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The bias shield

18 Post author: PhilGoetz 31 December 2011 05:44PM

A friend asked me to get her Bill O'Reilly's new book Killing Lincoln for Christmas.  I read its reviews on Amazon, and found several that said it wasn't as good as another book about the assassination, Blood on the Moon.  This seemed like a believable conclusion to me.  Killing Lincoln has no footnotes to document any of its claims, and is not in the Ford's Theatre national park service bookstore because the NPS decided it was too historically inaccurate to sell.  Nearly 200 books have been written about the Lincoln assassination, including some by professional Lincoln scholars.  So the odds seemed good that at least one of these was better than a book written by a TV talk show host.

But I was wrong.  To many people, this was not a believable conclusion.

(This is not about the irrationality of Fox network fans.  They are just a useful case study.)

One review ended like this:

I hope people are not writing off an honest review because they think I'm picking on O'Reilly. The only POSSIBLE reason that this book took off so fast on the bestseller lists is because it was publicized on the O'Reilly Factor, not because it was so much better than any of the other books written about the Lincoln assassination. There has been much back-and-forth about this for some time. Dishonest people who didn't read the book but hate O'Reilly gave it one-star reviews without ever opening it. O'Reilly fans have an attack of the vapors at anything less than a five-star review. The purpose of this review was to inform, not to express ideology. I stand by this review. If you don't like it, that's fine, but don't attack me simply because you're sticking up for Bill O'Reilly (a futile wish, apparently). Again -- I watch The O'Reilly Factor. I am also a Lincoln scholar. Take this review at face value.

And Amazon readers responded:

Ted says:  My guess is that your review is based on the same thing as every other liberal here... partisan hatred and the huffington post.

Robes says:  More time was spent by Mr. Ford going after O'Reilly than reviewing the merits of the content of the book itself. Smacks of a political agenda by Mr. Ford.  Best to leave serious reviews to the pro's.

(There was a further exchange where the reviewer gave his conservative credentials by saying he had worked for Sarah Palin, and readers responded by calling him a liar, but that has sadly been deleted by Amazon.)

Another review said in part:

It seems there are forces here who believe people who gave this book 1 star reviews are 1. anti-Bill O'Reilly, 2. never read the book, and 3. Partisan. One can follow this same illogical rationale and make the statement that the people who gave this book 5 star reviews are 1. pro-Bill O'Reilly, 2. never read the book, and 3. Partisan.

There are too many mistakes in this book for it to be considered factual. Mary Surratt was never aboard the Montauk; she was never hooded while imprisoned, etc. Daniel Sickles killed his wife's lover, not the husband of his mistress. Booth and Herold spent 5 days in the Pine Thicket, not 6. etc. Since this work is non-fiction, these cannot be cavalierly brushed off as nitpicking or minor details.

To which the readers responded:

Zascha Marie says:  Shoddy and lazy is writing a review on a book you did not read.

Greg E. Garcia says:  Swamp Poodle, please get a clue and try to not let your partisanship shine through. Just because you disagree with someone, does not mean that you have to make up false reviews and smear them. If the book is so bad, then why is it selling so well? I reply to your post not to get a no spin mug, but to try to bring reason into your partisan-addled brain.

Jscichi says:  Not a Verified Purchase so you didn't even read the book.. Typical Left Wing Loon!

Ted says:  Pitiful.... Nothing more than a vacuous, vapid, partisan rant, completely bereft of detail, analysis, example or reference.

The facts in contention were questions such as, When was the Oval Office built? What is the proper spelling of Der(r)inger?1  Was there a hole in the wall or in the door?  The readers defending O'Reilly made it clear that they were conservatives and considered anyone who disagreed with O'Reilly about these facts to be a liberal; yet a conservative bias would not cause one to believe these facts.2

Any of these readers would have been willing to believe that Bill O'Reilly had written a bad book, if they did not believe that Bill O'Reilly was strongly biased.  O'Reilly's strong bias about political matters made him more believable when writing about non-political matters.  He is invulnerable to criticism because he is known to be biased.  This is the Bias Shield:  Beyond some level of bias, the more biased you act, and the more publicly you do it, the more your statements will be taken by the audience you still have as objective and unbiased.  Not because they can't see your bias - because they can see it.  Any objections can be dismissed as ad hominem attacks by people who don't agree with your bias.  Claims by the criticizer to share the same bias will be dismissed as lies.

Optional mathy part

This is an opportunity to start thinking about how to model bias mathematically, for this and other problems.

  1. Take out an n-dimensional piece of graph paper, and label its axes with political, religious, or other biases that are largely independent, so that you can describe a person's biases with a single point.  Each axis will range from -1 to 1.  Person i's biases are described by a point pi.
  2. Define the agreement a(pi,pj) between points pi and pj as the square of the inverse of the length of the vector pipj (a(i,j) = 1 / |pipj|2).  (Having a function that approaches infinity is inconvenient for some purposes, but I can't think of anything better.  The squaring makes taking derivatives easier.)
  3. Plot the origin (0, 0) at the center of the paper.  Define the vector vi as the vector from 0 to pi.
  4. Define a weight wx for each axis x that is monotonic increasing in the average conditional prior P(opinioni | xi) people use to guess the opinion of person i based on their position on that axis, wx ≤ 1.  Define a function w(v) as v scaled along each dimension x by wx.  (Biases perceived as less-important will be scaled down by w.)
  5. Define a bias shield function s(w), which operates over vectors, and returns a multiplicative factor that is larger the larger |w| is.  We can start by assuming s(w) = |w|.
  6. Define the believability b(pi,pj), the probability a person at pi assigns to statements made by a person at pj, as a(w(pi),w(pj))(1 + s(w(vj))(vi·vj)).  The dot-product scales the bias shield function by the degree to which i and j are biased in the same direction (negative if in opposite directions).  The right-hand side is 1 + s(w(vj))(vi·vj) because agreement can go to infinity, while max(s(w(vj))(vi·vj)) = 1, and we don't want agreement to be infinitely larger than the bias shield.  Note that b(pi,-pi) = (1 - s(w(v)))/4|pi|, which is zero if |w(vi)| = 1.  b can be negative, which is a good property if you can disagree with someone so much that, hearing them assert P will make you consider P less likely.
  7. Shade each point p on the paper with the expected believability at that point, which is the sum over all i in the population of b(pi,p).

Believability is concentrated at the outer fringes of every dimension x where Σi δ(b(pi,x))/δx, evaluated at x = wx, is > 0.  That is, if even at the farthest fringe, moving further out on dimension x makes person j more believable on average.  (This is a sufficient but perhaps not necessary condition.)  This turns out to be the case... always, for every dimension.  That surprised me.  The figure you drew on your n-dimensional paper will look like a hollow ball.

However, the density of that hollow ball at the fringes can vary.  The degree to which being more biased makes someone more believable is proportional to the derivative given above.  With the function definitions given above, the partial derivative along dimension xd works out to be xd / [(wd - xd)2 + Σi≠dxi2].  If people's opinions are distributed randomly, and we assume no two people have exactly the same biases (to avoid infinities), the expected believability at <0, ... maxd, 0, 0, ...> is proportional to the multiple integral over all the dimensions of that partial derivative.  You can work it out if you want to, but the main observation is that the bias shield effect is nearly proportional to a non-linear increasing function of 1 /Σdwd2.  The effect is therefore huge if there is only one dimension, but becomes smaller when there are more dimensions, or when other dimensions have a larger wx.

In this model, there is no hump in the believability function at (0,0).  That says that people don't give speakers credit for being unbiased. Whether this is generally true is an important open question.

Conclusion (end of mathy part)

The take-home messages are

  1. When someone publicly displays a strong bias, it can have the counter-intuitive effect of giving them more credibility, not because people don't acknowledge the bias, but because a known bias can be used to justify ad hominem attacks on critics.
  2. This effect (and all bias effects, although that isn't established here) is strongly diminished if people categorize speakers in two rather than just one dimension.  It might be as simple a matter as reminding someone that Bill O'Reilly is a Harvard graduate, or Irish, just to get them thinking in more dimensions.

Footnotes

  1. O'Reilly was right on this one.  "Derringer" describes guns that are generic knock-offs of small pistols made by Henry Deringer.  Booth shot Lincoln with a Deringer.
  2. It's stranger than that, because the O'Reilly defenders seldom responded by defending his statements.  The facts did not seem important to them.  Sometimes they said so: "He is not writing to get his PhD in history or to impress a small group of academics with his erudition", "I don't think O'Reilly was targeting "serious historians" when he wrote the book," "Sounds like you are too educated to have wasted your time reading this book."

Comments (65)

Comment author: TheOtherDave 31 December 2011 09:43:25PM 17 points [-]

It seems an alternate conclusion from the data is:

1) When person X publicly asserts their allegiance to tribe Y, it increases the willingness of other members of Y to defend X.
2) If person Z subsequently attacks X, it increases the willingness of other members of Y to attack Z.
3) Beliefs about either the book or Lincoln are basically a distraction here.

Comment author: PhilGoetz 31 December 2011 09:59:33PM *  5 points [-]

If there's a distinction between these explanations, it's that I'm adding an extra cognitive step, and saying that people are engaged in critical thinking about arguments but use knowledge about bias to dismiss some opinions.

Goetz's Razor says, When comparing explanations for a group's behavior, choose the one that requires the least intelligence. TheOtherDave's explanation wins by that test. If we had to choose one or the other, I'd go with TheOtherDave's.

We don't have to choose one or the other. I think both things happen, but people are less aware of the one described in the post.

But, yeah, maybe the entire post should be replaced by your comment.

Comment author: thomblake 03 January 2012 07:32:57PM 0 points [-]

Goetz's Razor says, When comparing explanations for a group's behavior, choose the one that requires the least intelligence.

An excellent policy. I use this one for individuals too.

Comment author: Dr_Manhattan 03 January 2012 07:46:47PM 0 points [-]
Comment author: thomblake 03 January 2012 07:54:40PM 0 points [-]

Similar, but different emphasis. It's actually just an application of Occam's Razor. Use the accurate model with the fewest moving parts.

And I prefer this formulation of Hanlon's Razor:

Never attribute to malice that which can be adequately explained by intelligence

(with stupidity as a special case).

Comment author: shminux 31 December 2011 02:32:22AM *  9 points [-]

You are overthinking it. People are blind to the biases identical to their own, that is all (how many lefties said that it was an unbiased account?).

Comment author: PhilGoetz 31 December 2011 05:33:57PM *  5 points [-]

I don't think so. You don't see moderates behaving that way towards moderates. And most of the commenters (though not all) are well aware that BIll O'Reilly is a conservative. Most of the comments don't say something like, "Bill O'Reilly was correct about his claims about Edwin Stanton". They say "You disagree with Bill O'Reilly; this must be because you are a liberal; I am not even going to consider your claims."

The subject matter here was history about Lincoln, and the facts in contention were not political, but questions such as, When was the Oval Office built? What kind of gun did Booth use? Was there a hole in the wall or in the door? So people are not defending O'Reilly's book because they have a bias that makes them agree with him about the content of the book. (I think I'll add that to the post.)

Comment author: Oligopsony 31 December 2011 07:22:48PM 5 points [-]

I don't think so. You don't see moderates behaving that way towards moderates.

I don't think that's really true: the few people who do think about politics a lot and self-identify as "moderates" to the exclusion of other labels do seem very apt to explain disagreement by others' fanaticism, ideology, groupthink, extremism, or other ways of not being a moderate.

Comment author: Eugine_Nier 31 December 2011 06:36:35PM 1 point [-]

On the other hand a lefty who disagrees with O'Reilly might very well try to make this disagreement look like it's about something apolitical.

Comment author: shminux 31 December 2011 07:57:45PM *  0 points [-]

You don't see moderates behaving that way towards moderates.

Presumably by a "moderate" you mean someone who agrees that "politics is a mind-killer", not a DNC supporter, because those are not much better.

So people are not defending O'Reilly's book because they have a bias that makes them agree with him about the content of the book.

Of course not, for many of them O'Reilly can do no wrong, so anyone who attacks the book attacks their values. The content of the book is not relevant, only the fact that Bill O'Reilly wrote it is.

Comment author: wedrifid 31 December 2011 03:06:32AM -1 points [-]

People are blind to the biases identical to their own

Except when it takes one to know one.

Comment author: PhilGoetz 31 December 2011 05:49:26PM 6 points [-]

It was also interesting to find that there are a few people who, based on my extensive sampling, read most of the over 1,000 bad reader reviews on Amazon and made snarky comments on many of them, even though they had already formed their opinion of the book and had no reason to read more reviews of it (other than to reply with snarky comments).

Comment author: Will_Newsome 01 January 2012 02:49:01AM 3 points [-]

Maybe they were doing it out of altruism?

Comment author: MixedNuts 02 January 2012 04:31:08PM 5 points [-]

I don't think it's fair to describe O'Reilly as having a conservative bias. The useful usage of "bias" around here has roughly been "predictable error". If we had a perfect policy optimizer, we could compare it with O'Reilly and conclude that he systematically overestimates evidence for conservative policies (a conservative bias) or underestimates it (a liberal bias). But conservative supporters of O'Reilly certainly don't currently believe that he exhibits a conservative bias. They just think that he neutrally evaluates policies and rates conservative ones as better because they are in fact better. A battery has a bias for outputting current rather than consuming it, but that doesn't make it bad.

Comment author: PhilGoetz 03 January 2012 11:33:39PM 1 point [-]

Good point. "Bias" is the wrong word. The proposed reasoning is the same - the further away from the opinions of others that I know person X to be, the more I can dismiss the attitude of random person R towards X as being based on that distance.

Comment author: Nominull 31 December 2011 07:37:00PM 5 points [-]

Perhaps fans of Mr. O'Reilly are fans of his style of discourse, and so they would prefer to read a book written in that style over a book written in a stereotypical dry academic style, even if (especially because) Mr. O'Reilly does not footnote quite as scrupulously as some others?

History is not about history.

Comment author: jfm 05 January 2012 02:51:29PM 0 points [-]

This explanation seems quite likely to account for some of the positive ratings from O'Reilly fans, but does it really do anything to account for the vehemence of reactions to negative ratings?

Comment author: Locke 31 December 2011 02:55:29AM 13 points [-]

Forgive my confusion, but did you conclude that Killing Lincoln was or was not the best book on the Lincoln Assassination?

Comment author: PhilGoetz 31 December 2011 05:03:12PM 3 points [-]

No; I bought Blood on the Moon instead.

Comment author: Emile 31 December 2011 05:41:48PM 10 points [-]

Then I'm not sure I understood the second part of:

Nearly 200 books have been written about the Lincoln assassination, including some by professional Lincoln scholars. So the odds seemed good that at least one of these was better than a book written by a TV talk show host.

But I was wrong. This was not a believable conclusion.

... I expected the following to show that you were wrong, and that the O'Reilly book was indeed better than those written by Lincoln scholars.

Comment author: PhilGoetz 31 December 2011 06:02:00PM 2 points [-]

Oh. I meant there were many people to whom it was not believable. I'll clarify that.

Comment author: [deleted] 31 December 2011 07:30:11PM 6 points [-]

To me, it still reads like "I was wrong" means "I was wrong about thinking that at least one of these was a better book". Perhaps you could get rid of the sentence "I was wrong" entirely, and say something like "To my surprise, it turned out that for many people, this was not a believable conclusion."

Comment author: Louie 03 January 2012 12:54:17AM 2 points [-]

I also thought you meant that Bill O'Reilly had (surprisingly) written the best book ever on the Lincoln shooting when you said "But I was wrong."

Comment author: waveman 31 December 2011 09:17:27PM 0 points [-]

Presumably "but I was wrong" was meant to be ironic. But there is no way to know that. Maybe say "But I was wrong according to O'Reilly fanboyz".

Comment author: Sniffnoy 31 December 2011 07:55:12PM 0 points [-]

The "but I was wrong" part is still unclear (consider what it appears to refer to).

Comment author: Daniel_Burfoot 31 December 2011 03:03:17AM 20 points [-]

I think this is a deep insight. I'm not sure the math adds anything, though.

Comment author: Nymogenous 31 December 2011 05:11:17AM 10 points [-]

I feel like it may have even obscured the point...I spent more time wading through the math than I did thinking about the bias shield effect. Since it didn't really clarify anything, it came across as some kind of signalling...not sure if that's what it was, but it's certainly what it looks like.

Comment author: Daniel_Burfoot 31 December 2011 01:21:06PM 14 points [-]

Oh, I don't think it's signalling, I just think Phil really likes math :-)

Comment author: PhilGoetz 31 December 2011 05:06:44PM *  3 points [-]

The math doesn't add a lot at the moment, but I like to promote the idea that you can analyze the behavior of large groups of people mathematically. If you remove the "bias shield function" from the math, I think you have a framework that can be used for analyzing bias in other problems.

It does add the conclusion that categorizing people in more dimensions diminishes the effect, and that this effect may not ever be cancelled out entirely by adding more dimensions (I thought it would be). The particular functions I used are not well-motivated, so you can't draw the second conclusion with confidence.

Comment author: Eugine_Nier 31 December 2011 06:32:19PM 6 points [-]

The math doesn't add a lot at the moment, but I like to promote the idea that you can analyze the behavior of large groups of people mathematically. If you remove the "bias shield function" from the math, I think you have a framework that can be used for analyzing bias in other problems.

I disagree, adding a mathematical model where every aspect of it was pulled out of your ass, is more likely to give you a false sense of precision then help your analysis.

Comment author: PhilGoetz 31 December 2011 09:31:31PM *  9 points [-]

Most sociological mathematical models are formed by thinking about the relationships involved qualitatively, then building models that are simple, have those qualitative aspects, and can be worked with. Then you can gather data, compare the models to the data, and revise the models.

Take the Cobb-Douglas model of a production function. It was pulled out of someone's mind (not their ass) in just such a manner. Perhaps by luck, it never needed to be revised, because it turned out to model real data very well.

The use of a "bias function" is the really questionable thing here. Is what is really going on that people add imputed believability to someone who is biased, or can you get the same results just by saying that people agree with people who agree with them? You can't get the same results easily in this case, because moderates don't make the same kind impassioned defense of other moderates. You could suppose opinions are not evenly-distributed and conclude that moderates would do that, if there were more of them. Or you could explain the data as signalling group affiliation rather than having to do with belief.

Whatever explanation you prefer, putting it in math (when done well) distinguishes these different parts of the argument and makes these critical points more apparent.

(Some models are built by looking at a lot of data and noticing patterns, like Zipf's law. Some, more often found in physics, are built from the ground up, like E=mc^2. Those are better, if we can make them.)

Comment author: Eugine_Nier 31 December 2011 10:04:06PM 1 point [-]

Most sociological mathematical models are formed by thinking about the relationships involved qualitatively, then building models that are simple, have those qualitative aspects, and can be worked with.

Most sociological mathematical models are also total crap.

Then you can gather data, compare the models to the data, and revise the models.

Repeat until your model has enough free parameters to be unfalsifiable.

Comment author: daenerys 31 December 2011 11:13:44PM 2 points [-]

Most sociological mathematical models are also total crap.

Downvoted.

I am disappointed in the LW community every time I see a comment consisting primarily of: "<X> is crap."

It doesn't add anything useful to the conversation, or propose any solutions or ideas. It is just plain rude. Please stop.

We can do better than that.

Comment author: wedrifid 01 January 2012 12:31:13AM *  8 points [-]

It doesn't add anything useful to the conversation,

Yes it does. It is (often) a negation of the previous assertion - in this case just the implied suggestion that sociological mathematical models are useful. To the extent that said assertion can be said to be a contribution the rejection of it can too.

Comment author: hamnox 31 December 2011 06:58:07PM *  3 points [-]

Is there a graph for the mathy part? It sounds like there might be a (very, very rough) graph but I wouldn't have a clue how to make it. Displaying the maths visually might be useful for clarifying your point.

Comment author: PhilGoetz 31 December 2011 10:01:29PM *  10 points [-]

True. But according to my utility calculations, I should go to a New Year's Eve party instead of drawing a graph.

Comment author: peter_hurford 31 December 2011 07:23:23AM 3 points [-]

You completely lost me with the math, but I thought this post was exceptionally strong.

Comment author: PhilGoetz 30 December 2011 11:50:25PM *  5 points [-]

Anyone know why the website is removing spaces before and after every italicized word in the post, and how I can make it stop?

Comment author: Vladimir_Nesov 31 December 2011 12:03:03AM *  5 points [-]

Fixed. <div> tags present in HTML trigger this bug; there is an issue on LW bug tracker, but developers say it can't be easily fixed, so we are left with having to manually remove the offending tag each time it slips in.

Comment author: wedrifid 31 December 2011 12:17:58AM 1 point [-]

Fixed. <div> tags present in HTML trigger this bug, there is an issue on LW bug tracker, but developers say it can't be easily fixed, except by manually getting rid of the offending tag.

Not easily fixed? Nonsense. Not elegantly fixed perhaps. It can be easily fixed with, by way of proof of concept, regex and a cron job.

Comment author: Vladimir_Nesov 31 December 2011 12:23:31AM *  0 points [-]

Quite. To fix the more heavily misformatted posts, I use a script that in particular does just that for the <div> tags (replacing them with <p>), but sometimes it garbles/discards nonstandard ways of meaningfully marking up the text and needs to be corrected on case-to-case basis. Writing a script that automatically works in general would be harder, require more expertise with peculiarities of HTML, and ideally a test suite to make sure that it doesn't disrupt useful markup. (I fail to see where a cron job might enter the picture.)

Comment author: wedrifid 31 December 2011 12:35:20AM *  1 point [-]

(I fail to see where a cron job might enter the picture.)

Trike has knowledge about the codebase that I do not. It is theoretically possible (albeit unlikely) that the code is constructed such making the correction take place at an appropriate time does not qualify as 'easy'. However information that is available is that Vladimir_Nesov is able to log in and make such corrections via a html interface. To establish with confidence that correction (within the limits of the side effects you mentioned) would be easy without relying on any assumptions about the inner workings of the codebase it is simple to consider simply running a regexing script remotely using Vladimir_Nesov credentials. This establishes an upper bound.

Comment author: Vladimir_Nesov 31 December 2011 12:41:03AM *  0 points [-]

OK. In any case, that doesn't work, because of the mentioned difficulty with creating an automatic script that doesn't damage the text sometimes (it's generally irresponsible to edit a public archive without manually testing the results when probability of damage is nontrivial), though a script that alerts about the presence of <div>'s could ensure that corrections happen more reliably.

Comment author: PhilGoetz 31 December 2011 05:36:04PM 0 points [-]

Thanks! I see it didn't revert when I edited the post, which surprises me.

Comment author: CharlieSheen 31 December 2011 06:11:16PM 1 point [-]

I've encountered this bug too.

Comment author: Konkvistador 31 December 2011 05:35:21PM *  1 point [-]

It is an extremely annoying bug. It shows up not just with italicized words, but sometimes with words written in bold and most damning of all, links.

It really saps my will to try and make new posts, I have two or so posts languishing in draft form because I'm tired of working around the bug all the time.

Comment author: wedrifid 31 December 2011 05:55:58PM 0 points [-]

It is an extremely annoying bug. It shows up not just with italicized words, but sometimes with words written in bold and most damning of all, links.

A bug that affects italicized words also affects words written in bold. Inconceivable!

Comment author: CharlieSheen 31 December 2011 06:11:54PM *  4 points [-]

It is an extremely annoying bug. It shows up not just with italicized words, but sometimes with words written in bold and most damning of all, links.

Clearly he meant written in blood. Come on guys, if we are going to be serious about this doomsday cult thing we need to stop jumping each other because of typos, so chill wedrifid.

Comment author: Konkvistador 31 December 2011 06:05:43PM *  3 points [-]

Well of course it does, but my brain perceives these as separate separate insults to my weak determination to work through trivial inconveniences! In any case I don't share Eliezer's love for italics, but not being able to spam my articles with a dozen links, is plain unforgivable.

Tis is the season to air grievances after all. :P

Comment author: wedrifid 31 December 2011 06:11:41PM 4 points [-]

but not being able to spam my articles with a dozen links, is plain unforgivable.

Definitely! I love link spam. It formalizes how we use phrases as references to existing concepts and saves on google searches!

Comment author: arundelo 31 December 2011 06:30:11PM 1 point [-]

It may help to think of selecting pieces of text and clicking buttons to turn them into links as the inconvenient part. On the rare occasions when I've made top-level posts, I've edited the HTML source directly (in a real text editor). Happy Festivus!

Comment author: Jonathan_Graehl 31 December 2011 06:38:29PM *  1 point [-]

Great illustration. For vector similarity, I recommend cosine similarity, and if you want it to extend to infinity, the (negative) logarithm of something between 0 and 1 is good. However, I guess your model is pretty arbitrary already, so I can't vouch for my suggestion in the context of improving your model's predictiveness.

Disclaimer: since you're modeling intensity of belief, cosine similarity (which ignores vector length) alone wouldn't even apply. I guess since you're using dotproduct already, my suggestion missed the mark. It does seem like, by squaring, you're declaring diametrically opposed vectors the same as colinear ones (i.e. negative dotproduct becomes positive when squared).

Comment author: kilobug 10 January 2012 03:11:31PM 1 point [-]

I don't agree with the « not because people don't acknowledge the bias » idea (which is quite recurrent in the post). People who give credit to O'Reilly don't think he is biased. They think the rest have a "liberal bias", and that O'Reilly is himself neutral and accurate. To us it seems that they give credit to O'Reilly because he's openly biased, but to them they give credit to him because they consider him to be less biased than average.

Comment author: geebee2 06 January 2012 03:15:05PM *  1 point [-]

Any of these readers would have been willing to believe that Bill O'Reilly had written a bad book, if they did not believe that Bill O'Reilly was strongly biased.

Are you possibly confusing what these readers said with what they believe? I suspect many of these people had no well-founded opinion on the book, or may have privately thought it was a bad book, rather they were seeking to defend the author for political reasons.

So in this book review example, it's just that people who have a strong affiliation with a well known political figure will seek to defend him regardless. When we read messages like this, they tell us practically nothing about what the defenders really think about the book ( probably in this case they haven't even read it, there is nothing to tell ).

[ My first comment on this site, so be gentle - I'm just getting acquainted with the furniture, so I may well be wide of the mark. Reading further down, it seems others are thinking on similar lines (TheOtherDave), this was my reaction before I read that. ]

Comment author: dlthomas 04 January 2012 12:03:53AM *  1 point [-]

Killing Lincoln [..] is not in the Ford's Theatre national park service bookstore because the NPS decided it was too historically inaccurate to sell.

Last time I was there, the gift shop on Alcatraz was selling The Rock on DVD. It is also run by NPS.

Comment author: gwern 04 January 2012 01:00:49AM 0 points [-]

I was there in February and can confirm this, as I was amused for the same reason when I spotted it.

Comment author: Prismattic 04 January 2012 12:29:21AM 0 points [-]

The Rock doesn't make any claim to be a work of nonfiction, however.

Comment author: dlthomas 04 January 2012 12:41:33AM 0 points [-]

Granted, of course.

Comment author: Vaniver 04 January 2012 12:12:36AM 0 points [-]

I would imagine that fiction that claims to be fiction gets treated differently.

Comment author: dlthomas 04 January 2012 12:23:01AM 0 points [-]

That makes sense. I wasn't really trying to demonstrate anything; just amused.

Comment author: jimmy 01 January 2012 10:20:49PM *  1 point [-]

Maybe I'm not following right, but the math seems arbitrary in important conclusion changing ways.

Since the belief function goes with 1/distance, all the weight comes from the immediate neighborhood. The problem of finding the maximum expected believability reduces to finding the maximum product of radius^3*opinion density (under the simplest s(w) and w(v)), which obviously is strongest at the edges for uniform opinion density.

If you get rid of the radius dependence in the right side term or assume a gaussian distribution, the effect should go away.

Also, since we're using one number for "expected belief", there's no distinction between "a lot of people believe" and "a few believe super intensely", and this is setup for the latter to dominate. If you bounded agreement, it should shift the balance towards "a lot of people believe", and the conclusion about "total believability" would change.

Why these particular equations?

Comment author: PhilGoetz 03 January 2012 11:29:29PM 0 points [-]

You're right - using 1/distance makes all the weight come from the immediate neighborhood. I don't think it's a fatal problem. That's not why the effect is strongest at the edges. The addition of s(w) makes it work out that way.

I'm not following the bit about maximum product of radius^3*opinion density. Is that something that came out after integrating? I've thrown my work out by now, so can't check without redoing it.

The simplest w(v) is w(v) = v. The s(w) proposed is |w|. a(w(pi),w(pj))(1 + s(w(vj))(vi·vj)) when pj = <x, 0> is a(pi, <x,0>)(1 + |vj|)(vi · <x,0>) = (1+x)*x*xi / [(xi - x)^2 + yi^2]. The problem with x_i being close to x is the same everywhere in the plane.

Comment author: jimmy 04 January 2012 09:08:45PM 0 points [-]

You're right - using 1/distance makes all the weight come from the immediate neighborhood. I don't think it's a fatal problem. That's not why the effect is strongest at the edges. The addition of s(w) makes it work out that way.

Right, that alone doesn't make it strongest at the edges. It just means that you're measuring how strongly a tiny few people agree, not how believable it is to any significant fraction of people, and it didn't come across to me as clearly marked as such.

I'm not following the bit about maximum product of radius^3*opinion density.

Err, I misread parenthesis.

The simplest w(v) is w(v) = v. The s(w) proposed is |w|.

Right, so a(w(pi),w(pj))(1 + s(w(vj))(vi·vj)) simplifies to 1/distance(pi,pj)(1+|vj|)(vi·vj). Since we're only interested in a tiny neighborhood, we can throw away 1/distance as having the same effect everywhere, and we can reduce the rest to (1+|vi|)(vi)^2 (which, if you misread parenthesis, is (1+|vi|vi^2) or 1+r^3 and we can ignore the constant 1)

So if the expected agreement's position dependency is r^2+r^3, you get the highest values at the edges. But its not clear to me why s(w) should be |w| instead of some constant, and it's not clear to me why the (vi·vj) shouldn't be renormalized as (vi·vj)/(|vi|*|vj|), or why we shouldn't have a Gaussian distribution of opinions.

Comment author: Dagon 31 December 2011 06:28:26PM *  1 point [-]

I suspect that many dimensions of bias are highly corollated, including the dimensions of history and politics. This would seriously weaken the premise.

More importantly, do you have a model that distinguishes believability from accuracy, and values them separately? What led you to analyze the former rather than the latter?

Comment author: [deleted] 31 December 2011 12:02:26AM *  0 points [-]

It's not that the spaces would be removed. It's just that the italicized letters reach into the spaces. But I have no solution to that.