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The Phrase "Absence of Evidence Isn’t Evidence of Absence" is a Logical Fallacy


Dutch Schultz

So I’ve implied in all of my posts about Bayes Theorem that probabilities are like energy; they have to be conserved. In other words, when everything is added up it has to add up to 1.00:

P(H) + P(~H) = 1.00
P(E | H) + P(~E | H) = 1.00
P(E) + P(~E) = 1.00
P(H | E) + P(~H | E) = 1.00
etc.

Why is that? If these axioms are ignored, you could end up getting Dutch Booked:

Horse number Offered odds Implied
probability
Bet Price Bookie Pays
if Horse Wins
1 Even \frac{1}{1+1} = 0.5 $100 $100 stake + $100
2 3 to 1 against \frac{1}{3+1} = 0.25 $50 $50 stake + $150
3 4 to 1 against \frac{1}{4+1} = 0.2 $40 $40 stake + $160
4 9 to 1 against \frac{1}{9+1} = 0.1 $20 $20 stake + $180
Total: 1.05 Total: $210 Always: $200

[…]

In Bayesian probability, Frank P. Ramsey and Bruno de Finetti required personal degrees of belief to be coherent so that a Dutch book could not be made against them, whichever way bets were made. Necessary and sufficient conditions for this are that their degrees of belief satisfy the axioms of probability.

Why is this relevant? Well, there’s the common refrain that “absence of evidence isn’t evidence of absence” but if that were true, then your probabilities won’t be coherent and you’ll Dutch Book yourself above; your terms won’t equal 1.00.

In reality, the phrase “absence of evidence is not evidence of absence” is itself a logical fallacy. Or rather a probabilistic fallacy, since the person asserting it is trying to Dutch Book you. Let’s go over what would happen if absence of evidence indeed were not evidence of absence.

Say you have some hypothesis H that is a coin flip; you are agnostic about whether it’s true or false. A friend predicts that some evidence E increases the probability of H. You then rejoin that E is not found, so this must decrease the probability of H, but your friend asserts “absence of evidence isn’t evidence of absence!”. What is he actually saying by that? He’s saying that the absence of E doesn’t move H so it stays at 50%.

So the simple form of Bayes’ is P(H | E) = P(E | H) * P(H) / P(E).

In formulaic terms, your friend is saying P(H) = .5. P(H | E) > .5, yet P(H | ~E) = .5. We can model this by throwing in some values in Bayes’ Theorem. P(H) is already .5, so let’s say that P(E) is also .5 and P(E | H) is .6. This becomes:

Evidence is found:
P(H | E) = .6 * .5 / .5
P(H | E) = .6

Let’s try it for the absence of evidence. This would be P(H | ~E) = P(~E | H) * P(H) / P(~E). Again, P(H) is already .5, and to keep it simple let’s again make P(~E) equal .5. In order to make P(H) = P(H | ~E), the conditional probability P(~E | H) should also be .5:

Absence of evidence isn’t evidence of absence:
P(H | ~E) = .5 * .5 / .5
P(H | ~E) = .5

But wait, remember that P(E) + P(~E) = 1.00. This means that P(~E) is also necessarily .5. What’s the other axiom? P(E | H) + P(~E | H) should also equal 1.00. But in order to make sure that absence of evidence isn’t evidence of absence, we made P(E | H) = .6, and P(~E | H) = .5. That totals to 1.10: DUTCH BOOKED!!11!!

In reality, if P(E | H) is .6 then P(~E | H) is .4. Let’s run through the so-called “absence of evidence isn’t evidence of absence” part again:

P(H | ~E) = .4 * .5 / .5
P(H | ~E) = .4

As you can see, P(H) was .5 and the absence of the evidence that would have increased it to .6 probability, with its absence, decreases the probability to .4. This will always be easy to test. Since Bayes Theorem works for updating the probability of some hypothesis when encountering new evidence, Bayes Theorem also works for determining how the absence of said evidence would have affected the hypothesis due to the compliments rule above.

Bayes Theorem for updating upon new evidence: P(H | E) = P(E | H) * P(H) / P(E)
The compliment Bayes Theorem for absence of evidence: P(H | ~E) = P(~E | H) * P(H) / P(~E).

There’s another way of looking at it. Remember the Monty Hall problem? It’s an example of statistical independence. Independence is what happens when P(H) = P(H | E). By saying that absence of evidence isn’t evidence of absence, a person is implying that the evidence exists independently of the hypothesis. Again, if the absence of evidence doesn’t have any effect on the hypothesis, then finding the evidence would have the same effect unless they are trying to Dutch Book you.

So make no mistake. Anyone who asserts that absence of evidence isn’t evidence of absence is trying to Dutch Book you.

 
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Posted by on June 29, 2012 in Bayes

 

The Dark Arts

Over on Less Wrong, there is a wiki article about the Dark Arts. This is any form of persuation that takes advantage of a person’s cognitive biases to get them to do what you want them to do.

I’ve read a bit about cognitive science, and “hacking” my own brain to become more aware of my own biases seems like, at the least, harmless fun and at the most a massive supporting beam in the skyscraper of my self-improvement. However, there’s another side to that: Knowing that your own cognitive biases are also present in other people. It’s only a short leap to go from not telling people about those biases and simply using their own ignorance of their brainware to your advantage.

Yudkowsky’s allusion to the Dark Side of the Force seems apt.

For example, you can prime people to land on an answer you want by making them decide on something else totally unrelated. If you ask someone to tell you how many letters there are in some sentence, and then later ask them to guess how many countries are in Africa, they’ll guess a number pretty close to how many letters there are in the unrelated sentence.

That is a very simple, harmless application of brain-hacking. But what about more nefarious things? You could probably use something like the availability heuristic to scare someone or fool them into taking a position you currently hold or favor. I kinda did this to myself; a friend of mine is in Afghanistan and I would always read stories about all of the craziness going on over there, needlessly worrying myself.

On the other hand, I made it a point to not mention to her how bad sexual harrassment and rapes are over in Afghanistan with her being an attractive girl. She actually did ask me what I thought about her going over there, telling me how important it was that she go there to advance her career, so mentioning that might have scared her more than necessary. I’m ex-military, so I’ve had more briefings about how problematic sexual harrassment is over in theater than I can remember. But then again, that in and of itself might be an availability heuristic that the military forced on us guys to make us more cognizant of it!

Another example, I was trying to teach an ex-gf of mine about cold reading and how that plays into the leaps in inference that the brain makes. What brought this up was that she thought that astrology tapped into some sort of evil energy and I rejoined that no, astrology is bullshit because it’s a cold reading technique. So I did a little cold reading game on her, and she seemed a bit startled about how “accurate” it was. There’s one point in the cold read game where I describe her “ideal boyfriend”, and it would have been easy for me to manipulate the cold read to say “wow, your ideal boyfriend sounds like me!” but I didn’t do that since it seemed to be an abuse of my powers, so to say (especially with her being my ex and all).

Some further reading: http://lesswrong.com/lw/9iw/the_dark_arts_a_beginners_guide/

 
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Posted by on June 28, 2012 in cognitive science

 

Why Are Religions Misogynist?

Besides the obvious/superficial answer “religions were invented by premodern men”, take a look at this conversation over at Jason Rosenhouse’s blog that explains it a bit better:

P.Z. Myers:

Whenever I hear that tripe about the beneficial effects of religion on human cultural evolution, it’s useful to note that the world’s dominant faiths all hardcode directly into their core beliefs the idea that women are unclean, inferior, weak, and responsible for the failings of mankind…that even their omnipotent, all-loving god regards women as lesser creatures not fit to be intermediaries with him, and that their cosmic fate is to be subservient slaves to men, just as men are to be subservient slaves to capital-H Him.

David Sloan Wilson can argue all he wants that religion helped promote group survival in our evolutionary history, or that his group selectionist models somehow explain its origins, but it doesn’t matter. Here and now, everywhere, those with eyes to see can see for themselves that religion has for thousands of years perpetuated the oppression of half our species. Half of the great minds our peoples have produced have lived and died unknown and forgotten, their educations neglected, their lives spent doing laundry and other menial tasks for men — their merits unrecognized and buried under lies promulgated by religion, in cultures soaked in the destructive myths of faith which codify misogyny and give it a godly blessing.

Isn’t that reason enough to tear down the cathedrals — that with this one far-reaching, difficult change to our cultures, we double human potential?

David Sloan Wilson:

Myers the ideologue thinks that he can demonstrate the harmful effects of religion on human welfare with a single word — WOMEN. Here’s how a scientist would set about studying women in relation to men. The first step would be to ask what evolutionary theory predicts about male-female relationships and how the predictions are borne out in nonhuman species. That inquiry would show that sexual conflict is common in the animal world and that the kind of sexual equality that has become a virtue in contemporary western society evolves by genetic evolution only under special circumstances. Among the great apes, gibbons are monogamous, bonobos form female coalitions that resist domination by males, and males boss females around in all of the other species (and most other primate species). None of this variation can be explained by religion.

The second step would be to see if variation in male-female relations within the human species can be explained by the same evolutionary dynamics that explain cross-species variation. For example, it is likely that in both cases, the ability of males to control resources needed by females will result in sexual inequality. This is one reason why agricultural societies are more patriarchal than hunter-gatherer societies — regardless of their religions.

To measure the effect of a given religion on sexual inequality, that religion should be compared to the other cultural forms (religious and otherwise) that existed at the same time and place, such as early Christianity vs. Roman pagan society, early Islam vs. the many Arabic cultures of the region, or Christianity vs. scientific views about sexual equality in Britain during the Victorian era. I won’t try to second-guess the result of such an inquiry, but I do know this — it isn’t self-evident.

So why are religions misogynist? Because religions are a product of evolution, and evolution is “misogynist”.

As far as evolution is concerned — at least, what would be the most successful reproductive strategy based on the design and function of the sexes — women are valuable and men are expendable. Something that is valuable has to be kept safe and hidden away, lest you lose it. Something that is expendable is free to go out and get killed. This is built into our biology to make sure the species maximizes reproductive success.

That’s why I put “misogynist” in scare quotes.

If a man and a woman were both trapped in a building on fire, and the firefighters could only save one person before the building collapsed, who is the instinctual person most people would save? The woman. This makes sense from an evolutionary framework. If there were some apocalypse that reduced us to the stone age, the society with 100 women and 1 man (i.e. the society that kept women repressed and men maximally expendable) will outbreed the society that has 1 woman and 100 men. That’s why polygyny is the most common form of relationships in human history. That’s why we “slut shame”: Slut shaming — where men can sleep with as many women as they want while women are shamed for doing the same — assumes polygyny, since that’s the only arrangement where a man can sleep with a lot of women while women only sleep with one or very few men.

Thus, any religion that doesn’t reflect this evolutionary instincts to “value” women (that is a tricky word since I’m talking about reproductive value and not humanistic value, i.e. feminism) would not have the same reproductive success as a religion that did “value” women.

On the other hand, in societies that don’t care about reproductive success they will tend towards equality between the sexes. This has empirical verification; the most feminist societies generally also have the lowest rates of childbirth. The most religious societies/communities have the highest (cue the Duggars). That’s because it is in men’s benefit to make sure that women have very little reproductive rights; anti-abortion sentiment is a hard-wired male desire to have as many kids by as many different women as possible. This strategy maximizes the reproductive success of the species… “mother nature” — evolution — is actually a man.

Yet, ironically, evolution still wins in a sense; the individual men in those societies who reflect the sexist nature of nature (i.e. religion) will have the most “reproductive” success. Which might be one reason for the paradox that women are more religious than men.

Basically, the misogyny of religion is proof that religions are a product of evolution. Not a hand-me-down from some god (at least no god other than a truly alien god).

 
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Posted by on June 19, 2012 in Evolution

 

The Halo Effect and Super Happy Death Spirals

Recent posts have reminded me that no one is immune to Happy Death Spirals. I guess if I get nothing intelligent out of them (their point has already been addressed, reminding us that bad arguments are like roaches), those blogs are still a good example of what not to do from a cognitive science point of view; to remind myself that conclusions are never a good in and of themselves. That only correct methodology (the rules of logic and probability) is the “one true good”. Everything else is transitory:

Yesterday, I suggested that one key to resisting an affective death spiral is the principle of “burdensome details“—just remembering to question the specific details of each additional nice claim about the Great Idea. (It’s not trivial advice. People often don’t remember to do this when they’re listening to a futurist sketching amazingly detailed projections about the wonders of tomorrow, let alone when they’re thinking about their favorite idea ever.) This wouldn’t get rid of the halo effect, but it would hopefully reduce the resonance to below criticality, so that one nice-sounding claim triggers less than 1.0 additional nice-sounding claims, on average.

The diametric opposite of this advice, which sends the halo effect supercritical, is when it feels wrong to argue against any positive claim about the Great Idea. Politics is the mind-killer. Arguments are soldiers. Once you know which side you’re on, you must support all favorable claims, and argue against all unfavorable claims. Otherwise it’s like giving aid and comfort to the enemy, or stabbing your friends in the back.

If…

  • …you feel that contradicting someone else who makes a flawed nice claim in favor of evolution, would be giving aid and comfort to the creationists;
  • …you feel like you get spiritual credit for each nice thing you say about God, and arguing about it would interfere with your relationship with God;
  • …you have the distinct sense that the other people in the room will dislike you for “not supporting our troops” if you argue against the latest war;
  • …saying anything against Communism gets you stoned to death shot;

…then the affective death spiral has gone supercritical. It is now a Super Happy Death Spiral.

What happens if “the existence of Jesus” is the Great Idea? You have to decide which side of the war you’re on. And of course, you just have to be on the side of the Great Idea. Now arguments are soldiers fighting for the Great Idea; any argument for the historicity of Jesus has to be good and must be supported (even if under normal circumstances it is a bad argument). Any argument or potential argument that could be used against the historicity of Jesus must be bad, and has to be fought at all costs.

This is war, folks. There is no mercy, no quarter, no solace for the enemy. They must be crushed and driven before you. Followed only by the lamentations of their women.

It just so happens that this mentality is exactly what creates an impenetrable wall of unfalsifiability around the Great Idea, even if the Great Idea would not be unfalsifiable absent a Super Happy Death Spiral.

Without thinking that conclusions must be defended at all costs, you start to see that there actually are some scraps of evidence that make better sense for the opposition than for you. To me, it’s highly unlikely that an all powerful god exists or that Christianity is true. Yet I’m not afraid to admit that some things make more sense under Christianity than under atheism. I’m not afraid to admit that there are some things that might make more sense under Creationism than evolution (I don’t know what those would be, but I don’t reject it out of hand). I’m not afraid of those sorts of evidences because resting an entire conclusion on them is the Base Rate Fallacy. P(E | H) is not P(H | E).

Similarly, there are some things that make more sense under Jesus mythicism than historicism (like Paul’s silence). And there are also things that make more sense under historicism than under mythicism (like brother of the Lord). Someone who is defending arguments with soldiers could never admit this, because it is giving ammunition to “the enemy”. Which leads to all types of sophistry. The true strength lies in methodology; of accumulating all evidence and seeing which hypothesis wins when all cards are put on the table.

But that cannot be done if the entire reason for the existence of your arguments is that they are soldiers fighting on the side of the Great Idea. This sort of thing leads to the absurdity of rejecting the laws of logic and probability and replacing it with your own flawed intuition.

 
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Posted by on June 15, 2012 in cognitive science, jesus myth

 

Jerry Coyne: Belief in God Plummets Among Millennials

According to the latest Pew Research poll numbers posted by Jerry Coyne, there is a massive difference in doubting god between my generation (born 1965 – 1980) and “Millennials” (born 1981+).

The Millennials are the biggest users of the Internet, and this fits my prediction.

 
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Posted by on June 14, 2012 in economics/sociology

 

Infinite Certainty

There is a small battle brewing about the application of Bayes Theorem over at R. Joseph Hoffmann’s blog. The main confusion seems to be about the difference between objective and subjective probability. Hoffmann, an anti-mythicist (note: I think there’s a difference between being a Jesus historicist and an anti-mythicist, much like the difference between atheist and anti-theist), thinks that Richard Carrier is attempting to bluff mathematical precision onto the conclusion that Jesus didn’t exist by way of Bayes.

This is a fundamental misunderstanding of two different types of probability. Objective probability is what one thinks of when one thinks of mathematical precision. Subjective probability confers no such precision. There can be 100% objective mathematical probability, but 100% subjective probability is, at the least, unreasonable. I’ll quote a few words of Grand Bayesian Eliezer Yudkowsky:

Infinite Certainty:

Yet the map is not the territory: if I say that I am 99% confident that 2 + 2 = 4, it doesn’t mean that I think “2 + 2 = 4” is true to within 99% precision, or that “2 + 2 = 4” is true 99 times out of 100.  The proposition in which I repose my confidence is the proposition that “2 + 2 = 4 is always and exactly true”, not the proposition “2 + 2 = 4 is mostly and usually true”.

I hope that impresses upon you. The “map”, that is, our model of the world, cannot have 100% certainty. However, the “territory” can. There is objectively 100% probability that 2 + 2 = 4; that is, from a Frequentist perspective, every single time we’ve put 2 and 2 together we get 4. But from a subjective point of view, asserting 100% probability that 2 + 2 = 4 would translate into infinite certainty. And it would take infinitely strong evidence to convince someone otherwise (how to convince Yudkowsky that 2 + 2 = 3). This is explained in a following post of Yudkowsky’s:

0 and 1 Are Not Probabilities:

In the usual way of writing probabilities, probabilities are between 0 and 1.  A coin might have a probability of 0.5 of coming up tails, or the weatherman might assign probability 0.9 to rain tomorrow.

This isn’t the only way of writing probabilities, though.  For example, you can transform probabilities into odds via the transformation O = (P / (1 – P)).  So a probability of 50% would go to odds of 0.5/0.5 or 1, usually written 1:1, while a probability of 0.9 would go to odds of 0.9/0.1 or 9, usually written 9:1.  To take odds back to probabilities you use P = (O / (1 + O)), and this is perfectly reversible, so the transformation is an isomorphism—a two-way reversible mapping.  Thus, probabilities and odds are isomorphic, and you can use one or the other according to convenience.

[…]

Why am I saying all this?  To show that “odd ratios” are just as legitimate a way of mapping uncertainties onto real numbers as “probabilities”.  Odds ratios are more convenient for some operations, probabilities are more convenient for others.  A famous proof called Cox’s Theorem (plus various extensions and refinements thereof) shows that all ways of representing uncertainties that obey some reasonable-sounding constraints, end up isomorphic to each other.

Why does it matter that odds ratios are just as legitimate as probabilities?  Probabilities as ordinarily written are between 0 and 1, and both 0 and 1 look like they ought to be readily reachable quantities—it’s easy to see 1 zebra or 0 unicorns.  But when you transform probabilities onto odds ratios, 0 goes to 0, but 1 goes to positive infinity.  Now absolute truth doesn’t look like it should be so easy to reach.

A representation that makes it even simpler to do Bayesian updates is the log odds—this is how E. T. Jaynes recommended thinking about probabilities.  For example, let’s say that the prior probability of a proposition is 0.0001—this corresponds to a log odds of around -40 decibels.  Then you see evidence that seems 100 times more likely if the proposition is true than if it is false.  This is 20 decibels of evidence.  So the posterior odds are around -40 db + 20 db = -20 db, that is, the posterior probability is ~0.01.

When you transform probabilities to log odds, 0 goes onto negative infinity and 1 goes onto positive infinity.  Now both infinite certainty and infinite improbability seem a bit more out-of-reach.

This makes sense. If I have a prior probability of 0 for some hypothesis, what sort of evidence could move it beyond 0? Anything multiplied by 0 is still 0.

But it doesn’t seem to be sticking, the distinction between objective and subjective probability. Just like an unflinching Frequentist, Hoffmann claims that subjective probability has no utility in formulating arguments and only objective probability is the “true” probability. In his mind, 100% “certainty” is possible, because he is only thinking of probability in terms of a Frequentist. And why not? 100% is a valid type of mathematical precision; 2 + 2 being equal to 4 has happened 100% of the time.

Richard Carrier’s point in introducing Bayes Theorem to the study of the historical Jesus (and history in general) isn’t mathematical precision or the illusion of it. Carrier’s point is that historians should follow the rules of logic when constructing arguments. The rules of probability follow from the rules of logic, thus historians should also follow the rules of probability when constructing arguments. The easiest way to do that is Bayes Theorem. Both objective and subjective probability have to follow the rules of probability, just like real premises and hypothetical premises have to follow the rules of logic when constructing arguments.

Now I’m no psychologist, but I think something a bit more nefarious is going on. There is a lot of bad blood between Carrier and Hoffmann. What I think is happening is that Hoffmann thinks that any argument that is being used by Carrier or for mythicism must be wrong, because Jesus existed. That, of course, is a logical fallacy. And I’m not even sure it’s the type of logical fallacy that is weak Bayesian evidence. 

 
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Posted by on June 5, 2012 in Bayes

 
 
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