one witness has a 50% chance of being right all on his own. Two witnesses saying the same thing, even with this default position of scepticism, raise the chance to 75%.

That happens to be the level of 'clear and convincing evidence' under US law. It isn't 'beyond reasonable doubt' yet, but more than good enough for most legal situations (you can lose your house and livelihood in a civil court on a 51% probability, the 'balance of probabilities')

This simply models common sense. If one person says something then they may be telling the truth, or they may not. If two people say the same thing, then your willingness to accept the story is raised and you are right to be more credulous. If three people, yet more. In fact, one should cross the 90% threshold very rapidly with only a few more corroborating sources, if one starts with a .5 level of neutrality.

Again, this simply models common sense. If you had five witnesses to a domestic murder, assuming no contradictory evidence - i.e. forensics were not incompatible with the witness claims and the defense could muster no alibi or exculpatory evidence - then this is beyond reasonable doubt.

Under old Roman law, the standard was two witnesses. While open to potential abuses - which is why we ask for other corroboration in the form of forensics, etc - probability theory shows this is not such a bad standard. The same with journalists' two-source rule.


ElMondoHummus

Aren't we misapplying things? Probablity is a method of analyzing random, non-deterministic events. Human behavior has some rather strong deterministic aspects, such as motive, belief systems, prior experiences, etc.

Actually, there are whole branches of philosophy which investigate these issues using probabilistic methods (not least the prisoner's dilemma, but more importantly Bayesianism); the methodology has also been applied to the legal system. A fair bit of psychology and neuroscience has used quantitative methods to reach its conclusions, and thus generates probabilities. Take the cliched pop psychology research study cited in the newspaper, 'scientists have found that men are more likely than women to do x'. That falls under probability.

I think the examples above are useful at the extremes. pomeroo's example was taking an extreme-sceptical position and applying it to the probability that a conspiracy could keep its secrets.

It is true that human behaviour has strong deterministic aspects such as motive, but it is also true that determinism pure and simple is fallacious. There is no such thing as a 100% determinant in human behaviour of the kind we are discussing. Someone will always break the mould sooner or later.

Ron's example took an extreme-sceptical position and applied it to the probability that a conspiracy could keep its secrets. He showed what is already known, that conspiracies do not keep their secrets very well. It is improbable that one can find several 100 conspirators to commit a crime without news of it leaking out in some form.

This is borne out by any adult consideration of American contemporary history, which is littered with examples of whistleblowers and revelations of things that a particular administration intended to keep secret: Watergate is the classic instance, and here we have the quoted figure of 12 conspirators, one whistleblower, which means 8.3% of the conspirators broke solidarity and secrecy.

 

[ElMondoHummus]

ElMondoHummus



Actually, there are whole branches of philosophy which investigate these issues using probabilistic methods (not least the prisoner's dilemma, but more importantly Bayesianism); the methodology has also been applied to the legal system. A fair bit of psychology and neuroscience has used quantitative methods to reach its conclusions, and thus generates probabilities. Take the cliched pop psychology research study cited in the newspaper, 'scientists have found that men are more likely than women to do x'. That falls under probability.

I think the examples above are useful at the extremes. pomeroo's example was taking an extreme-sceptical position and applying it to the probability that a conspiracy could keep its secrets.

It is true that human behaviour has strong deterministic aspects such as motive, but it is also true that determinism pure and simple is fallacious. There is no such thing as a 100% determinant in human behaviour of the kind we are discussing. Someone will always break the mould sooner or later.

Ron's example took an extreme-sceptical position and applied it to the probability that a conspiracy could keep its secrets. He showed what is already known, that conspiracies do not keep their secrets very well. It is improbable that one can find several 100 conspirators to commit a crime without news of it leaking out in some form.

This is borne out by any adult consideration of American contemporary history, which is littered with examples of whistleblowers and revelations of things that a particular administration intended to keep secret: Watergate is the classic instance, and here we have the quoted figure of 12 conspirators, one whistleblower, which means 8.3% of the conspirators broke solidarity and secrecy.

Huh. Okay... I can accept that. I guess my problem with grasping this application of probability is that I'm only used to rather simple, very non-deterministic systems: Coin tosses, dice rolls, and card playing, specifically the various poker games (i.e. what odds are to draw to a given hand, etc.). I never stopped and thought about how probability applies to human actions, given that the degrees of freedom a human has to react to a given situation can be insanely broad, and very difficult to know ahead of time. Coins have 2 sides, dice have 6 (unless you're playing D&D ), there are 52 cards in a deck, those are all fixed ranges, and you can list all the possible outcomes and assign probabilities to them. But I lose comprehension fast when we're talking about human actions in given situations. It's not like you can definitively say a result will fall between 1 and 6 inclusively when the human element of reactions in given situations is discussed.

Yes, poker has a human element too; I never said I was any good at that aspect . That's probably why I'm better at the more "mechanical" variants - i.e. fixed limit games, stud games, etc. - than I am at something like No Limit Hold 'Em. On top of that, a poker game is sort of a closed system; there are only so many reactions a human can make that actually matter.

Anyway, thanks for the info.

 

[maxpower1227]

The condition is that both have told the same story, and one is starting with a default 50/50 position of neutrality, neither wholly credulous nor wholly sceptical. In this default position of .5, one witness has a 50% chance of being right all on his own. Two witnesses saying the same thing, even with this default position of scepticism, raise the chance to 75%.

I still disagree. When dealing with discrete probability spaces, the probability of any event is simply the sum of the individual probabilities that make up the event in question. In this case your sample space should only have two possible outcomes: both told the truth, or both lied. By symmetry those should have equal probabilities.

If you contend that the probability is 75%, can you show me the mathematical justification for that? What would the probability be if 3 people told the same story? Of is n people told the same story? I guess you're taking the probability to be 1-(1/2)^n?