is there a simple explanation of the conflict between bayesianism and frequentialism? I have sort of a feel for it from reading background materials but a specific example where they yield different predictions would be awesome. has such already been posted before?
Eliezer's views as expressed in Blueberry's links touch on a key identifying characteristic of frequentism: the tendency to think of probabilities as inherent properties of objects. More concretely, a pure frequentist (a being as rare as a pure Bayesian) treats probabilities as proper only to outcomes of a repeatable random experiment. (The definition of such a thing is pretty tricky, of course.)
What does that mean for frequentist statistical inference? Well, it's forbidden to assign probabilities to anything that is deterministic in your model of reality. So you have estimators, which are functions of the random data and thus random themselves, and you assess how good they are for your purpose by looking at their sampling distributions. You have confidence interval procedures, the endpoints of which are random variables, and you assess the sampling probability that the interval contains the true value of the parameter (and the width of the interval, to avoid pathological intervals that have nothing to do with the data). You have statistical hypothesis testing, which categorizes a simple hypothesis as “rejected” or “not rejected” based on a procedure assessed in terms of the sampling probability of an error in the categorization. You have, basically, anything you can come up with, provided you justify it in terms of its sampling properties over infinitely repeated random experiments.
Here is a more general definition of "pure frequentism" (which includes frequentists such as Reichenbach):
Consider an assertion of probability of the form "This X has probability p of being a Y." A frequentist holds that this assertion is meaningful only if the following conditions are met:
The speaker has already specified a determinate set X of things that actually have or will exist, and this set contains "this X".
The speaker has already specified a determinate set Y containing all things that have been or will be Ys.
The assertion is true if the proportion of elements of X that are also in Y is precisely p.
A few remarks:
The assertion would mean something different if the speaker had specified different sets X and Y, even though X and Y aren't mentioned explicitly in the assertion.
If no such sets had been specified in the preceding discourse, the assertion by itself would be meaningless.
However, the speaker has complete freedom in what to take as the set X containing "this X", so long as X contains X. In particular, the other elements don't have to be exactly like X, or be generated by exactly the same repeatable procedure,
Wait - Bayesians can assign probabilities to things that are deterministic? What does that mean?
Absolutely!
The Bayesian philosophy is that probabilities are about states of knowledge. Probability is reasoning with incomplete information, not about whether an event is "deterministic", as probabilities do still make sense in a completely deterministic universe. In a poker game, there are almost surely no quantum events influencing how the deck is shuffled. Classical mechanics, which is deterministic, suffices to predict the ordering of cards. Even so, we have neither sufficient initial conditions (on all the particles in the dealer's body and brain, and any incoming signals), nor computational power to calculate the ordering of the cards. In this case, we can still use probability theory to figure out probabilities of various hand combinations that we can use to guide our betting. Incorporating knowledge of what cards I've been dealt, and what (if any) are public is straightforward. Incorporating player's actions and reactions is much harder, and not really well enough defined that there is a mathematically correct answer, but clearly we should use that knowledge ...
...shades into decision theory...Models that take longer to compute data probabilities should similarly have a probability penalty, not simply because they're implausible, but because we don't want to use them unless the data force us to.
Whoa! that sounds dangerous! Why not keep the beliefs and costs separate and only apply this penalty at the decision theory stage?
Nice explanation. My only concern is that by the opening statement "aiming low". It makes it difficult to send this article to people without them justifiably rejecting it out of hand as a patronizing act. When the intention for aim low is truly noble, perhaps it is just as accurately described as writing clearly, writing for non-experts, or maybe even just writing an "introduction".
Great, great post. I like that it's more qualitative and philosophical than quantitative, which really makes it clear how to think like a Bayesian. Though I know the math is important, having this kind of intuitive, qualitative understanding is very useful for real life, when we don't have exact statistics for so many things.
Re: "Core tenet 1: For any given observation, there are lots of different reasons that may have caused it."
This seems badly phrased. It is normally previous events that cause observations. It is not clear what it means for a reason to cause something.
I don't know if it belongs here or in a separate post but afaik there is no explanation of the Dutch book argument on Less Wrong. It seems like there should be. Just telling people that structuring your beliefs according to Bayes Theorem will make them accurate might not do the trick for some. The Dutch book argument makes it clear why you can't just use any old probability distribution.
Thanks Kaj,
As I stated in my last post, reading LW often gives me the feeling that I have read something very important, yet I often don't immediately know why what I just read should be important until I have some later context in which to place the prior content.
Your post just gave me the context in which to make better sense of all of the prior content on Bayes here on LW.
It doesn't hurt that I have finally dipped my toes in the Bayesian Waters of Academia in an official capacity with a Probability and Stats class (which seems to be a prerequisite for s...
Possible typo:
A theory about the laws of physics governing the motion of planets, devised by Sir Isaac Newton, or a theory simply stating that the Flying Spaghetti Monster pushes the planets forward>s< with His Noodly Appendage.
In the spirit of aiming low, I don't think you aimed nearly low enough. If I hadn't already read a small amount from the sequences I wouldn't have been able to pick up too much from this article. This reads as a great summary; I am not convinced it is a good explanation.
The rest of this comment is me saying the above in mo...
A frequentist asks, "did you find enough evidence?" A Bayesian asks, "how much evidence did you find?"
Frequentists can be tricky, by saying that a very small amount of evidence is sufficient; and they can hide this claim behind lots of fancy calculations, so they usually get away with it. This makes for better press releases, because saying "we found 10dB of evidence that X" doesn't sound nearly as good as saying "we found that X".
I recently started working through this Applied Bayesian Statistics course material, which has done wonders for my understanding of Bayesianism vs. the bag-of-tricks statistics I learned in engineering school.
Bayesianism is more than just subjective probability; it is a complete decision theory.
A decent summary is provided by Sven Ove Hansson:
- The Bayesian subject has a coherent set of probabilistic beliefs.
- The Bayesian subject has a complete set of probabilistic beliefs.
- When exposed to new evidence, the Bayesian subject changes his (her) beliefs in accordance with his (her) conditional probabilities.
- Finally, Bayesianism states that the rational agent chooses the option with the highest expected utility.
The book this quote is taken from can be downloaded for free here.
"A might be the reason for symptom X, then we have to take into account both the probability that X caused A"
I think you have accidentally swapped some variables there
It seems there are a few meta-positions you have to hold before taking Bayesianism as talked about here; you need the concept of Winning first. Bayes is not sufficient for sanity, if you have, say, an anti-Occamian or anti-Laplacian prior.
What this site is for is to help us be good rationalists; to win. Bayesianism is the best candidate methodology for dealing with uncertainty. We even have theorems that show that in it's domain it's uniquely good. My understanding of what we mean by Bayesianism is updating in the light of new evidence, and updating correctly within the constraints of sanity (cf Dutch books).
The penultimate paragraph about our beliefs isn't about Bayesianism so much as heuristics and biases. Unless you were a Bayesian from birth, for at least part of your life your beliefs evolved in a crazy fashion not entirely governed by Bayes' theorem. It is for this reason that we should be suspicious of the beliefs based on assumptions we've never scrutinized.
Or take the debate we had on 9/11 conspiracy theories. Some people thought that unexplained and otherwise suspicious things in the official account had to mean that it was a government conspiracy. Others considered their prior for "the government is ready to conduct massively risky operations that kill thousands of its own citizens as a publicity stunt", judged that to be overwhelmingly unlikely, and thought it far more probable that something else caused the suspicious things.
Don't forget the prior: "The official account of big conflicts...
Others considered their prior for "the government is ready to conduct massively risky operations that kill thousands of its own citizens as a publicity stunt", judged that to be overwhelmingly unlikely,
Here I have to take objection: you framed it as a publicity stunt but actually 9-11 has shaped everything in the USA: domestic policies, foreign policies, military spending the identity of the nation as a whole(It's US vs. THEM) etc... So there is a lot at stake.
Btw, as far as the willingness of the government to kill its own citzens goes, more...
I think this parenthetical statement should maybe be a footnote or something, because it makes the and part of the sentence too far away from the both part. Or maybe put it in the following sentence? I got a little lost.
Doesn't "Bayesianism" basically boil down to the idea that one can think of beliefs in terms of mathematical probabilities?
Core tenet 3: We can use the concept of probability to measure our subjective belief in something. Furthermore, we can apply the mathematical laws regarding probability to choosing between different beliefs. If we want our beliefs to be correct, we must do so.
Frequently misunderstood. E.g. you have propositions A and B , you mistakenly consider that probably either one of them will happen, and you may give me money if you judge P(A)/P(B) > some threshold.
If both A and B happen to be unlikely, I can use that to make arguments which only prompt you to...
Sub-tenet 1: If you experience something that you think could only be caused by cause A, ask yourself "if this cause didn't exist, would I regardless expect to experience this with equal probability?" If the answer is "yes", then it probably wasn't cause A.
I don't understand this at all - if you experience something that you think could only be caused by A, then the question you're supposed to ask yourself makes no sense whatsoever: absent A, you would expect to never experience this thing, per the original condition! And if the a...
. Further suppose that there are two reasons for why people get headaches: they might have a brain tumor, or they might have a cold.
Or, if you're very unlucky, you could have a headache and a brain tumor.... :3
A brain tumor always causes a headache, but exceedingly few people have a brain tumor. In contrast, a headache is rarely a symptom for cold, but most people manage to catch a cold every single year. Given no other information, do you think it more likely that the headache is caused by a tumor, or by a cold?
Given no other information, we don't know which is more likely. We need numbers for "rarely", "most", and "exceedingly few". For example, if 10% of humans currently have a cold, and 1% of humans with a cold have a heada...
You're missing the point. This post is suitable for an audience whose eyes would glaze over if you threw in numbers, which is wonderful (I read the "Intuitive Explanation of Bayes' Theorem" and was ranting for days about how there was not one intuitive thing about it! it was all numbers! and graphs!). Adding numbers would make it more strictly accurate but would not improve anyone's understanding. Anyone who would understand better if numbers were provided has their needs adequately served by the "Intuitive" explanation.
Agreed, I did not find the "Intuitive Explanation" to be particularly intuitive even after multiple readings. Understanding the math and principles is one thing, but this post actually made me sit up and go, "Oh, now I see what all the fuss is about," outside a relatively narrow range of issues like diagnosing cancer or identifying spam emails.
Now I get it well enough to summarize: "Even if A will always cause B, that doesn't mean A did cause B. If B would happen anyway, this tells you nothing about whether A caused B."
Which is both a "well duh" and an important idea at the same time, when you consider that our brains appear to be built to latch onto the first "A" that would cause B, and then stubbornly hang onto it until it can be conclusively disproven.
That's a "click" right there, that makes retroactively comprehensible many reams of Eliezer's math rants and Beisutsukai stories. (Well, not that I didn't comprehend them as such... more that I wasn't able to intuitively recreate all the implications that I now think he was expecting his readers to take away.)
So, yeah... this is way too important of an idea to have math associated with it in any way. ;-)
Okay, I'm rising to the bait here...
I would really appreciate it if people would be more careful about passing on memes regarding subjects they have not researched properly. This should be a basic part of "rationalist etiquette", in the same way that "wash your hands before you handle food" is part of common eating etiquette.
I say this because I'm finding myself increasingly irritated by casual (and ill-informed) snipes at the 9/11 Truth movement, which mostly tries very hard to be rational and evidence-based:
...Or take the debate we had
Keeping my comments on topic:
may believe it likely that the government did something horrendous, but we realize the evidence is weak and circumstantial
Did you read the actual post about Bayesianism? Part of the point is you're not allowed to do this! One can't both think something is likely and think the evidence is weak and circumstantial! Holding a belief but not arguing for it because you know you don't have the evidence is a defining example of irrationality. If you don't think the government was involved, fine. But if you do you're obligated to defend your belief.
Off Topic: I'm not going to go through every one of your positions but... how long have you been researching the issue? I haven't looked up the answer for every single thing I've heard truthers argue- I don't have the time. But every time I do look something up I find that the truthers just have no idea what they're talking about. And some of the claims don't even pass the blush test. For example, your first "unanswered" question just sounds crazy! I mean, HOLY SHIT! the hijackers names aren't on the manifest! That is huge! And yet, of course they absolutely are on the flight manifests and, indeed, they flew under their own names. Indeed, we even have seating charts. For example, Mohamed Atta was in seat 8D. That's business class, btw.
For example, your first "unanswered" question just sounds crazy! I mean, HOLY SHIT! the hijackers names aren't on the manifest! That is huge! And yet, of course they absolutely are on the flight manifests and, indeed, they flew under their own names. Indeed, we even have seating charts. For example, Mohamed Atta was in seat 8D. That's business class, btw.
This is a crowning moment of awesome.
Well, the main thing that'd cause me to mistrust your judgment there, as phrased, is A8. Pre-9/11, airlines had an explicit policy of not resisting hijackers, even ones armed only with boxcutters, because they thought they could minimize casualties that way. So taking over an airplane using boxcutters pre-9/11 is perfectly normal and expected and non-anomalous; and if someone takes exception to that event, it probably implies that in general their anomaly-detectors are tuned too high.
I also suspect that some of these questions are phrased a bit promptingly, and I would ask others, like, "Do you think that malice is a more likely explanation than stupidity for the level of incompetence displayed during Hurricane Katrina? What was to be gained politically from that? Was that level of incompetence more or less than the level of hypothesized government incompetence that you think is anomalous with respect to 9/11?" and so on.
The problem you have is the one shared by everyone from devotees of parapsychology to people who believe Meredith Kercher was killed in an orgy initiated by Amanda Knox: your prior on your theory is simply way too high.
Simply put, the events of 9/11 are so overwhelmingly more likely a priori to have been the exclusive work of a few terrorists than the product of a conspiracy involving the U.S. government, that the puzzling details you cite, even in their totality, fail to make a dent in a rational observer's credence of (more or less) the official story.
You might try asking yourself: if the official story were in fact correct, wouldn't you nevertheless expect that there would be strange facts that appear difficult to explain, and that these facts would be seized upon by conspiracy theorists, who, for some reason or another, were eager to believe the government may have been involved? And that they would be able to come up with arguments that sound convincing?
I want to stress that it is not the fact that the terrorists-only theory is officially sanctioned that makes it the (overwhelming) default explanation; as the Kercher case illustrates, sometimes the official story is an impl...
"Not silencing skeptical inquiry" is a great-sounding applause light
The main issue with it has been noted multiple times by people like Dawkins: there is an effort asymmetry between plucking a false but slightly believable theory out of thin air, and actually refuting that same theory. Making shit up takes very little effort, while rationally refuting random made-up shit takes the same effort as rationally refuting theories whose refutation yields actual intellectual value. Creationists can open a hundred false arguments at very little intellectual cost, and if they are dismissed out of hand by the scientific establishment they get to cry "suppression of skeptical inquiry".
This feels related to pjeby's recent comments about curiosity. The mere feeling that "there's something odd going on here", followed by the insistence that other people should inquire into the odd phenomenon, isn't valid curiosity. That's only ersatz curiosity. Real curiosity is what ends up with you actually constructing a refutable hypothesis, and subjecting it to at least the kind of test that a random person from the Internet would perform - before actually publishing your hypothesis, and insisting that others should consider it carefully.
Inflicting random damage on other people's belief networks isn't promoting "skeptical inquiry", it's the intellectual analogue of terrorism.
Oh, and to try and make this vaguely on topic: say I was trying to do a Bayesian analysis of how likely woozle is to be right. Should I update on the fact that s/he is citing easily debunked facts like "the hijackers weren't on the passenger manifest", as well as on the evidence presented?
I am "happy to take it as fact" until I find something contradictory. When that happens, I generally make note of both sources and look for more authoritative information. If you have a better methodology, I am open to suggestions.
So your standard of accepting something as evidence is "a 'mainstream source' asserted it and I haven't seen someone contradict it". That seems like you are setting the bar quite low. Especially because we have seen that your claim about the hijackers not being on the passenger manifest was quickly debunked (or at least, contradicted, which is what prompts you to abandon your belief and look for more authoritative information) by simple googling. Maybe you should, at minimum, try googling all your beliefs and seeing if there is some contradictory information out there.
I wasn't intending to be snide; I apologize if it came across that way. I meant it sincerely: Jack found an error in my work, which I have since corrected. I see this as a good thing, and a vital part of the process of successive approximation towards the truth.
I suggest that a better way to convey that might have been "Sorry, I was wrong" rather than &qu...
I guess this is the wrong place for this comment but i don't know where else to put it and after reading the extensive threads on 9/11 below i felt this was a valid point. If someone objects to this being here i'll move it to somewhere more appropriate. It looks like i'm a bit out of date with the discussion anyway.
Firstly I should say i'm still very undecided on the matter. Iv'e heard a lot of convincing evidence for both sides of the story, and I know many intelligent people who's opinion i respect on both sides of the fence. I do however think that it...
This article is an attempt to summarize basic material, and thus probably won't have anything new for the hard core posting crowd. It'd be interesting to know whether you think there's anything essential I missed, though.
You've probably seen the word 'Bayesian' used a lot on this site, but may be a bit uncertain of what exactly we mean by that. You may have read the intuitive explanation, but that only seems to explain a certain math formula. There's a wiki entry about "Bayesian", but that doesn't help much. And the LW usage seems different from just the "Bayesian and frequentist statistics" thing, too. As far as I can tell, there's no article explicitly defining what's meant by Bayesianism. The core ideas are sprinkled across a large amount of posts, 'Bayesian' has its own tag, but there's not a single post that explicitly comes out to make the connections and say "this is Bayesianism". So let me try to offer my definition, which boils Bayesianism down to three core tenets.
We'll start with a brief example, illustrating Bayes' theorem. Suppose you are a doctor, and a patient comes to you, complaining about a headache. Further suppose that there are two reasons for why people get headaches: they might have a brain tumor, or they might have a cold. A brain tumor always causes a headache, but exceedingly few people have a brain tumor. In contrast, a headache is rarely a symptom for cold, but most people manage to catch a cold every single year. Given no other information, do you think it more likely that the headache is caused by a tumor, or by a cold?
If you thought a cold was more likely, well, that was the answer I was after. Even if a brain tumor caused a headache every time, and a cold caused a headache only one per cent of the time (say), having a cold is so much more common that it's going to cause a lot more headaches than brain tumors do. Bayes' theorem, basically, says that if cause A might be the reason for symptom X, then we have to take into account both the probability that A caused X (found, roughly, by multiplying the frequency of A with the chance that A causes X) and the probability that anything else caused X. (For a thorough mathematical treatment of Bayes' theorem, see Eliezer's Intuitive Explanation.)
There should be nothing surprising about that, of course. Suppose you're outside, and you see a person running. They might be running for the sake of exercise, or they might be running because they're in a hurry somewhere, or they might even be running because it's cold and they want to stay warm. To figure out which one is the case, you'll try to consider which of the explanations is true most often, and fits the circumstances best.
Core tenet 1: Any given observation has many different possible causes.
Acknowledging this, however, leads to a somewhat less intuitive realization. For any given observation, how you should interpret it always depends on previous information. Simply seeing that the person was running wasn't enough to tell you that they were in a hurry, or that they were getting some exercise. Or suppose you had to choose between two competing scientific theories about the motion of planets. A theory about the laws of physics governing the motion of planets, devised by Sir Isaac Newton, or a theory simply stating that the Flying Spaghetti Monster pushes the planets forwards with His Noodly Appendage. If these both theories made the same predictions, you'd have to depend on your prior knowledge - your prior, for short - to judge which one was more likely. And even if they didn't make the same predictions, you'd need some prior knowledge that told you which of the predictions were better, or that the predictions matter in the first place (as opposed to, say, theoretical elegance).
Or take the debate we had on 9/11 conspiracy theories. Some people thought that unexplained and otherwise suspicious things in the official account had to mean that it was a government conspiracy. Others considered their prior for "the government is ready to conduct massively risky operations that kill thousands of its own citizens as a publicity stunt", judged that to be overwhelmingly unlikely, and thought it far more probable that something else caused the suspicious things.
Again, this might seem obvious. But there are many well-known instances in which people forget to apply this information. Take supernatural phenomena: yes, if there were spirits or gods influencing our world, some of the things people experience would certainly be the kinds of things that supernatural beings cause. But then there are also countless of mundane explanations, from coincidences to mental disorders to an overactive imagination, that could cause them to perceived. Most of the time, postulating a supernatural explanation shouldn't even occur to you, because the mundane causes already have lots of evidence in their favor and supernatural causes have none.
Core tenet 2: How we interpret any event, and the new information we get from anything, depends on information we already had.
Sub-tenet 1: If you experience something that you think could only be caused by cause A, ask yourself "if this cause didn't exist, would I regardless expect to experience this with equal probability?" If the answer is "yes", then it probably wasn't cause A.
This realization, in turn, leads us to
Core tenet 3: We can use the concept of probability to measure our subjective belief in something. Furthermore, we can apply the mathematical laws regarding probability to choosing between different beliefs. If we want our beliefs to be correct, we must do so.
The fact that anything can be caused by an infinite amount of things explains why Bayesians are so strict about the theories they'll endorse. It isn't enough that a theory explains a phenomenon; if it can explain too many things, it isn't a good theory. Remember that if you'd expect to experience something even when your supposed cause was untrue, then that's no evidence for your cause. Likewise, if a theory can explain anything you see - if the theory allowed any possible event - then nothing you see can be evidence for the theory.
At its heart, Bayesianism isn't anything more complex than this: a mindset that takes three core tenets fully into account. Add a sprinkle of idealism: a perfect Bayesian is someone who processes all information perfectly, and always arrives at the best conclusions that can be drawn from the data. When we talk about Bayesianism, that's the ideal we aim for.
Fully internalized, that mindset does tend to color your thought in its own, peculiar way. Once you realize that all the beliefs you have today are based - in a mechanistic, lawful fashion - on the beliefs you had yesterday, which were based on the beliefs you had last year, which were based on the beliefs you had as a child, which were based on the assumptions about the world that were embedded in your brain while you were growing in your mother's womb... it does make you question your beliefs more. Wonder about whether all of those previous beliefs really corresponded maximally to reality.
And that's basically what this site is for: to help us become good Bayesians.
Your standard of verification seems to be the Wikipedia standard - if you can find a "mainstream" source saying something, then you are happy to take it as fact (provided it fits your case).
I am "happy to take it as fact" until I find something contradictory. When that happens, I generally make note of both sources and look for more authoritative information. If you have a better methodology, I am open to suggestions.
The "Wikipedia standard" seems to work pretty well, though -- didn't someone do a study comparing Wikipedia's accuracy with Encyclopedia Britannica's, and they came out about even?
you rather snidely reply "You win a cookie!", before conceding that it only took a bit of research to find out that the supposed "anomaly" never existed. But then, instead of considering what this means for the quality of all your other evidence, you then sarcastically cite the factoid that "6 of the alleged hijackers have turned up alive" as another killer anomaly, completely ignoring the possibility of identity theft/forged passports!
I wasn't intending to be snide; I apologize if it came across that way. I meant it sincerely: Jack found an error in my work, which I have since corrected. I see this as a good thing, and a vital part of the process of successive approximation towards the truth.
I also did not cite the 6 living hijackers as a "killer anomaly" but specifically said it didn't seem to be worth worrying about -- below the level of my "anomaly filter".
Just as an example of my thought-processes on this: I haven't yet seen any evidence that the "living hijackers" weren't simply people with the same names as some of those ascribed to the hijackers. I'd need to see some evidence that all (or most) of the other hijackers had been identified as being on the planes but none of those six before thinking that there might have been an error... and even then, so what? If those six men weren't actually on the plane, that is a loose end to be explored -- why did investigators believe they were on the plane? -- but hardly incriminating.
If you made a good-faith attempt to verify ALL the facts you rely on (rather than jumping from one factoid to another), I'm confident you would find that most of the "anomalies" have been debunked.
I verify when I can, but I am not paid to do this. This is why my site (issuepedia.org) is a wiki: so that anyone who finds errors or omissions can make their own corrections. I don't know of any other site investigating 9/11 which provides a wiki interface, so I consider this a valuable service (even if nobody else seems to).
For example, you seem to think it's unlikely that hijackers could take over a plane using "only box-cutters", because the pilots were "professionals" who were somehow "trained" to fight and might not have found a knife sufficiently threatening. So you think two unarmed pilots would resist ten men who had knives and had already stabbed flight attendants to show they meant business? Imagine yourself actually facing down ten fanatics with knives.
The idea that this is unlikely is one I have seen repeatedly, and it makes sense to me: if someone came at me with a box-cutter, I'd be tempted to laugh at them even if I wasn't responsible for a plane-load of passengers -- and I've never been good at physical combat. Furthermore, the "Pilots for 9/11 Truth" site -- which is operated by licensed pilots (it has a page listing its members by name and experience) -- backs up this statement.
And that's the best authority I can find. If you can find me an experienced pilot (or a military veteran, for that matter) who thinks that this is nonsense, I would very much like to hear from them.
The rest of your arguments that don't rely on debunked facts are about framing perfectly reasonable trains of events in terms to make them seem unlikely - in Less Wrong terms, "privileging the hypothesis". "How likely is that...
I did that precisely as a counter to someone who was doing the same thing in the other direction -- to show that if you accepted "how likely..." as a valid form of argument, then the case is just as strong (if not stronger) for a conspiracy as it is against.
I do not accept "apparent likeliness" as a valid form of argument, and have said so elsewhere.
Well, since it was piloted by a suicidal hijacker who had been trained to fly a plane, I guess it's not unlikely that it would manouevre to hit the building.
You're missing the point; it would have been much easier to hit the other side, the one that wasn't heavily reinforced -- which would have caused more damage, too. On top of that, the maneuver necessary to turn around and hit the reinforced side was, to all accounts, an extremely difficult one which many experienced pilots would hesitate to attempt.
(I suppose one might argue that he overshot and had to turn around; not being skilled, he didn't realize how dangerous this was... so he missed that badly on the first attempt, and yet he was skillful enough to bullseye on the second attempt, skimming barely 10 feet above the ground without even grazing it?)
But that's just one of the "how likely"s, and I shouldn't even be rising to the bait of responding; it's not essential to my main point...
...which, as I have said elsewhere, is this: 9/11 "Truthers" may be wrong, but they are (mostly) not crazy. They have some very good arguments which deserve serious consideration.
Maybe each of their arguments have been successfully knocked down, somewhere -- but I have yet to see any source which does so. All I've been able to find are straw-man attacks and curiosity-stoppers.
I am "happy to take it as fact" until I find something contradictory. When that happens, I generally make note of both sources and look for more authoritative information. If you have a better methodology, I am open to suggestions.
So your standard of accepting something as evidence is "a 'mainstream source' asserted it and I haven't seen someone contradict it". That seems like you are setting the bar quite low. Especially because we have seen that your claim about the hijackers not being on the passenger manifest was quickly debunked...