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 p_i.
2. Define the agreement a(p_i,p_j) between points p_i and p_j as the square of the inverse of the length of the vector p_ip_j (a(i,j) = 1 / |p_ip_j|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 v_i as the vector from 0 to p_i.
4. Define a weight w_x for each axis x that is monotonic increasing in the average conditional prior P(opinion_i | x_i) people use to guess the opinion of person i based on their position on that axis, w_x ≤ 1.  Define a function w(v) as v scaled along each dimension x by w_x.  (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(p_i,p_j), the probability a person at p_i assigns to statements made by a person at p_j, as a(w(p_i),w(p_j))(1 + s(w(v_j))(v_i·v_j)).  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(v_j))(v_i·v_j) because agreement can go to infinity, while max(s(w(v_j))(v_i·v_j)) = 1, and we don't want agreement to be infinitely larger than the bias shield.  Note that b(p_i,-p_i) = (1 - s(w(v)))/4|p_i|, which is zero if |w(v_i)| = 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(p_i,p).

Believability is concentrated at the outer fringes of every dimension x where Σ_i δ(b(p_i,x))/δx, evaluated at x = w_x, 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 x_d works out to be x_d / [(w_d - x_d)² + Σ_i≠dx_i²].  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, ... max_d, 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 /Σ_dw_d².  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 w_x.

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."