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I am currently reading up on Bayesian inference in Richard Carrier’s book about proving history. I have known Bayesian’s approach for over a decade by now, mostly from my work in the natural sciences, and I was curious about how one would apply Bayesian to disciplines that are far less math-laden and that more often than not do not rely on measurements. I am quite aware of the flaws of this book, some of which have been addressed elsewhere. However, despite these flaws, I quite like Carrier’s Book, mostly since it has increased my appreciation for the utility of Bayesian inference, especially when applying it to foundational questions one typical asks oneself when debating a topic. Examples for such topics are:

  • How many times do I need to show that the opponent’s hypothesis does not track reality?
  • In case my opponent is hedging her bet, does this count against her hypothesis? And if yes, how much so?

Note well that I presume the reader’s general familiarity with Bayes’s formula and how to use it. In its short form it reads

P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)},

where

  • H stands for a hypothesis than can be affected by evidence;
  • E is new evidence, i.e. data that was not available when forming and perhaps testing H the last time;
  • P(H|E) is the probability of H given the new evidence.
  • P(E|H) is the probability of observing given H. It indicates how likely the evidence is given the hypothesis at hand;
  • P(H) is the probability of being true before E is observed;
  • P(E) is the probability of the evidence.

For a debate, it is often sufficient to assume that only two hypotheses, viz. yours (H_0) and that of the opponent (H_1), need to be considered. In that case, the above equation can be written in a very handy way:

\frac{P(H_0|E)}{P(H_1|E)}=\frac{P(E|H_0)}{P(E|H_1)}\cdot \frac{P(H_0)}{P(H_1)}.

Note well that P(H_1)=1-P(H_0).

In this equation, each ratio represents odds. For instance, {P(H_0)} / {P(H_1)} represents the odds for your hypothesis over that of the opponent before new evidence is presented. Let me illustrate this with a hypothetical example, where your hypothesis is very unlikely: P(H_0)= 10^{-3}. In this case P(H_0)/P(H_1) = 10^{-3}/0.999 \approx 1:1000, i.e. your hypothesis is about 1000 times less likely than that of your opponent.

For brevity, let us write the above equation as

O(H|E) = O(E|H) \cdot O(H),

where O(H) stands for P(H_0)/P(H_1), etc.

With the above equations at hand we are almost ready to address the afore stated questions. Before addressing the above questions, let me illustrate the usage of the “odds” equation with an example: the quest for extraterrestrial life on exoplanets. In this hypothetical example, the upper bound of O(life exists on a “Goldilock” planet) was set to 1:1000, based on theoretical arguments and the failure of finding life on “Goldilock” planets in the past. Let us now assume that, somewhat later, indications for life on a new exoplanet are found. However, initially, the strength of the evidence is rather weak: it makes it only two times more likely that the new planet harbours life than not. In this case

O(H|E) = 2 \cdot 10^{-3}.

In other words, the likelihood of extraterrestrial life has only increased by a factor of 2. However, let us assume that later investigations with independent methods reduce P(E|H_1) (the likelihood of the evidence if there is no life on the new exoplanet) to 1:1000, then

O(H|E) \approx 2,

i.e. it is now somewhat more likely that life exists on exoplanets than not.

This example confirms one’s intuition, that one needs very strong evidence when starting from very unlikely hypotheses.

Now we are ready to address the afore stated questions.

Question 1: How much evidence does it take?

Let us assume that a set of independent pieces of evidence = 1, 2, 3 … is available and that their respective odds are O(E_j|H). Applying the above odds equation iteratively, one can show that

O(H|E) \approx O(H) \cdot \prod_j\ O(E_j|H),

where E represents the sequential application of all pieces of evidence, and \prod is the symbol for product.

The fact that the odds for the evidence enter the above Bayesian equation as a product is very important: it does not take much independent evidence to increase the odds beyond reasonable doubt. Let me illustrate this with an example from the contemporary “debate” of biological evolution versus creationism. In this example I assume, for simplicity sake, that the odds for all pieces of evidence presented are the same. Note well that this approximation underestimates O(H|E), since O(E_{j+1}|H) > O(E_j|H). The latter is true since the hypothesis actually gets more likely with the sequential application of each piece of evidence, which makes consequent, supporting pieces of evidence more likely.

In this case,

O(H|E) = O(H) \cdot O(E_{\rm individual\ evidence}|H)^n

for pieces of evidence.

In a hypothetical public debate, you are defending biological evolution, but each piece of evidence is not overwhelmingly strong, for instance O(E_{\rm individual\ evidence}|H) = 10. Let us also assume, that you charitably agreed on O(H) = 1, i.e. that both hypotheses are assumed to be equally likely at the beginning of the debate. How many pieces of evidence do you need to present to make the odds for biological evolution overwhelmingly large, for instance one billion to one? The answer is nine pieces of evidence. Note well that more pieces of evidence are needed in case the “evidence odds” are lower, but as long as they are larger than one, a surprisingly low number of pieces of evidence will make the odds for your hypothesis overwhelmingly large. For instance, if  O(E_{\rm individual\ evidence}|H) = 2, thirty pieces of evidence are needed, and if O(E_{\rm individual\ evidence}|H) = 11/10 (very weak evidence!), two hundred eighteen pieces of evidence are needed.

Also notice that one does not need to start with O(H) = 1. Let us assume that the creationist opponent thinks that the odds for biological evolution being right are one in a trillion, and that you want to indulge your opponent. Given O(E_{\rm individual\ evidence}|H) = 10, the odds for biological evolution are again overwhelming after the presentation of merely twenty-one independent pieces of evidence. This might sound like a tough challenge, but it is not.

Question 2: How often may one hedge the bet?

Examples for this can again be found in the above “debate”, i.e. that of biological evolution versus creationism. One piece of evidence that was presented in the Dover trial was the number difference of chromosomes between humans and our genetic cousins (Bonobos and Chimpanzees); our genetic cousins have one more pair of chromosomes. Assuming that biological evolution is true, and considering the short time since the three species parted, the only viable explanation in the evolutionary context is that two chromosomes fused to end up with the 23 chromosome pairs we have today. So, when evidence for fusion was found, P(E|H_{\rm evolution} \approx 1. At the same time P(E|H_{\rm creation} = 1:2 since, from a creationist point of view, the evidence could have gone either way (fusion – no fusion). So, in light of this new evidence, O(E_{\rm new}|H) = 2, i.e. the odds for biological evolution have only increased by a factor of two,and the evidence presented is thus not as strong as often suggested by the proponents of Darwinian evolution in the Dover context. However, as we have seen above, this “game” only needs to be repeated some few times. Each time the evidence could have gone the other way from a creationist point of view, i.e. each time creationism does not produce an a-priori prediction of how the evidence should turn out, the odds for biological evolution are increased by a factor of about two. So, as in the above example about how much evidence it does take, the odds for biological evolution increases exponentially with the number of pieces of evidence that did not falsify biological evolution.

Summary

While Bayesian inference confirms one’s gut feeling that one only needs a very limited amount of independent evidence to decisively tilt the odds in one’s favour, it also shows that a single “slam-dunk” case does not suffice in case the opponent is hedging the bet. But, luckily, even in this case only a very limited amount of “slam-dunk” independent pieces of evidence are needed to make one’s argument overwhelmingly strong.

 

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