The crazy bookmaker and the Cult of probability

A Critique of the Dutch Book Argument

 

Many neutral observers concur into thinking we are assisting to the formation of a new religion among hopelessly nerdy people.

The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy (Unabridged)

I’m thinking of course on what has been called hardcore Bayesianism, the epistemology according to which each proposition (“Tomorrow it’ll rain”, “String theory is the true description of the world”, “There is no god” etc.) has a probability which can and should be computed under almost every conceivable circumstance.

In a previous post I briefly explained the two main theories of probabilities, frequentism and Bayesianism. In another post, I laid out my own alternative view called “knowledge-dependent frequentism” which attempts at keeping the objectivity of frequentism while including the limited knowledge of the agent. An application to the Theory of Evolution can be found here.

It is not rare to hear Bayesians talk about their own view of probability as a life-saving truth you cannot live without, or a bit more modestly as THE “key to the universe“.

While trying to win new converts, they often put it as if it were all about accepting Bayes theorem whose truth is certain since it has been mathematically proven. This is a tactic I’ve seen Richard Carrier repeatedly employing.

http://www.utne.com/~/media/Images/UTR/Editorial/Articles/Online%20Articles/2012/07-01/Quest%20for%20the%20Historical%20Jesus/u-proving-history.jpg

 

I wrote this post as a reply for showing that frequentists accept Bayes theorem as well, and that the matter of the dispute isn’t about its mathematical demonstration but about whether or not one accepts that for every proposition, there exists a rational degree of belief behaving like a probability.

 

Establishing the necessity of probabilistic coherence

 

One very popular argument aiming at establishing this is the “Dutch Book Argument” (DBA). I think it is no exaggeration to state that many committed Bayesians venerate it with almost the same degree of devotion a Conservative Evangelical feels towards the doctrine of Biblical inerrancy.

FOBTs on Green Lanes

Put forward by Ramsey and De Finetti, it defines a very specific betting game whose participants are threatened by a sure loss (“being Dutch booked”) if the amounts of their odds do not fulfill the basic axioms of probabilities, the so-called Kolmogorov’s axioms (I hope my non-geeky readers will forgive me one day for becoming so shamelessly boring…):

1) the probability of an event is always a real positive number

2)  the probability of an event regrouping all possibilities is equal to 1

3) the probability of the sum of disjoint events is equal to the sum of the probability of each event

 

The betting game upon which the DBA lies is defined as follows: (You can skip this more technical green part whose comprehension isn’t necessary for following the basic thrust of my criticism of the DBA).

 

 

A not very wise wager

 

Let us consider an event E upon which it must be wagered.

The bookmaker determines a sum of money S (say 100 €) that a person R  (Receiver) will get from a person G (Giver) if E comes true. But the person R  has to give p*S to the person G beforehand.

The bookmaker determines himself who is going to be R and who is going to be G.

 

Holding fast to these rules, it’s possible to demonstrate that a clever bookmaker can set up things in such a way that any better not choosing p respecting the laws of probabilities will lose money regardless of the outcome of the event.

Let us consider for example that a better wagers upon the propositions 

1) “Tomorrow it will snow” with P1 = 0.65  and upon

2) “Tomorrow it will not snow” with P2 = 0.70.

P1 and P2 violate the laws of probability because the sum of the probabilities of these two mutually exclusive events should be 1 instead of 1.35

In this case, the bookmaker would choose to be G and first get P1*S + P2*S = 100*(1.135) = 135 €  from his better R. Afterwards, he wins in the two cases:

– It snows. He must give 100 € to R because of 1).  The bookmaker’s gain is  135 € – 100 = 35 €

– It doesn’t snow. He must give 100 € to R because of 2).  The bookmaker’s gain is also 135 € – 100 = 35 €

 

Let us consider the same example where this time the better comes up with P1 = 0.20 and P2 = 0.3 whose sum is largely inferior to 1.

The Bookmaker would choose to be R giving 0.20*100 = 20 € about the snow and 0.3*100 = 30 € about the absence of snow. Again, he wins in both cases:

– It snows. The better must give 100 € to R (the bookmaker) because of 1).  The bookmaker’s gain is -30 – 20 +100 = 50 €

– It does not snows. The better must give 100 € to R (the bookmaker) because of 2).  The bookmaker’s gain is  -30 – 20 +100 = 50 €

 

In both cases, P1 and P2 having fulfilled the probability axioms would have been BOTH a necessary and sufficient condition for keeping the sure loss from happening.

The same demonstration can be generalized to all other basic axioms of probabilities.

 

The thrust of the argument and its shortcomings

https://i1.wp.com/i-love-my-life.com/wp-content/uploads/2011/08/Irrational.jpg

 

The Dutch Book Argument can be formulated as follows:

1) It is irrational to be involved in a bet where you’re bound to lose

2) One can make up a betting game such that for every proposition, you’re doomed to lose if the sums you set do not satisfy the rules of probabilities. In the contrary case you’re safe.

3) Thus you’d be irrational if the amounts you set broke the rules of probabilities.

4) The amounts you set are identical to your psychological degrees of belief

5) Hence you’d be irrational if your psychological degrees of beliefs do not behave like probabilities

 

Now I could bet any amount you wish there are demonstrably countless flaws in this reasoning.

 

I’m not wagering

One unmentioned premise of this purely pragmatic argument is that the agent is willing to wager in the first place. In the large majority of situations where there will be no opportunity for him to do so, he wouldn’t be irrational if his degrees of beliefs were non-probabilistic because there would be no monetary stakes whatsoever.

Moreover, a great number of human beings always refuse to bet by principle and would of course undergo no such threat of “sure loss”.

Since it is a thought experiment, one could of course modify it in such a way that:

“If you don’t agree to participate, I’ll bring you to Guatemala where you’ll be water-boarded until you’ve given up”. 

But to my eyes and that of many observers, this would make the argument look incredibly silly and convoluted.

 

I don’t care about money

Premise 1) is far from being airtight.

Let us suppose you’re a billionaire who happens to enjoy betting moderate amounts of money for various psychological reasons. Let us further assume your sums do not respect the axioms of probabilities and as a consequence you lose 300 €, that is 0.00003% of your wealth while enjoying the whole game. One must use an extraordinarily question-begging notion of rationality for calling you “irrational” in such a situation.

 

Degrees of belief and actions

It is absolutely not true that our betting amounts HAVE to be identical or even closely related to our psychological degree of beliefs.

Let us say that a lunatic bookie threatens to kill my children if I don’t accept to engage in a series of bets concerning insignificant political events in some Chinese provinces I had never heard of previously.

Being in a situation of total ignorance, my psychological degree of beliefs are undefined and keep fluctuating in my brain. But since I want to avoid a sure loss, I make up amounts behaving like probabilities which will prevent me from getting “Dutch-booked”, i.e. amounts having nothing to do with my psychology.

So I avoid sure loss even if my psychological states didn’t behave like probabilities at any moment.

 

Propositions whose truth we’ll never discover

There are countless things we will never know (at least assuming atheism is true, as do most Bayesians.)

Let us consider the proposition: “There exists an unreachable parallel universe which is fundamentally governed by a rotation between string-theory and loop-quantum gravity and many related assertions.

Let us suppose I ask to a Bayesian friend: “Why am I irrational if my corresponding degrees of belief in my brain do not fulfill the basic rules of probability?”

The best thing he could answer me (based on the DBA) would be:

“Imagine we NOW had to set odds about each of these propositions. It is true we’ll never know anything about that during our earthly life. But imagine my atheism was wrong: there is a hell, we are both stuck in it, and the devil DEMANDS us to abide by the sums we had set at that time.

You’re irrational because the non-probabilistic degrees of belief you’re having right now means you’ll get dutch-booked by me in hell in front of the malevolent laughters of fiery demons.”

Now I have no doubt this might be a good joke for impressing a geeky girl being not too picky (which is truly an extraordinarily unlikely combination).

But it is incredibly hard to take this as a serious philosophical argument, to say the least.

 

 

A more modest Bayesianism is probably required

 

To their credits, many more moderate Bayesians have started backing away from the alleged strength and scope of the DBA and state instead that:

“First of all, pretty much no serious Bayesian that I know of uses the Dutch book argument to justify probability. Things like the Savage axioms are much more popular, and much more realistic. Therefore, the scheme does not in any way rest on whether or not you find the Dutch book scenario reasonable. These days you should think of it as an easily digestible demonstration that simple operational decision making principles can lead to the axioms of probability rather than thinking of it as the final story. It is certainly easier to understand than Savage, and an important part of it, namely the “sure thing principle”, does survive in more sophisticated approaches.”

 

Given that Savage axioms rely heavily on risk assessment, they’re bound to be related to events very well treatable through my own knowledge-dependent frequentism, and I don’t see how they could justify the existence and probabilistic nature of degree of beliefs having no connection with our current concerns (such as the evolutionary path through which a small sub-species of dinosaurs evolved countless years ago).

 

To conclude, I think there is a gigantic gap between:

– the fragility of the arguments for radical Bayesianism, its serious problems such as magically turning utter ignorance into specific knowledge.

and

– the boldness, self-righteousness and terrible arrogance of its most ardent defenders.

 

I am myself not a typical old-school frequentist and do find valuable elements in Bayesian epistemology but I find it extremely unpleasant to discuss with disagreeable folks who are much more interested in winning an argument than in humbly improving human epistemology.

 

 

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On the probability of evolution

 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.

https://lotharlorraine.files.wordpress.com/2014/03/geek-zone.gif

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”.

 

 

Stephen Jay Gould

He wrote:

We are glorious accidents of an unpredictable process with no drive to complexity, not the expected results of evolutionary principles that yearn to produce a creature capable of understanding the mode of its own necessary construction.
 
“We are here because one odd group of fishes had a peculiar fin anatomy that could transform into legs for terrestrial creatures; because the earth never froze entirely during an ice age; because a small and tenuous species, arising in Africa a quarter of a million years ago, has managed, so far, to survive by hook and by crook. We may yearn for a ‘higher answer’– but none exists”

“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

evolution

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.

Bayesian interpretation

Image Of Thomas Bayes

 

 

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.

 

Traditional frequentism

 

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).

 

Knowledge-dependent frequentism

 

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.

Dino

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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John Loftus, probabilities and the Outsider Test of Faith

John Loftus is a former fundamentalist who has become an outspoken opponent of Christianity which he desires to debunk.

He has created what he calls the “Outsider Test of Faith” which he described as follows:

“This whole inside/outside perspective is quite a dilemma and prompts me to propose and argue on behalf of the OTF, the result of which makes the presumption of skepticism the preferred stance when approaching any religious faith, especially one’s own. The outsider test is simply a challenge to test one’s own religious faith with the presumption of skepticism, as an outsider. It calls upon believers to “Test or examine your religious beliefs as if you were outsiders with the same presumption of skepticism you use to test or examine other religious beliefs.” Its presumption is that when examining any set of religious beliefs skepticism is warranted, since the odds are good that the particular set of religious beliefs you have adopted is wrong.”

But why are the odds very low (instead of unknown) to begin with? His reasoning seems to be as follows:

1) Before we start our investigation, we should consider each religion to possess the same likelihood.

2) Thus if there are (say) N = 70000 religions, the prior probality of a religion being true is 1/70000 p(R), p(R) being the total probability of a religious worldview being true.

(I could not find a writing of Loftus explicitly saying that but it seems to be what he means. However I could find one of the supporters of the OST taking that line of reasoning).

 

Objective Bayesianism and the principle of indifference

 

This is actually a straightforward application of the principle of indifference followed by objective Bayesians:

In completely unknown situations, every rational agent should assign the same probability to all outcomes or theory he is aware of.

While this principle can seem pretty intuitive to many people, it is highly problematic.

In the prestigious Standford Encyclopedia of philosophy, one can read in the article about Bayesian epistemology :

“it is generally agreed by both objectivists and subjectivists that ignorance alone cannot be the basis for assigning prior probabilities.”

To illustrate the problem,  I concocted the following story.

Once upon a time, king Lothar of Lorraine had 1000 treasures he wanted to share with his people. He disposed of 50000 red balls and 50000 white balls.

Frederic the Knight (the hero of my trilingual Christmas tale) has to choose one of those in the hope he would get one of the“goldenen Wundern”.

On Monday, Lothar distributes his treasures in a perfectly random fashion.
Frederic knows that the probability of finding the treasure in a red or in a white ball is the same: p(r) = p(w) = 0.5

On Tuesday, the great king puts 10% of the treasure within red balls and 90% within white ones.

Frederic  knows that the probabilities are   p(r) = 0.10   and    p(w) = 0.90

On Wednesday, the sovereign lord of Lorraine puts 67% of the treasures in red balls and 33% in white ones.

Frederic knows that the probabilities are p(r) = 0.67 and p(w) = 0.33

On Thursday, Frederic does not know what the wise king did with his treasure. He could have distributed them in the same way he did during one of the previous days but also have chosen a completely different method.

Therefore Frederic does not know the probabilities;   p(r) = ?  and p(w) = ?

According to the principle of indifference, Fred would be irrational because he ought to believe that p(r) = 0.5 and p(w) = 0.5 on the grounds it is an unknown situation.

This is an extremely strong claim and I could not find in the literature any hint why Frederic would be irrational by accepting his ignorance of the probabilities.

Actually, I believe that quite the contrary is the case.

If the principle of indifference were true, Fred should reason like this:

“I know that on Monday my Lord mixed the treasures randomly so that p(r) = p(w) = 0.5
I know that on Tuesday He distributed 10% in the white ones and 90% in the red ones so that p(w) = 0.10 and p(r) = 0.90
I know that on Wednesday He distributed 67% in the white ones and 33% in the red ones so that p(w) = 0.67 and p(r) = 0.33
AND
I know absolutely nothing what He did on Thursday, therefore I know tthat the probabilities are p(r) = p(w) = 0.5 exactly like on Monday. “

Now I think that this seems intuitively silly and even absurd to many people. There seems to be just no way how one can transform an utter ignorance into a specific knowledge.

Degrees of belief of a rational agent

More moderate Bayesians will probably agree with me that it is misguided to speak of a knowledge of probabilities in the fourth case. Nevertheless they might insist he should have the same confidence that the treasure is in a white ball as in a red one.

I’m afraid this changes nothing to the problem. On Monday Fred has a perfect warrant for feeling the same confidence.
How can he have the same confidence on Thursday if he knows absolutely nothing about the distribution?

So Frederic would be perfectly rational in believing that he does not know the probabilities p(r) = ? and p(w) = ?

Likewise, an alien having just landed on earth would be perfectly rational not to know the initial likelihood of the religions:
p(Christianity) = ?     p(Islam) = ?     p(Mormonism) = ? and so on and so forth.

But there is an additional problem here.

The proposition “the religion x is true one” is not related to any event and it is doubted by non-Bayesian (and moderate Bayesian) philosophers that is warranted to speak of probabilities in such a situation.

Either x is true or false and this cannot be related to any kind of frequency.

The great science philosopher Elliot Sobert (who is sympathetic to Bayesian epistemology) wrote this about the probability of a theory BEFORE any data has been taken into account:

Newton’s universal law of gravitation, when suitably supplemented with plausible background assumptions, can be said to confer probabilities on observations. But what does it mean to say that the law has a probability in the light of those observations? More puzzling still is the idea that it has a probability before any observations are taken into account. If God chose the laws of nature by drawing slips of paper from an urn, it would make sense to say that Newton’s law has an objective prior. But no one believes this process model, and nothing similar seems remotely plausible.”

He rightly reminds us t the beginning of his article that “it is not inevitable that all propositions should have probabilities. That depends on what one means by probability, a point to which I’ll return. The claim that all propositions have probabilities is a philosophical doctrine, not a theorem of mathematics.” l

So, it would be perfectly warranted for the alien to either confess his ignorance of the prior likelihoods of the various religions or perhaps even consider that these prior probabilities do not exist, as Elliot Sober did with the theory of gravitation.

In future posts, I will lay out a non-Bayesian way to evaluate the goodness of theory which only depends on the set of all known facts and don’t assume the existence of a prior probability before any data has been considered.

As we shall see, many of the probabilistic challenges of Dr. Richard Carrier against Christianity kind of dissolves if one drops the assertion that all propositions have objective prior probabilities.

To conclude, I think I have shown in this post that the probabilistic defense of the Outsider Test of Faith is unsound and depends on very questionable assumptions.

I have not, however, showed at all that the OST is flawed for it might very well be successfully defended based on pragmatic grounds. This will be the topic of future conversations.

Knowledge-dependent frequentist probabilities

 

This is going to be a (relatively) geeky post which I tried to make understandable for lay people.

Given the important role than epistemological assumptions play in debate between theists and atheists, I deemed it necessary to first write a groundwork upon which more interesting discussions (about the existence of God, the historicity of Jesus, miracles, the paranormal…) will lie.

Bayesianism, Degrees of belief

In other posts I explained why I am skeptical about the Bayesian interpretation of probabilities as degrees of belief. I see no need to adjust the intensity of our belief in string theory (which is a subjective feeling) in order to do good science or to avoid irrationality.

Many Bayesians complain that if we don’t consider subjective probabilities, a great number of fields  such as economy, biology, geography or even history would collapse.
This is a strong pragmatic ground for being a Bayesian I hear over and over again.

Central limit theorem and frequencies

I don’t think this is warranted for I believe that the incredible successes brought about by probabilistic calculations concern events which are (in principle) repeatable and therefore open to a frequentist interpretation of the related likelihoods.

According to a knowledge-dependent interpretation of frequentism I rely on the probability of an event is its frequency if the known circumstances were to be repeated an infinite number of times.

Let us consider an ideal dice which is thrown in a perfectly random way. Obviously we can only find approximations of this situation in the real world, but a computer can reasonably do the job.

In the following graphics, I plotted the results for five series of trials.

FrequencyNormal

FrequencyLog

The frequentist probability of the event is defined as

formel1

,

that is the limit of the frequency of “3” when the number of trials becomes close to infinity.

This is a mathematical abstraction which never exists in the real world, but from the 6000-th trial onward the frequency is a very good approximation of the probability which will converge to the probability according to the central limit theorem.

Actually my knowledge-dependent frequentist interpretation allows me to consider the probability of unique events which have not yet occurred.

For example, a Bayesian wrote that “the advantage of this view over the frequency interpretation is that it can deal with cases where there is no relative frequency to draw on: for example, Gigerenzer mentions the first ever heart transplant patient who was given a 70% chance of survival by the surgeon. Under the frequency interpretation that statement made no sense, because there had never actually been any similar operations by then.“

HeartTransplant

I think there are many confusions going on here.
Let us call K the total knowledge of the physician which might include the different bodily features of the patient, the state of his organs and the hazard of the novel procedure.

The frequentist probability would be defined as the ratio of surviving patients divided by the total number of patients undergoing the operation if the known circumstances underlying K were to be repeated a very great (actually infinite) number of times.formel2Granted, for many people this does not seem as intuitive as the previous example with the dice.
And it is obvious there existed for the physician no frequency he could have used to directly approximate the probability.
Nevertheless, this frequentist interpretation is by no means absurd.

The physician could very well have used Bayes’s theorem to approximate the probability while having only used other frequentist probabilities, such as the probability that the body reacting in a certain way would be followed by death or the probability that introducing a device in some organs could have lethal consequences.

Another example is the estimation of the probability it is going to rain tomorrow morning as you will wake up.

Wetter
While the situation you are confronted with might very well be unique in the whole history of mankind, the probability is well defined by the frequency of rain if all the circumstances you know of were to be repeated an extremely high number of times.

Given this extended, knowledge-dependent variant of frequentism, the probabilities of single events are meaningful and many fields considered as Bayesian (such as economical simulations, history or evolutionary biology) could be as well interpreted according to this version of frequentism.

It has a great advantage: it allows us to bypass completely subjective degrees of belief and to focus on an objective concept of probability.

Now, some Bayesians could come up and tell me that it is possible that the frequentist probabilities of the survival of the first heart transplant patient or of the weather does not exist: in other words, if the known circumstances were to be repeated an infinite number of times, the frequency would keep oscillating instead of converging to a fixed value (such as 1/6 for the dice).

Fluctuations

This is a fair objection, but such a situation would not only show that the frequentist probability does not exist but that the Bayesian interpretation is meaningless as well.

It seems utterly nonsensical to my mind to say that every rational agent ought to have a degree of belief of (say) 0.45 or 0.87 if the frequency of the event (given all known circumstances) would keep fluctuating between 0.01 and 0.99.
For in this case the event is completely unpredictable and it seems entirely misguided to associate a probability to it.

Another related problem is that in such a situation a degree of belief could be no nothing more than a pure mind state with no relation to the objective world whatsoever.

As professor Jon Williamson wrote:
Since Bayesian methods for estimating physical probabilities depend on a given prior probability function, and it is precisely the prior that is in question here, this leaves classical (frequentist) estimation methods—in particular confidence interval estimation methods—as the natural candidate for determining physical probabilities. Hence the Bayesian needs the frequentist for calibration.”

But if this frequentist probability does not exist, the Bayesian has absolutely no way to relate his degree of  belief to reality since no prior can be defined and evaluated.

Fortunately, the incredible success of the mathematical treatment of uncertain phenomenons (in biology, evolution, geology, history, economics and politics to name only a few) show that we are justified in believing in the meaningfulness of the probability of the underlying events, even if they might be quite unique.

In this way, I believe that many examples Bayesians use to argue for the indispensability of their subjectivist probabilistic concept ultimately fail because the same cases could have been handled using the frequentist concept I have outlined here.

However this still leaves out an important aspect: what are we to do about theories such as the universal gravitation, string theory or the existence of a multiverse?
It is obvious no frequentist interpretation of their truth can be given.
Does that mean that without Bayesianism we would have no way to evaluate the relative merits of such competing models in these situations?
Fortunately no, but this will be the topic of a future post.
At the moment I would hate to kill the suspense 🙂

A mathematical proof of Bayesianism?

This is going to be another boring post (at least for most people who are not nerds).

However before approaching interesting questions such as the existence of God, morality and history a sound epistemology (theory of knowledge) must already be present. During most (heated) debates between theists and atheists, people tend to take for granted many epistemological principles which are very questionable.

This is why I spend a certain amount of my time exploring such questions, as a groundwork for more applied discussions.

I highly recommand all my reader to first read my two other posts on the concept of probability before reading what follows.

Bayesianism is a theory of knowledge according to which our degrees of belief in theories are well defined probabilities taking on values between 0 and 1.

According to this view, saying that string theory has a probability of 0.2 to be true is as meaningful as saying that a normal dice randomly thrown has a probability of 1/6 to produce a “3”.

Bayesians like asserting over and over again that it is mathematically proven to say we ought to compute the likelihood of all beliefs according to the laws of probability and first and foremost Bayes formula:

BayesGeneral

Here I want to debunk this popular assertion. Bayes theorem can be mathematically proven for frequential probabilities but there is no such proof that ALL our degrees of belief behave that way.

Let us consider (as an example) the American population (360 millions people) and two features a person might have.

CE (Conservative Evangelical): the individual believes that the Bible contains no error.

God-said-it-believe-it-1

FH (Fag Hating): the individual passionately hates gay people.

homobashing

Let us suppose that 30% of Americans are CE and that 5.8% of Americans hate homosexuals.

The frequencies are f(CE) = 0.30 and f(FH) = 0.058

Let us now consider a random event: you meet an American by chance.
What is the probability that you meet a CE person and what is the probability that you meet a FH individual?
According to a frequentist interpretation, the probability equals the frequency of meeting such kinds of persons given a very great (actually infinite) number of encounters.
From this it naturally follows that p(CE) = f(CE) = 0.30 and p(FH) = f(FH) = 0.058

Let us now introduce the concept of conditional probability: if you meet a Conservative Evangelical, what is the probability that he hates faggots p(FH|CE)? (the | stands for „given“).

If you meet a fag-hating person, what is the probability that he believes in Biblical inerrancy p(CE|FH)?

To answer these questions (thereby proving Bayes theorem) it is necessary to get back to our consideration of frequencies.

Let us consider that 10% of all Conservative Evangelicals and 4% of people who are not CE hate faggots: f(FH/CE) = 0.1 and f(FH/CE) = 0.04. The symbol ⌐ stands for the negation (denial) of a proposition.

The proportion of Americans who are both conservative Evangelicals and fag-haters is f(FHCE) = f(FH/CE)*f(CE) = 0.1*0.3 = 0.03.

The proportion of Americans who are NOT conservative Evangelicals but fag-haters is f(FH∩⌐CE) = f(FH/⌐CE)*f(⌐CE) = 0.04*0.7 = 0.028.

Logically the frequency of fag-haters in the whole American population is equal to the sum of the two proportions:

f(FH) = f(FHCE) + f(FH∩⌐CE) = 0.03 + 0.028 = 0.058

But what if we are interested to know the probability that a person is a conservative Evangelical IF that person hates queers p(CE|FH)?

This corresponds to the frequency(proportion) of Conservative Evangelicals among Fag-Haters: f(CE|FH).

We know that f(FHCE) = f(CE∩FH) = f(CE|FH)*f(FH)

Thus f(CE|FH) = f(FH∩CE) / f(FH)

FagBayesFrequencies

Given a frequentist interpretation of probability, this entails that

FagBayes

which is of course Bayes theorem. We have mathematically proven it in this particular case but the rigorous mathematical demonstration would be pretty much the same given events expressable as frequencies.

If you meet an American who hates gays, the probability that he is a Conservative Evangalical is 51.72% (given the validity of my starting values above).

But let us now consider the Bayesian interpretation of probability (our degree of confidence in a theory) in a context having nothing to do with frequencies.

Let S be “String theory is true“ and UEP “an Undead Elementary Particle has been detected during an experience in the LHC“.

StringTheory

In that context, the probabilities correspond to our confidence in the truth of theories and hypotheses.

We have no compelling grounds for thinking that

StringBayes

, that is to say that is the way our brains actually work or ought to work that way in order to strive for truth.

The mathematical demonstration used to prove Bayes theorem relies on related frequencies and cannot be employed in a context where propositions (such as S and UEP) cannot be understood as frequencies.
Considering ALL our degrees of beliefs like probabilities is a philosophical decision and not an inevitable result of mathematics.

I hope that I have been not too boring for lay people.

Now I have a homework for you: what is the probability that Homeschooling Parents would like to employ my post as an introduction to probability interpretation, given that they live in the Bible Belt  p(HP|BB)?

Image Of Thomas Bayes

Naked Calvinism: on the sinful nature of man and Genesis

Youtube Version

Ever since Saint Augustine, the Western Church has always taught that
 adam_and_eve
1) man was created perfect, that is to say without any moral flaw
2) following the advice of the snake, he chose to eat the wrong fruit
3) God cursed him and he inherited a sinful nature, making hatred, lies, adultery, selfishness and many other related evils inevitable for him and all his descendents.
Most Christians, Atheists and even Muslims I talked to told me they view that as an essential Christian doctrine which can be found from the first pages of the Bible.
It might come as a surprise to many people that the Eastern Orthodox Church denies this radical change of nature following the Fall.
And if we look at the text of Genesis, which describes the very event responsible for our alleged sinful nature, we realize that they are not less Biblical than the Conservative Protestants who view them as heathens.
Now, let us take a look at a text many of us grew up with.
Genesis 3
The snake was sneakier than any of the other wild animals that the Lord God had made. One day it came to the woman and asked, “Did God tell you not to eat fruit from any tree in the garden?”
   2 The woman answered, “God said we could eat fruit from any tree in the garden, 3 except the one in the middle. He told us not to eat fruit from that tree or even to touch it. If we do, we will die.”
   4 “No, you won’t!” the snake replied. 5 “God understands what will happen on the day you eat fruit from that tree. You will see what you have done, and you will know the difference between right and wrong, just as God does.”
   6 The woman stared at the fruit. It looked beautiful and tasty. She wanted the wisdom that it would give her, and she ate some of the fruit. Her husband was there with her, so she gave some to him, and he ate it too. 7 Right away they saw what they had done, and they realized they were naked. Then they sewed fig leaves together to make something to cover themselves.
   8 Late in the afternoon a breeze began to blow, and the man and woman heard the Lord God walking in the garden. They were frightened and hid behind some trees.
9 The Lord called out to the man and asked, “Where are you?”
   10 The man answered, “I was naked, and when I heard you walking through the garden, I was frightened and hid!”
   11 “How did you know you were naked?” God asked. “Did you eat any fruit from that tree in the middle of the garden?”
   12 “It was the woman you put here with me,” the man said. “She gave me some of the fruit, and I ate it.”
   13 The Lord God then asked the woman, “What have you done?”
   “The snake tricked me,” she answered. “And I ate some of that fruit.”
   14 So the Lord God said to the snake:
   “Because of what you have done,
   you will be the only animal
      to suffer this curse—
   For as long as you live,
   you will crawl on your stomach
      and eat dirt.
15 You and this woman
            will hate each other;
      your descendants and hers
         will always be enemies.
      One of hers will strike you
         on the head,
      and you will strike him
         on the heel.”
16 Then the Lord said to the woman,
   “You will suffer terribly
         when you give birth.
   But you will still desire
   your husband,
      and he will rule over you.”
17 The Lord said to the man,
   “You listened to your wife
         and ate fruit from that tree.
   And so, the ground
   will be under a curse
      because of what you did.
   As long as you live,
   you will have to struggle
      to grow enough food.
18 Your food will be plants,
     but the ground will produce
         thorns and thistles.
19 You will have to sweat
         to earn a living;
      you were made out of soil,
      and you will once again
         turn into soil.”
   20 The man Adam named his wife Eve because she would become the mother of all who live.
   21 Then the Lord God made clothes out of animal skins for the man and his wife.
22 The Lord said, “These people now know the difference between right and wrong, just as we do. But they must not be allowed to eat fruit from the tree that lets them live forever.” 23 So the Lord God sent them out of the Garden of Eden, where they would have to work the ground from which the man had been made. 24 Then God put winged creatures at the entrance to the garden and a flaming, flashing sword to guard the way to the life-giving tree.”
Imagine now that you are a space alien reading this text for the very first time and trying to understand its meaning.
It is truly remarkable that we find absolutely no evidence of a dramatic psychological (or even biological) change turning the first morally perfect humans into atrociously wicked, greedy and selfish creatures.
The only possible reference to a psychological consequence is  “But you will still desire, your husband, and he will rule over you.” which is pretty ambiguous and falls infinitely short of describing a radical and inheritable psychological transformation.
The verse “These people now know the difference between right and wrong, just as we do. But they must not be allowed to eat fruit from the tree that lets them live forever.” is very profound and enigmatic but if one takes it at face value, it teaches that people became efficient moral realists, not psychopathic murderers!
There are situations where the absence of evidence is evidence of absence IF one would clearly expect to find certain things given the truth of a theory.
If the author(s) of Genesis really believed in the doctrine of the sinful nature, they would have clearly expressed it using sentences such as: “Curse on both of you! From now on, you and your offspring won’t stop cultivating wicked thoughts within your hearts.”
Yet this is clearly not what one finds in Genesis 3.
If one reads the sad story of Cain murdering his brother Abel, one fails to see any evidence of this sinful nature making evil deeds inevitable unless God intervenes.
According to Calvinism, God should have said: “Boy, sin flows in your very blood so that you have no other choice than sinning and committing atrocities. It is up to me (and my sovereign grace) to decide if you will kill your brother or not.”
 cainkillsabel
But what do we read in the text?
“This made Cain so angry that he could not hide his feelings. 
6 The Lord said to Cain: 
What’s wrong with you? Why do you have such an angry look on your face? 7 If you had done the right thing, you would be smiling.[c] But you did the wrong thing, and now sin is waiting to attack you like a lion. Sin wants to destroy you, but don’t let it!
 “
God warns Cain about the possible consequences of his state of mind but emphasized that it is up to him to overcome the temptation. There is no indication whatsoever that before the fall, humankind could not have been confronted with such heinous thoughts.
In order to prove the doctrine of Total depravity in their TULIP, Calvinism like to quote God’s description of the state of mankind before the flood broke in:
“The Lord saw how bad the people on earth were and that everything they thought and planned was evil. 6 He was very sorry that he had made them, 7 and he said, “I’ll destroy every living creature on earth! I’ll wipe out people, animals, birds, and reptiles. I’m sorry I ever made them.”
But if this wickedness was a consequence of the fall of Adam and Eve, this is not what we would read. The text says that such misbehaviour and misdeeds were a (not necessarily inevitable) consequence of human nature as it was originally made by God.
Let us suppose it was a consequence of the Fall itself, and that man was previously morally perfect. This would be of uttermost importance and the writer would have said:
“He was very sorry that he left them the choice between the two fruits and that they made the wrong decision.”
Once one has taken all of this into account, how likely is it that the author of Genesis had a Calvinist understanding of the Fall?
And how likely is that later theologians read into the text something which was never there?

Why probabilities matter

 

images

In real life, it’s pretty rare (some would even say utterly impossible) to be sure of anything at all, like knowing it’s going to rain in one hour, that a conservative president is going to be elected, that you will be happily married in two years and so on and so forth.

We all recognize that it is only meaningful to speak of the probability or likelihood of each of these events.

The question of how to interpret their profound nature (ontoloy) is however, far from being an easy one.

I will use the basic proposition: if I roll the dice, there is a probability of 1/6 I will get a 3 in order to illustrate the two main interpretation of the probability concept out there.

1. Frequentism

According to this interpretation, the probability of an event equals its frequency if it is repeated an infinite number of times. If you roll a dice a great number of time, the frequency of the event (that is the number of 3s divided by the total number of rollings) will converge towards 1/6.

Mathematically it is a well defined concept and in many cases it can be relatively easily approximated. One of the main difficulties is that it apparently fails to account for the likelihood of unique situations, such as that (as far as we know in 2013) the Republicans are going to win the next American elections.

This brings us to the next popular interpretation of probability.

2. Bayesianism

For Bayesians, probabilities are degrees of belief and each degree of belief is a probability.

My degree of belief that the dice will fall onto 3 is 1/6.

But what is then a „degree of belief“? It is a psychological mind state which is correlated with a certain readiness for action.

According to many proponents of Bayenianism, degrees of belief are objective in so far that every rational creature disposing of a set of information would have exactly the same.

While such a claim is largely defensible for many situations such as the rolling of dices, the spread of a disaease or the results of the next elections, there are cases where it does not seem to make any sense at all.

Take for exampling the young Isaac Newton who was considering his newly developed theory of universal gravitation. What value should his degree of belief have taken on BEFORE he had begun to consider the first data of the real world?

applenewton1

And what would it mean ontologically to say that we have a degree of belief of 60% that the theory is true? What is the relation (in that particular situation) between the intensity of certain brain processes and the objective reality?

Such considerations have led other Bayesians to give up objectivity and define „degrees of belief“ as subjective states of mind, which might however be objectively constrained in many situations.

Another criticism of (strong) Bayesianism is that it ties the concept of probability to the belief of intelligent creatures. Yet it is clear that even in an universe lacking conscious beings, the probability of the decay of an atom and of more fundamental quantum processes would still exist and be meaningful.

For completeness, I should mention the propensity interpretation of Karl Popper who viewed the likelihood of an event as an intrinsic tendency of a physical system to tend towards a certain state of affairs.

 

So this was my completely unbiased (pun intended!) views on probabilities.

When debating (and fighting!) each other, theists and atheists tend to take their own epistemology (theory of knowledge) as granted.

This often leads to fruitless and idle discussions.

This is why I want to take the time to examine how we can know, what it means to know, before discussing what we can (and cannot) know.

 theres-probably-no-god.jpg?w=500&h=283

Thematic list of ALL posts on this blog (regularly updated)

My other blog on Unidentified Aerial Phenomena (UAP)

 

 

Next episod: Naked Bayesianism.