### Evidence, Probability and Belief : Introducing Reverend Bayes And His Remarkable Idea

What does the internet's greatest search engine, email technology's hottest spam filters, the location of a missing nuclear submarine (the USS Scorpion), the location of a missing H bomb, and a Presbyterian minister who lived in London in the mid 18th century all have in common?

A little mathematical theorem that took the name of its discoverer : Reverend Thomas Bayes. The theorem, known as Bayes Theorem, is making inroads in science, technology, philosophy ... and yes, apologetics.

Bayes Theorem allows one to calculate something known as a posterior probability. A posterior probability is a revised probability conditional on evidence and the likelihood of that evidence being observed in two competing hypotheses. In other words, it allows us to revise our belief (expressed as a probability) in light of evidence. That is why it has such a wide application. Apologetics certainly deals with evidence, likelihoods and degrees of belief.

For starters, Dr. Alvin Plantinga uses a Bayesian argument in his famous Evolutionary Argument Against Naturalism. Plantinga examines the probability of human cognitive faculties being reliable, given the theory that human cognitive faculties have been produced by evolution. I'll give you a hint: the probability of our cognitive faculties being reliable if evolution is true ... is low.

Dr. JP Moreland uses Bayes Theorem in his argument for design offered in The Creation Hypothesis. Moreland uses Bayes to calculate a positive posterior probability that a theistic designer likely exists. This blogger does a nice write up on Moreland's use of Bayes.

Then we have Dr. Robin Collins in his fine essay, God, Design, and Fine-Tuning. He applies Bayesian thinking in comparing two competing hypotheses: the atheist single-universe hypothesis versus the theistic universe hypothesis. Collins says,

"The prime principle of confirmation is a general principle of reasoning which tells us when some observation counts as evidence in favor of one hypothesis over another. Simply put, the principle says that whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability (or is the least improbable). (Or, put slightly differently, the principle says that whenever we are considering two competing hypotheses, H1 and H2, an observation, O, counts as evidence in favor of H1 over H2 if O is more probable under H1 than it is under H2.) Moreover, the degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which the observation is more probable under the one hypothesis than the other."
What Dr. Collins is talking about are conditional probabilities given evidence : once again, Bayes is used to show how evidence favors one hypothesis versus another, and to what degree it supports one hypothesis over another. Good stuff.

Finally, we have the famous "Bart's Blunder" comment in the Craig-Ehrman debate. Craig shows that Ehrman makes probabilistic claims based on conditional probabilities, but makes an egregious error by failing to use Bayes Thereom to do it. Ehrman walks into a buzz saw. Perhaps if he had seen this blog post he could have avoided an embarrassing moment in his recent debate.

Bayes Theorem has done more than make Google's founders billionaires. It offers a compelling tool to use in your next apologetic encounter.

I have done a series on Bayes Theorem if you wish to learn more.

slaveofone said…
I've been thinking along these lines for years and never knew it had a name!

A problem occurs, however, when Positivists approach it, because Positivists believe in "the objective" and "facts" and so forth...but this theory doesn't account for absolutes like that--it deals with probabilty instead--i.e., it is not Positivistic. But neither is it Relativistic or Phenominalistic. I believe this theory is the way for us to move out of Modernism's Positivism and Post-Modernism's Phenominalism into a new and better epistemology.
The positivist's approach is self-refuting anyway. Their standard of truth is empirical in nature -- but there is no empirical proof that positivism is true -- therefore, positivism is false. ;)

The Phenomenonalists are self-refuting in a different sort of way -- they seem to punt objective knowledge away altogether -- except of course, they require that you objectively know their epistemology in order to do this. Knowing that you can't truly know anything is self-refuting.

The Bayesian approach presupposes realism because we are making inferences about a world that is really there. We trust our perceptions. In that sense, the positivist's have little to complain about.

Taking a Bayesian approach also avoids dogmatic assertions about certainty. Claims are made, but in a probabilistic way. Beliefs are most likely true. The grow in strength as evidence grows. In this sense, the post modernists have little to complain about.

Like you said, it offers somewhat of a middle ground.

I think Bayesian arguments dovetail nicely with the "justified true belief" view of knowledge. The probabilities offer the justification -- or rationality -- for the true beliefs.
Siamang said…
The positivist's approach is self-refuting anyway. Their standard of truth is empirical in nature -- but there is no empirical proof that positivism is true -- therefore, positivism is false. ;)

That's a joke, right?

I thought positivism was merely a method, not a proposition.

I have to say, the articles linked have some pretty sloppy logic tied in with the math expressions. They presume faulty presumptions, so like any math, garbage in-garbage out.

But to your writing, are you actually arguing:

1: Design advocates use the Bayes Theorem
2: Google uses the Bayes Theorem
4: Therefore the Bayes Theorem is real scientific math that disproves non-theistic evolution!
Siamang said…
Hi Dawn,

I notice on your other blog you note Dembski's criticism of Bayes. You rightly note it, but say you like it anyway.

Do you not see his point? These articles you link to front load their answer in their supposition. Bayes is useful, but only if used with reliable values. Putting God in and supposing He absolutely can do X overwhelms the rest of the equation.

Can't you see that those arguments are mathematically unsound?
"I thought positivism was merely a method, not a proposition."

It is a philosophy ... and one that refutes itself.

"1: Design advocates use the Bayes Theorem"

Actually, I don't know of any design advocates that use Bayes Theorem. I know of some philosophers who do. I think Bayes Theorem could potentially be used in arguing design, however.

"2: Google uses the Bayes Theorem"

Yes ... at least some form of it. Bayes is used in search algorithms and it is well known that Google relies on Bayes in some fashion.

True.

"4: Therefore the Bayes Theorem is real scientific math that disproves non-theistic evolution!"

Nope. Not my argument. I don't reject naturalistic evolution based on Bayes Theorem. I reject it for other reasons. But that was not the thrust of this post.

"Do you not see his point?"

I do see Dembski's point. But I thought I responded adequately to his argument in my post.

For example, one of the areas that I think Bayes Theorem may have immediate application is in lowering the probability of belief in the hypothesis that earth - sun - solar system - and galaxy formed through naturalistic means. Our estimates on the likelihoods of finding the perfect conditions for life are quite good because we have a fair amount of observational evidence from astronomy and in particular, the SETI project. Our prior probabilities and conditional probabilities are not the things of mere guesswork -- but based on what we can see.

The competing hypotheses are natural occurrence versus non-natural occurrence. Given what we observe in our own solar system and given what we observe in the rest of the universe, what are the likelihoods of a natural explanation versus a non-natural explanation. I can't see why these revised probabilities given those conditional probabilities should be totally unsound. The result is simply a revised likelihood of one hypothesis versus another.
Siamang said…
"The competing hypotheses are natural occurrence versus non-natural occurrence. Given what we observe in our own solar system and given what we observe in the rest of the universe, what are the likelihoods of a natural explanation versus a non-natural explanation."

You don't know the likelihoods. You make up some numbers weighted by your beliefs and stick them in an equation to make your arguments sound scientifically sound. They aren't. Bayes mathematics was not designed to evaluate philosophical or metaphysical ideas. It utterly fails because loading one side of the equation with an absolute (like God with a 1/1 probability) completely overwhelms the equation.

Your thumb is on the scale.
Let's run with your thumb on the scale idea.

Set up the Bayesian equation that you think I am proposing so that I can better follow your argument.
Siamang said…
P (T/E) =(P (E/T) x P (T))/P (E)

Now the numbers plugged in in the allyourmind.net link you link above are:

The liklihood of a designer God P (T/E)

equals the odds in a christian designer worldview of God creating the universe we see P(E/T) according to Careys summation of Moreland, that value is 1/1. That times the possibility for God apart from the design argument P(T). Since you believe in God, that possibility may be 1/1 as well, but at least a positive value because you DO believe it. Your thumb is on this side of the scale, you plug in a positive value, based on your own beliefs.

Now the denominator is the likelihood of our current universe if God didn't make it P(E). You assume that number is less than 1/1. In fact, because of your beliefs, you are going to assign a number less than your belief in God.

So you win the measurement. But all you did is argue that "design implies a designer" and assign numbers to it, and pretend you've sat back and folded your hands and said "this is Google's math, it made Google founders rich. Isn't it USEFUL?!?!"

You belive God exists. That's great. But what you're doing is assigning your own number to what you think your own confidence in God is, then comparing it to a different number that corresponds to your confidence in a nontheistic worldview. Well, duh, yes, we do assume that you would assign a higher number to the first term and a lower number to the second. That proves nothing but the fact that you believe one proposition over the other.

Putting it in a piece of algebra is just dressing that belief up in a miniskirt.

All this equation proves is that your BELIEF in the possibility of a designer God is stronger than your BELIEF in the universe springing forth without a designer.

The use of fancy math and stories of billionaires seems meant to impress, not enlighten.
Siamang said…
FWIW, Good Math Bad Math performed a devistating take-apart on the Collins article here:

Your sarcasm and snippy remarks notwithstanding, I appreciate you taking the time to explain yourself.

It looks like you think I am presenting Moreland's argument for design using Bayes.

I am actually exploring the use of Bayes to revise beliefs about the likelihood of pure chance as a sufficient explanation for the conditions suitable for life in the universe. Comparing pure chance and ~ pure chance.

We have gathered a fair amount of data in our exploration of our universe (thanks in large part to Hubble and SETI) -- enough to begin to make some inferences. We know what kinds of galaxies will work, and what kind won't. We know what kinds of stars will work, and which ones won't. The same is true for solar systems and planets and orbital distances and so on and so on.

Since we know what won't work, and we can observe what is out there (at least galaxies and stars and we are getting better at detecting planets in other systems), it seems we have an opportunity to form some probabilities.

It seems quite plausible that we ought to be able to build some decent models with this data and then revise our models as we observe more things ... and basically follow the data where it leads -- either towards pure chance as a likely possibility, or away from it.

Could either model be wrong? In theory, yes. Even Bayes Theorem can be used to disprove a model. I doubt that will happen in the case of theories about the origin of the universe.

Given that falsifiability (in the Popperian sense) is not really an alternative, it seems Bayes would be a better way to process the observational evidence.
Siamang said…
Without knowing what environments can and cannot support life, you cannot make any judgements. The inherant presumption is "life within an environment close to that which we find on earth."

We do not know which environments cannot support life. Only those which cannot support our general type of life.

Arguments about a fine-tuned universe presume that we know which kind of universes can't support life. But I think you know that already.

"It looks like you think I am presenting Moreland's argument for design using Bayes."

Well, you didn't advance it personally. But you linked to 3 arguments that use the same faulty scale-tipping. And you speak of them warmly and with some appreciation. As you wrote about the Plantinga piece:

"I'll give you a hint: the probability of our cognitive faculties being reliable if evolution is true ... is low."

No, Plantinga front-loads his answer. He supposed a 100% success rate should God have wanted to imbue is with reliable faculties, and he supposes a lower success rate should humans have evolved them. By stacking the odds, God wins. But that's just the nature of omnipotence, you can't plug that number into an equation without overwhelming it.

I'll stand his argument on its head:

1. If God wanted to imbue us with reliable faculties the probability of them being reliable would be 1:1, since He would not error in bestowing it.
2. Our ability to trust our faculties is nowhere near 1:1, any optical illusion proves this. Every animal that uses camoflague proves this.
3. Therefore the odds are that God didn't imbue us with reliable faculties.

It's merely faith beliefs. He hobbles evolution's chances at the same time he gives God a head start of half a block. No wonder he comes to his desired conclusion.

Which is my point about bayes. Without useful values, you're just plugging in your own beliefs and assigning them numbers.

"Given that falsifiability (in the Popperian sense) is not really an alternative, it seems Bayes would be a better way to process the observational evidence."

Can you show me where someone you linked to has brought any observational evidence and plugged in the numbers? Can you also show where they have done that and refrained from plugging in a specious value for something there is no observational evidence for?

Because I DIG the actual numbers you got for breast cancer and false-positives. Those are real numbers.

But the odds for an expecteness of a universe with life apart from a creator are impossible to calculate, since the only known numerator is 1 and the denominator is a mystery.
"Without knowing what environments can and cannot support life, you cannot make any judgements. The inherant presumption is "life within an environment close to that which we find on earth."

We do not know which environments cannot support life. Only those which cannot support our general type of life."

We know what is required to support carbon based life. So far, there is no other molecule with the properties of carbon to support complex chains. Silicon is a distant second, and there is no distant third that I am aware of.

So yes ... given what we know ... acting on what knowledge we have of life chemistry, we are quite capable of determining which types of environments are incapable of sustaining carbon based life forms.

To speculate that life may exist without organic chemistry is fanciful thinking that has gained little popular support among those who are serious about finding life "out there".

So you are right ... I am acting on knowledge we have versus than on knowledge we don't have. I think most would agree that is reasonable.
"No, Plantinga front-loads his answer. He supposed a 100% success rate should God have wanted to imbue is with reliable faculties, and he supposes a lower success rate should humans have evolved them."

I disagree. Plantinga's argument is about evolution / naturalism. He doesn't front load it with any numbers about God. He isn't comparing the God hypothesis -vs- the evolution hypothesis. Plantinga's argument is that if naturalistic evolution is true, then we have no reason to trust our cognitive abilities -- therefore, naturalistic evolution can't be true because those who claim it is true are trusting their cognitive abilities in making their case.

Therefore, naturalistic evolution is false. Another theory is needed to explain cognitive abilities. Pure survival instinct is insufficent to explain why we should trust our beliefs.

Plantinga never presents an argument that God imbues humans with reliable cognitive faculties. He is offering a defeater for naturalistic evolution -- not a positive argument for the existence of God.
"Can you show me where someone you linked to has brought any observational evidence and plugged in the numbers? Can you also show where they have done that and refrained from plugging in a specious value for something there is no observational evidence for?"

No. But I am not sure what that gains you, to be honest. Those who calculate the odds of self-assembly of nucleotide chains and protein chains using non-Bayesian approach (i.e. traditional Fischerian or frequentist methodology) are not taken seriously either.

I have found that those are hardened to the God question are simply not interested in any approach -- no matter how rational or irrational.

I happen to believe that there are those who are seriously interested in exploring life's weighty questions and looking for whatever tools are available in their question. They want to make sense out of life -- our universe -- our existence -- our purpose etc. They hold rational beliefs and hold to the correspondence view of truth.

The Bayesian approach to acknowledging prior beliefs and incorporating evidence so as to revise beliefs seems to mirror the way rational creatures such as ourselves process information and form beliefs.

I would like to see more work done in this area -- hence my blogging efforts to raise awareness.
I re-read Moreland's argument to see if you were characterizing it fairly.

I don't think Moreland's argument is very convincing ... but not for the reasons you state. In fact, I don't think you truly understood what his argument was based on your comments.

For example, you said P(E) was "the likelihood of our current universe if God didn't make it".

That is not right. Go back and re-read the blog I linked to.

It says clearly that P(E) is the "likelihood of the appearance of design without the existence of God". In other words, if our universe is random, and the processes which caused the universe to begin and life to exist were all random, then the probability that our universe would have the appearance of design would be low. I think Moreland's premise is reasonable -- I don't know why you would expect chaos to produce order -- that seems counter intuitive. At any rate, you misunderstood or misstated what the P(E) term was.

Where I personally think Moreland's argument would be contested is that the P(T) would be a big number. P(T) is the theory that God exists without any evidence of design at all. Now I happen to agree that there is plenty of evidence outside of design evidence to support that claim -- the resurrection of Christ supports it -- the moral argument supports it -- the Kalaam Cosmological argument supports it ... and so on. Atheists would of course dispute all of that.

Still, Moreland's claim at the end of the day is quite modest.

"If the probability that T is true given E is greater than the probability that T is true without E, P (T/E) > P (T), then the evidence E offers positive support for T."

In other words, if the probability of a designer given evidence of design is greater than the evidence of that God exists with no evidence of design, then any postivie design evidence offers positive support. His argument is not that God exists - case closed. It would be that design evidence strengthens the belief.

In my opinion, this is not shocking news. For that reason, I would not present Moreland's argument and expect anyone to suddenly renounce atheism.
Siamang said…
"We know what is required to support carbon based life. So far, there is no other molecule with the properties of carbon to support complex chains. Silicon is a distant second, and there is no distant third that I am aware of."

So you assume that other universes have the same chemicals that we do, and follow the same chemical laws? And that life is limited to that? That's a mighty big assumption. Can't God create life any way He sees fit?

I touched on this on the latest Cadre post about fine tuning the universe. My premise is this, if God had to fine-tune the universe to create life, then that means that God didn't create the laws of physics in the first place, he's merely following a recipe. In order to argue a fine-tuned universe, you have to say that God had no choice in the laws of physics. An all-powerful God could create any universe He wanted, and life in any form He required.

In order to argue that other possible universes cannot contain life, you limit God's power. You must say that God couldn't possibly create life in a universe with different physical values for gravity, etc. Limiting God's power is a very strange way to argue for God.

"To speculate that life may exist without organic chemistry is fanciful thinking that has gained little popular support among those who are serious about finding life "out there".

First, it's quite reasonable that SETI is looking at a very particular type of life... it's where they're most likely to find something.
Second, it does not follow that since that's where SETI has the best odds, that no such outside possibilities exist.
Third, life in our universe doesn't say anything about what life in other possible universes are. I think you're conflating the two.

"Therefore, naturalistic evolution is false. Another theory is needed to explain cognitive abilities. Pure survival instinct is insufficent to explain why we should trust our beliefs.

Plantinga never presents an argument that God imbues humans with reliable cognitive faculties. He is offering a defeater for naturalistic evolution -- not a positive argument for the existence of God."

Now I do see that he doesn't offer positive argument for God, but I do think he does bring God into the talk as a ready clean-up hitter on-deck.

I'll just note that his story of Paul and the tiger is laughable. One mistaken impression of the tiger and that story should have ended, Paul gets eaten and his faulty fearless genes die with him without being passed on. Fear is an instinct, not a belief-desire.

I am amazed at how philosophers can surmise things about biology while sitting back in their armchairs smoking their pipes and stroking their beards. Any naturalist or any physiologist will tell him EXACTLY where the "fight or flight" response occurrs in the nervous system. It's not in the higher rational centers of the brain. Animals can be specially bred to be fearless, which shows it's in the genes, not in the ideas. Natural selection made sure "fearless Pauls" don't exist today in any jungle with tigers.

That Plantinga seems to be oblivious to this tells me something.

"I have found that those are hardened to the God question are simply not interested in any approach -- no matter how rational or irrational."

I wish I had a dollar for every theist I've asked for evidence who has then asserted that my heart is hardened and I wouldn't believe it anyway even if they HAD the evidence.

His argument is not that God exists - case closed. It would be that design evidence strengthens the belief.

I'm willing to stipulate that I may not have understood the terms correctly. I also will accept that his argument that "design evidence strengthens the belief" as long as we're talking about his belief. Those of us who do not see design evidence will say the opposite, and that's where we're talking about subjective matters.

I do not see the utility in assigning numbers to these beliefs... except of course the overall desire to make these beliefs look "sciencey" and therefore more likely to be objective.
"So you assume that other universes have the same chemicals that we do, and follow the same chemical laws? And that life is limited to that? That's a mighty big assumption."

A big assumption? I would call it a rational assumption. Rather than building a case on knowledge we don't have, let's build one on what we know. We have no evidence for another universe. We have no evidence for life without carbon. Atheists and theists agree on those two points. I think it is irrational to build a worldview on things that have no evidence -- that would arguing from ignorance.

I prefer to build models on knowledge we have -- not on flights of fancy.

"An all-powerful God could create any universe He wanted, and life in any form He required."

Agreed. So far as we know, he didn't. Let's focus on this universe and these laws since it is all we can observe.

"First, it's quite reasonable that SETI is looking at a very particular type of life... it's where they're most likely to find something. Second, it does not follow that since that's where SETI has the best odds, that no such outside possibilities exist. Third, life in our universe doesn't say anything about what life in other possible universes are."

I find it ironic that you took a shot at my ostensible lack of evidence above (with your "if I had a dollar for everytime..." comment) and you seem to be arguing that life could exist in different forms and other universes could exist and so on ... though there is a complete lack of any evidence to support this argument. You told me to stick to evidence ... shouldn't we both play by those rules?

"Fear is an instinct, not a belief-desire."

You are supporting Plantinga's argument. An reasonable argument can be made that natural selection selects based on instincts. So far as I can tell, only a weak argument at best can be offered that natural selection selects based on belief-desires. Since belief in evolution is a belief not an instinct, why trust it?

"I do not see the utility in assigning numbers to these beliefs... except of course the overall desire to make these beliefs look "sciencey" and therefore more likely to be objective."

Except sciency beliefs are no more objective than non-sciency beliefs. Beliefs are beliefs -- they are based on inferences and reason. The question is, are they true.

I simply like Bayes Theorem because it presents beliefs in a left-brained, mathematical paradigm. For right-brained people, that is not terribly helpful. For left-brained people, however, it is a useful heuristic. A tool.

On a different note, I read your bio. I see you are an animator from my old home town, Los Angeles. My daughter wants to become an animator. i have great respect for people with your kind of talent. What sorts of things do you animate? Just curious.

Have a blessed day.
Siamang said…
Dawn Treader wrote: "A big assumption? I would call it a rational assumption. Rather than building a case on knowledge we don't have, let's build one on what we know. We have no evidence for another universe. We have no evidence for life without carbon. Atheists and theists agree on those two points. I think it is irrational to build a worldview on things that have no evidence -- that would arguing from ignorance."

I'm sorry if I mistook your reference to the conditions for life as being a reference to the Fine Tuning argument in the Collins essay. I thought you were arguing from that point. A "fine-tuned" universe argument begs the question, "what would an un-fine-tuned universe look like?" Not that we have any evidence that any such universe exists. But to make a fine-tuning assertion, we must imagine what the alternative would be.

"and you seem to be arguing that life could exist in different forms and other universes could exist and so on ... though there is a complete lack of any evidence to support this argument. You told me to stick to evidence ... shouldn't we both play by those rules?"

Sorry, it's hypothetical. As I said, to argue that this universe is fine-tuned, one must describe what an un-fine-tuned universe would look like. I make no positive assertion of either multiple universes or different forms of life. But someone saying that our universe is fine-tuned is assuming that a universe with different constants cannot support life. If all universes support life, then none of them are fine-tuned. To stick to the evidence: Since all the universes (1) we know of DO support life, none of them are fine-tuned. ;-)

"Since belief in evolution is a belief not an instinct, why trust it?"

Since belief in a human being by the nickname "Dawn Treader" is a belief not an instinct, why trust it? The answer is obvious to everyone except philosophers. ;-)

(Because the evidence points more to it than away from it.)

Except sciency beliefs are no more objective than non-sciency beliefs. Beliefs are beliefs -- they are based on inferences and reason.

And EVIDENCE! Darn that pesky evidence!

The question is, are they true.

I don't know. Let's test them.

" I see you are an animator from my old home town, Los Angeles. My daughter wants to become an animator. i have great respect for people with your kind of talent. What sorts of things do you animate? Just curious."

I'll send you private mail. I guard my privacy when posting online. Hope you don't mind. Thanks for the compliment.
I'm sorry if I mistook your reference to the conditions for life as being a reference to the Fine Tuning argument in the Collins essay.

No problem. I am not defending the FTA put forth by Collins. There is a whole different class of anthropic arguments from cosmology that focus more on Earth as a privileged planet. Those have an advantage over Collin's FTA argument. Finiteness. We can actually observe other galaxies for example. We know what is out there for the most part -- at least we can see pretty far now. We know what kind of galaxy we are in. Running the probability is therefore quite easy. Collins is arguing a completely different point and it gets too complex to discuss in this forum. For example, Collins argument faces the problem of non-normalizability of a uniform density
function over an infinite set ... that is heavy stuff and I am not qualified to make much of a contribution to that discussion.

"And EVIDENCE! Darn that pesky evidence!"

And PHILOSOPHICAL PRESUPPOSITIONS! Darn those philosphical presuppositions! :-)

"I don't know. Let's test them."

Agreed. I suppose we need to decide whose philosophical presuppositions to use when we test them ... d'oh! ;-)

"I guard my privacy when posting online. Hope you don't mind."

Gotcha. Didn't mean to pry. I just think animators rock.