As an aspiring scientist, I hold the Truth above all. As Hodgell once said, "That which can be destroyed by the truth should be." But what if the thing that is holding our pursuit of the Truth back is our own system? I will share an example of an argument I overheard between a theist and an atheist once - showing an instance where human intuition might fail us.

*General Transcript*

Atheist: Prove to me that God exists!

Theist: He obviously exists – can’t you see that plants growing, humans thinking, [insert laundry list here], is all His work?

Atheist: Those can easily be explained by evolutionary mechanisms!

Theist: Well prove to me that God doesn’t exist!

Atheist: I don’t have to! There may be an invisible pink unicorn baby flying around my head, there is probably not. I can’t prove that there is no unicorn, that doesn’t mean it exists!

Theist: That’s just complete reductio ad ridiculo, you could do infrared, polaroid, uv, vacuum scans, and if nothing appears it is statistically unlikely that the unicorn exists! But God is something metaphysical, you can’t do that with Him!

Atheist: Well Nietzsche killed metaphysics when he killed God. God is dead!

Theist: That is just words without argument. Can you actually…..

As one can see, the biggest problem is determining burden of proof.  Statistically speaking, this is much like the problem of defining the null hypothesis.

A theist would define: H₀ : God exists. H_a: God does not exist.

An atheist would define: H₀: God does not exist. H_a God does exist.

Both conclude that there is no significant evidence hinting at H_a over H­₀. Furthermore, and this is key, they both accept the null hypothesis. The correct statistical term for the proper conclusion if insignificant evidence exists for the acceptance of the alternate hypothesis is that one fails to reject the null hypothesis. However, human intuition fails to grasp this concept, and think in black and white, and instead we tend to accept the null hypothesis.

This is not so much a problem with statistics as it is with human intuition. Statistics usually take this form because simultaneous 100+ hypothesis considerations are taxing on the human brain. Therefore, we think of hypotheses to be defended or attacked, but not considered neutrally.

Considered a Bayesian outlook on this problem.

There are two possible outcomes: At least one deity exists(D). No deities exist(N).

Let us consider the natural evidence (Let’s call this E) before us.

P(D+N) = 1. P[(D+N)|E] = 1. P(D|E) + P(N|E) = 1. P(D|E) = 1- P(N|E).

Although the calculation of the prior probability of the probability of god existing is rather strange, and seems to reek of bias, I still argue that this is a better system of analysis than just the classical H₀ and H_a, because it effectively compares the probability of D and N with no bias inherent in the brain’s perception of the system.

Example such as these, I believe, show the flaws that result from faulty interpretations of the classical system. If instead we introduced a Bayesian perspective – the faulty interpretation would vanish.

This is a case for the expanded introduction of Bayesian probability theory. Even if cannot be applied correctly to every problem, even if it is apparently more complicated than the standard method they teach in statistics class ( I disagree here), it teaches people to analyze situations from a more objective perspective.

And if we can avoid Truth-seekers going awry due to simple biases such as those mentioned above, won’t we be that much closer to finding Truth?

Galton Visualizing Bayesian Inference (article @ CHANCE)

Excerpt:

What does Bayes Theorem look like? I do not mean what does the formula—

—look like; these days, every statistician knows that. I mean, how can we visualize the cognitive content of the theorem? What picture can we appeal to with the hope that any person curious about the theorem may look at it, and, after a bit of study say, “Why, that is clear—I can indeed see what is happening!”

Francis Galton could produce just such a picture; in fact, he built and operated a machine in 1877 that performs that calculation. But, despite having published the picture in Nature and the Proceedings of the Royal Institution of Great Britain, he never referred to it again—and no reader seems to have appreciated what it could accomplish until recently.

Schematics for the machine and its algorithm can be found at the link. This is a really cool design, and maybe it can aid Eliezer's and others' efforts to help people understand Bayes' Theorem.

Allow me to propose a thought experiment.  Suppose you, and you alone, were to make first contact with an alien species.  Since your survival and the survival of the entire human race may depend on the extraterrestrials recognizing you as a member of a rational species, how would you convey your knowledge of mathematics, logic, and the scientific method to them using only your personal knowledge and whatever tools you might reasonably have on your person on an average day?

When I thought of this question, the two methods that immediately came to mind were the Pythagorean Theorem and prime number sequences.  For instance, I could draw a rough right triangle and label one side with three dots, the other with four, and the hypotenuse with five.  However, I realized that these are fairly primitive maths.  After all, the ancient Greeks knew of them, and yet had no concept of the scientific method.  Would these likely be sufficient, and if not what would be?  Could you make a rough sketch of the first few atoms on the periodic table or other such universal phenomena so that it would be generally recognizable? Could you convey a proof of rationality in a manner that even aliens who cannot hear human vocalizations, or see in a completely different part of the EM spectrum?  Is it even in principle possible to express rationality without a common linguistic grounding?

In other words, what is the most rational thought you could convey without the benefit of common language, culture, psychology, or biology, and how would you do it?

Bonus point: Could you convey Bayes' theorem to said ET?

(Is Bayesianism even a word?  Should it be?  The suffix "ism" sets off warning lights for me.)

Visitors to LessWrong may come away with the impression that they need to be Bayesians to be rational, or to fit in here.  But most people are a long way from the point where learning Bayesian thought patterns is the most time-effective thing they can do to improve their rationality.  Most of the insights available on LessWrong don't require people to understand Bayes' Theorem (or timeless decision theory).

I'm not calling for any specific change.  Just to keep this in mind when writing things in the Wiki, or constructing a rationality workbook.

In Defense of Objective Bayesianism by Jon Williamson was mentioned recently in a post by lukeprog as the sort of book that should be being read by people on Less Wrong. Now, I have been reading it, and found some of it quite bizarre. This point in particular seems obviously false. If it’s just me, I’ll be glad to be enlightened as to what was meant. If collectively we don’t understand, that’d be pretty strong evidence that we should read more academic Bayesian stuff.

Williamson advocates use of the Maximum Entropy Principle. In short, you should take account of the limits placed on your probability by the empirical evidence, and then choose a probability distribution closest to uniform that satisfies those constraints.

So, if asked to assign a probability to an arbitrary A, you’d say p = 0.5. But if you were given evidence in the form of some constraints on p, say that p ≥ 0.8, you’d set p = 0.8, as that was the new entropy-maximising level. Constraints are restricted to Affine constraints. I found this somewhat counter-intuitive already, but I do follow what he means.

But now for the confusing bit. I quote directly;

“Suppose A is ‘Peterson is a Swede’, B is ‘Peterson is a Norwegian’, C is ‘Peterson is a Scandinavian’, and ε is ‘80% of all Scandinavians are Swedes’. Initially, the agent sets P(A) = 0.2, P(B) = 0.8, P(C) = 1 P(ε) = 0.2, P(A & ε) = P(B & ε) = 0.1. All these degrees of belief satisfy the norms of subjectivism. Updating by maxent on learning ε, the agent believes Peterson is a Swede to degree 0.8, which seems quite right. On the other hand, updating by conditionalizing on ε leads to a degree of belief of 0.5 that Peterson is a Swede, which is quite wrong. Thus, we see that maxent is to be preferred to conditionalization in this kind of example because the conditionalization update does not satisfy the new constraints X’, while the maxent update does.”

p80, 2010 edition. Note that this example is actually from Bacchus et al (1990), but Williamson quotes approvingly.

His calculation for the Bayesian update is correct; you do get 0.5. What’s more, this seems to be intuitively the right answer; the update has caused you to ‘zoom in’ on the probability mass assigned to ε, while maintaining relative proportions inside it.

As far as I can see, you get 0.8 only if we assume that Peterson is a randomly chosen Scandinavian. But if that were true, the prior given is bizarre. If he was a randomly chosen individual, the prior should have been something like P(A & ε) = 0.16 P(B & ε) = 0.04 The only way I can make sense of the prior is if constraints simply “don’t apply” until they have p=1.

Can anyone explain the reasoning behind a posterior probability of 0.8?

Could you use Bayes Theorem to figure out whether or not a given war is just?

If so, I was wondering how one would go about estimating the prior probability that a war is just.

Thanks for any help you can offer.