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

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On the ontology of the objective Bayesian probability interpretation

Warning: this post is going to analyse mathematical concepts and will most likely cause intense headaches to non-mathematical brains.

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 At the beginning I wanted to make it understandable for lay people before I realized I am not the right man for such a huge task.

I considered it necessary to write it since Bayesian considerations plays a very important role in many scientific and philosophical fields, including metaphysic problems such as the existence of God.

Basically, objective Bayesianism is a theory of knowledge according to which probabilities are degrees of belief (and vice-versa) whose values can be objectively identified by every rational agent disposing of the same information.

It stands in opposition to frequentism which stipulates that the probability of an event is identical with the frequency of a great (nearly infinite) number of events.

I illustrated how this plays out in a previous post.

The name of the philosophy stems from Bayes theorem which stipulates that

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where P(A|B) is the probability of an event A given an event B, B the probability of the event B given the event A, P(A) and P(B) the total probabilities of the event A and B, respectively.

At that point, it is important to realize that the Bayesian identification of these probabilities with degrees of belief in the hypotheses A and B is a philosophical decision and not a mathematical result, as many Bayesians seem to believe.

Bayes theorem is utilized to actualize the probability of the theory A as new data (the truth of B) come in. Unless one believes in infinite regress, there is going to be basic probabilities called priors which cannot themselves be deduced from former probabilities or likelihoods.

Here I want to go into two closely related problems of Bayesian epistemology, namely those of the ontological nature of these probabilities and the values one objectively assigns to them.

Let us consider that I throw a coin in the air. My degree of belief (1/2) it will land on heads is a subjective brain state which may (or should) be related to a frequency of action if betting money is involved.

But let us now consider 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?

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The great science philosopher Elliot Sobert wrote this about this particular situation:

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.”

Frequentism provides us with well-defined probabilities in many situations. The likelihood of getting a coin coming down as heads is identical with the frequency of this event if I were to repeat it an infinite number of times and the central limit theorem guarantees that one gets an increasingly better approximation of this quantity with a growing number of trials.

But what does the likelihood of the theory of universal gravitation being 2%, 5% or 15% mean?

And once one has come up with a definition one thinks to be valid, what is the objective value for the probability prior to any observation being taken into account?

I could not find any answer in the Bayesian papers I have read until now, these questions are apparently best ignored. But to my mind they are very important if you pretend to be building up a theory of knowledge based on probabilities.

 

Next episode: a mathematical proof of Bayesianism?

 

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Why probabilities matter

 

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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

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Next episod: Naked Bayesianism.