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:
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.
FH (Fag Hating): the individual passionately hates gay people.
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(FH∩CE) = 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(FH∩CE) + 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(FH∩CE) = f(CE∩FH) = f(CE|FH)*f(FH)
Thus f(CE|FH) = f(FH∩CE) / f(FH)
Given a frequentist interpretation of probability, this entails that
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.
In that context, the probabilities correspond to our confidence in the truth of theories and hypotheses.
We have no compelling grounds for thinking that
, 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)?
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