Chromosome 2 Fusion and Bayes Theorem: Addendum

In the final hours of 2016, I published a blog post discussing the fusion of human chromosome 2 and its implications for evolution versus creationism in the light of Bayesian probabilities. I walked through the simple maths demonstrating that contrary to what might seem intuitive, the fusion does in fact lend evidentiary support in favour of evolution (specifically common ancestry of humans and the other Great Apes).

At the time I was learning about Bayes Theorem purely in terms of phylogenetics (which is deserving of its own post one day), rather than from a more general philosophical perspective, but since then I’ve done a little more reading on the subject so thought it would be worth making a short addendum to accompany my original post.

Originally, I was getting familiar with Bayes Theorem as a concept when I suddenly realised one of its implications out of the blue, and that formed the basis for my first blog post of the subject after some back-of-the-envelope calculations showed that my realisation was correct. In a nutshell, what I realised was that Bayes Theorem entails that data favours hypotheses that more strongly predict that data. It sounds simple enough, but when you apply this reasoning to examples it becomes much less intuitive, at least in my opinion.

Alas, what I thought was a great epiphany at the time turns out to be extremely trivial, a central insight of Bayes Theorem that is well known by philosophers. I discovered this (slightly ego-bruising) fact when I found the Stanford Encyclopedia of Philosophy’s article on Bayes Theorem, which is definitely worth a read.

Below are some selected quotes that illustrate just how fundamental this insight is:

Indeed, the Theorem’s central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology.

“Subjectivist methodology” here refers to the subjectivist approach to epistemology, which relies on weighing probability statements to inform rational beliefs.

The ratio of probabilities for two hypotheses conditional on a body of data is equal to the ratio their unconditional probabilities multiplied by the degree to which the first hypothesis surpasses the second as a predictor of the data.

In other words, if the unconditional probabilities for two opposing hypotheses are the same, as I assumed they were in my original blog post and example of human chromosome 2 fusion (P(H)=0.5), then their ratio will be 1. Since this is then multiplied by the degree to which the first hypothesis surpasses the second as a predictor of the data to find the ratio of probabilities for the two hypotheses, it is the degree to which one hypothesis surpasses the other as a predictor of the data that is the real determining factor of which hypothesis is more likely to be true.

(2.1c) If H entails E, then E incrementally confirms H.

(2.1c) provides a subjectivist rationale for the hypothetico-deductive model of confirmation. According to this model, hypotheses are incrementally confirmed by any evidence they entail. While subjectivists reject the idea that evidentiary relations can be characterized in a belief-independent manner — Bayesian confirmation is always relativized to a person and her subjective probabilities — they seek to preserve the basic insight of the H-D model by pointing out that hypotheses are incrementally supported by evidence they entail for anyone who has not already made up her mind about the hypothesis or the evidence. More precisely, if H entails E, then PE(H) = P(H)/P(E), which exceeds P(H) whenever 1 > P(E), P(H) > 0. This explains why scientists so often seek to design experiments that fit the H-D paradigm. Even when evidentiary relations are relativized to subjective probabilities, experiments in which the hypothesis under test entails the data will be regarded as evidentially relevant by anyone who has not yet made up his mind about the hypothesis or the data. The degree of incremental confirmation will vary among people depending on their prior levels of confidence in H and E , but everyone will agree that the data incrementally supports the hypothesis to at least some degree.

As the quote says, this is why the hypothetico-deductive model of confirmation is so important, to the point where it has just about become synonymous with the “scientific method”, although obviously the philosophy of science on this goes much deeper and way beyond my knowledge. Scientists come up with a hypothesis and then design an experiment to test that hypothesis by deriving a set of outcomes that the hypothesis predicts compared to an opposing or null hypothesis. If the first experiments returns data that is consistent with the predictions of the first hypothesis instead of the opposing or null hypothesis, then the first hypothesis gains evidentiary support.

(2.1e) Weak Likelihood PrincipleE provides incremental evidence for H if and only if PH(E) > P~H(E). More generally, if PH(E) > PH*(E) and P~H(~E) ≥ P~H*(~E), then E provides more incremental evidence for H than for H*.

(2.1e) captures one core message of Bayes’ Theorem for theories of confirmation. Let’s say that H is uniformly better than H* as predictor of E‘s truth-value when (a) H predicts E more strongly than H* does, and (b) ~H predicts ~E more strongly than ~H* does. According to the weak likelihood principle, hypotheses that are uniformly better predictors of the data are better supported by the data. For example, the fact that little Johnny is a Christian is better evidence for thinking that his parents are Christian than for thinking that they are Hindu because (a) a far higher proportion of Christian parents than Hindu have Christian children, and (b) a far higher proportion of non-Christian parents than non-Hindu parents have non-Christian children.

Indeed, the weak likelihood principle must be an integral part of any account of evidential relevance that deserves the title “Bayesian”. To deny it is to misunderstand the central message of Bayes’ Theorem for questions of evidence: namely, that hypotheses are confirmed by data they predict.

I just wanted to drive the point home one last time. According to the “weak likelihood principle”, the hypothesis that predicts the data more strongly is better supported by the data when it comes in. I’ll adapt the “little Johnny” scenario to relate back to the fusion resulting in human chromosome 2. Unfortunately point (b) comes out a little convoluted because (2.1e) and the “little Johnny” scenario is referring to a case there there are multiple options available: the hypothesis (H), not the hypothesis (~H), the alternate hypothesis (H*), and not the alternative hypothesis (~H*), while I set up common ancestry vs separate ancestry (“creationism”) as a true dichotomy for the sake of simplicity.  If it helps, you can convert the “little Johnny” scenario into a dichotomy by replacing “Hindu” with “non-Christian”, and “non-Hindu” with “Christian”.

The fact that a fusion produced human chromosome 2 (little Johnny is a Christian) is better evidence for thinking that humans the other Great Apes share a common ancestry (his parents are Christian) than for thinking that they don’t share common ancestry (they are Hindu) because (a) common ancestry predicts the fusion (having Christian parents is a good predictor of having a Christian child), and (b) having separate ancestry predicts a lack of fusion more than common ancestry does (having non-Christian parents predicts having a non-Christian child more than having non-Hindu parents does).

I’ll let Stanford’s Encyclopedia of Philosophy summarise it one final time:

Though a mathematical triviality, the Theorem’s central insight — that a hypothesis is supported by any body of data it renders probable — lies at the heart of all subjectivist approaches to epistemology, statistics, and inductive logic.

 

I have to confess that until recently I was very dismissive of philosophy, at least as it compared to science, and I fear that this is a sentiment shared by the majority of scientists, or at least the younger ones. I didn’t see the influence that philosophy has on science and the rationale behind the scientific method, whereas now I think I’m starting to get a glimpse as I slowly read more on the topic. It’s more interesting than I thought it would be. What seems obvious to me now is that there should be more of an emphasis on the philosophy of science in the first semesters of undergraduate science education, if not earlier. I for one didn’t have any training in it whatsoever (maybe one 1 hour lecture if that, I can’t remember), and my understanding is that this is fairly common, at least in the UK. It’s quite scandalous really, undergraduate degree programs are where the next generation of scientists really begin their training but a large proportion of them aren’t being given the foundational understanding of the scientific method. Sure, we learn what the scientific method broadly-speaking is, how to formulate good hypotheses and design good experiments etc, but I think it’s equally important to understand the underlying reasoning behind it all. Some great minds have pondered these questions over the course of the last few hundred years, the least we could do would be to spend a few hours learning about the conclusions they came to and how they came to them.

 

Comments and queries are welcome.

-RM

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