In the following post, I won’t try to calculate specific values but rather to explicate my own Knowledge-dependent frequentist probabilities by using particular examples.
I strongly encourage every reader new to this topic to first read my groundwork (Click here).
The great evolutionary biologist Stephen Jay Gould was famous for his view that Evolution follows utterly unpredictable paths so that the emergence of any species can be viewed as a “cosmic accident”.
“Homo sapiens [are] a tiny twig on an improbable branch of a contingent limb on a fortunate tree.”
Dr. Stephen Jay Gould, the late Harvard paleontologist, crystallized the question in his book ”Wonderful Life.” What would happen, he asked, if the tape of the history of life were rewound and replayed? For many, including Dr. Gould, the answer was clear. He wrote that ”any replay of the tape would lead evolution down a pathway radically different from the road actually taken.”
You’re welcome to complement my list by adding other quotations. 🙂
Evolution of man
So, according to Stephen Jay Gould, the probability that human life would have evolved on our planet was extremely low, because countless other outcomes would have been possible as well.
Here, I’m interested to know what this probability p(Homo) means ontologically.
For a Bayesian, p(Homo) means the degree of belief we should have that a young planet having exactly the same features as ours back then would harbor a complex evolution leading to our species.
Many Bayesians like to model their degrees of belief in terms of betting amount, but in that situation this seems rather awkward since none of them would still be alive when the outcome of the wager will be known.
Let us consider (for the sake of the argument) an infinite space which also necessarily contain an infinite number of planets perfectly identical to our earth (according to the law of the large numbers.)
According to traditional frequentism, the probability p(Homo) that a planet identical to our world would produce mankind is given as the ratio of primitive earths having brought about humans divided by the total number of planets identical to ours for a large enough (actually endless) number of samples:
p(Homo) ≈≈ f(Homo) = N(Homo) / N(Primitive_Earths).
According to my own version of frequentism, the planets considered in the definition of probability do not have to be identical to our earth but to ALL PAST characteristics of our earth we’re aware of.
Let PrimiEarths‘ be the name of such a planet back then.
The probability of the evolution of human life would be defined as the limit p'(Homo) of
f'(Homo) = N'(Homo) / N(PrimiEarths‘)
whereby N(PrimiEarths‘) are all primitive planets in our hypothetical endless universe encompassing all features we are aware of on our own planet back then and N'(Homo) is the number of such planets where human beings evolved.
It is my contention that if this quantity exists (that is the ratio converges to a fixed value whereas the size of the sample is enlarged), all Bayesians would adopt p'(Homo) as their own degree of belief.
But what if there were no such convergence? In other words, while one would consider more and more N(PrimiEarths‘) , f'(Homo) would keep fluctuating between 0 and 1 without zooming in to a fixed value.
If that is the case, this means that the phenomenon “Human life evolving on a planet gathering the features we know” is completely unpredictable and cannot therefore be associated to a Bayesian degree of belief either, which would mean nothing more than a purely subjective psychological state.
Evolution of bird
I want to further illustrate the viability of my probabilistic ontology by considering another evolutionary event, namely the appearance of the first birds.
Let us define D as : “Dinosaurs were the forefathers of all modern birds”, a view which has apparently become mainstream over the last decades.
For a Bayesian, p(D) is the degree of belief about this event every rational agent ought to have.
Since this is an unique event of the past, many Bayesians keep arguing that it can’t be grasped by frequentism and can only be studied if one adopts a Bayesian epistemology.
It is my contention this can be avoided by resorting to my Knowledge-Dependent Frequentism (KDF).
Let us define N(Earths’) the number of planets encompassing all features we are aware of on our modern earth (including, of course, the countless birds crowding out the sky, and the numerous fossils found under the ground).
Let us define N(Dino’) as the number of these planets where all birds originated from dinosaurs.
According to my frequentism, f(D) = N(Dino’) / N(Earths’), and p(D) is the limit of f(D) as the sample is increasingly enlarged.
If p(D) is strong, this means that on most earth-like planets containing birds, the ancestors of birds were gruesome reptilians.
But if p(D) is weak (such as 0.05), it means than among the birds of 100 planets having exactly the known features of our earth, only 5 would descend from the grand dragons of Jurassic Park.
Again, what would occur if p(D) didn’t exist because f(d) doesn’t converge as the sample is increased?
This would mean that given our current knowledge, bird evolution is an entirely unpredictable phenomenon for which there can be no objective degree of belief every rational agent ought to satisfy.
A physical probability dependent on one’s knowledge
In my whole post, my goal was to argue for an alternative view of probability which can combine both strengths of traditional Frequentism and Bayesianism.
Like Frequentism, it is a physical or objective view of probability which isn’t defined in terms of the psychological or neurological state of the agent.
But like Bayesianism, it takes into account the fact that the knowledge of a real agent is always limited and include it into the definition of the probability.
To my mind, Knowledge-Dependent Frequentism (KDF) seems promising in that it allows one to handle the probabilities of single events while upholding a solid connection to the objectivity of the real world.
In future posts I’ll start out applying this concept to the probabilistic investigations of historical problems, as Dr. Richard Carrier is currently doing.
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