Monday, June 07, 2010

Bayesian inference

One thing that we keep tripping over in these conversations is the nature of "evidence". When can you say that a certain observation is "evidence" for a certain hypothesis, for the superiority of one model over another?

I tend to promote the view that almost anything can be "evidence" for an a-priori belief. If you really want to believe in a kind of system, then anything you observe can be explained in terms of that system (or you can trust that your system will probably have a way to explain it, even if you can't come up with the explanation yourself) and this reinforces the completeness and explanatory power of your system. Now, I don't REALLY believe in this, not as extremely as I make it sound. I offer this position in part to play Devil's Advocate, and in part because it is a view that is woefully under-represented in the theism-atheism debate.

I do honestly believe that, when people say "I see evidence for God" or "There is no evidence for God but there's tons of evidence for naturalism", this shows that they haven't stopped to think about what "evidence" means, or about when it is that you can or can't say that an observation supports a claim.

Instead of deconstructing their argument, or constructing a logically rigorous method for evaluating this kind of thing mathematically, allow me to just give an example to get you thinking and to help us move forwards.


Say I put several hundred golf balls and ping-pong balls into a box. Some are white and the others are yellow. Say that 70% of the golf balls are white and 30% yellow, and 20% of the ping-pong balls are white and 80% yellow. So, most of the golf balls are white (and the rest of them are yellow), and most of the ping-pong balls are yellow (and the rest are white). With me so far?

Say I mix things around for a while, to make sure everything is evenly distributed. I then reach into the box blindly, and randomly pick up a ball. I tell you that it is yellow. And now I ask: Is it more likely to have been a golf ball or a ping-pong ball?

You might say it’s probably a ping-pong ball, since the chances of a ping-pong ball being yellow (80%) are greater than the chances of a golf ball being yellow (30%).

But that's not necessarily correct. In fact, you simply do not have enough data to answer my question, not even probabilistically. And I can prove it to you.

What if I now tell you that only a tiny fraction of the balls in the box are ping-pong balls? If the box contains a thousand golf balls but only ten ping-pong balls, then three hundred of the balls are yellow golf balls and only eight are yellow ping-pong balls.This means that (despite the fact that the probability of a ping-pong ball being yellow is higher than the probability of a golf ball being yellow) most of the yellow balls in the box are golf balls, since by far most of the balls in the box are golf balls.

So, in this case, my yellow ball is probably a golf ball.

And until I told you how many golf balls there are, compared with how few ping pong-balls there are, you could not have reached this answer. You would just have been guessing (maybe based on the unfounded [and in this case, incorrect] assumption that the total number of golf balls is similar to the total number of ping-pong balls).


When you understand the thought experiment I just described, we can move on to a more rigorous explanation, and to its implications in the God-vs-no-God debate.


Mathematically: Just because X-given-A has a higher probability than X-given-B, this does not mean that A-given-X is more likely than B-given-X.

Ok, maybe I skipped a step. Bear with me.

"A" means "It's a ping-pong ball" and "B" means "It's a golf ball". "X" means "It's yellow" and "Y" means "It's white".

Just because Yellow-given-PingPongBall has a higher probability (80%) than Yellow-given-GolfBall (30%), this says nothing about the relative likelihood of GolfBall and PingPongBall (which could be, say, 99% and 1%, or anything else). This is even true if we know that we have a Yellow ball. The relative probability of GolfBall and PingPongBall is determined by a totally different set of factors (how many of each kind of ball I put in the box) than the factors that determine the probability of the color of each ball (what fraction of each kind of ball is colored each color).

In summary:

Just because Yellow-given-PingPongBall has a higher probability than Yellow-given-GolfBall, this does not mean that PingPongBall-given-Yellow is more likely than GolfBall-given-Yellow.

If we substitute "A" for "It's a ping-pong ball" and "B" for "It's a golf ball", and "X" for "It's yellow" and "Y" for "It's white", we can generalize:

Just because X-given-A has a higher probability than X-given-B, this does not mean that A-given-X is more likely than B-given-X.

Unless you can prove that A is impossible, and unless you can prove that B is impossible, then both A and B are possible. I can't know what kind of ball I have (I can't even know what kind of ball I PROBABLY have) just from looking at its color. The relative probability of A and B simply cannot be learned by comparing the relative probabilities of X-given-A and X-given-B. They can only be learned by taking multiple samples and seeing whether A or B come up more often, or by observing that A is often associated with certain kinds of observations and B is associated with certain other observations (e.g. when I draw a golf ball from this box it tends to be white, and when I draw a ping pong ball from this box it tends to be yellow).

Here's the important part.

You might say "In a world guided by a God, it is probable that intelligent life would be created. In a world not guided by a God, it is extremely unlikely that intelligent life would form". I could dispute both of these assumptions, but let's assume that they are correct for now. (And you can feel free to insert anything you like instead of "intelligent life", such as "beings who believe in God", or "cosmological constants that allow for chemistry").

Saying that the probability of IntelligentLife-given-God is higher than the probability of IntelligentLife-given-NoGod DOES NOT allow you to logically infer that the probability of God-given-IntelligentLife is higher than the probability of NoGod-given-Intelligent life.

In other words, saying “The evolution of intelligence [or any other "evidence for God" you may wish to insert here] is unlikely in a world without God and likely in a world with God, and we see intelligence, therefore God is more likely than No God” is logically equivalent to saying "The ball is yellow, therefore it's probably a ping-pong, ball since most ping-pong balls are yellow". Both statements are fallacious: it's possible that most yellow balls are golf balls, even if most ping-pong balls are be yellow.

This should allow us to see that an evidence-based approach to this God debate is hopeless. No matter how much we learn about the world and the different ways it might have become as it currently is, this knowledge will never tell us about the relative probability of World-given-“God” versus that of World-given-“no God”. All that each person can say is “I don’t need God to satisfactorily explain what I see around me”, or “I need God to satisfactorily explain what I see around me”, both of which are valid depending on what questions you’re asking. (I think that the questions that require God to be satisfactorily explained are not particularly meaningful, but I can see that this is a matter of personal taste).

Holders of different axioms will interpret the evidence so as to make their axioms and models seem, to them, more likely than an opponent’s axioms and models. But as elementary Statistics reveal, this is a fallacy. Besides, the bottom line is, even if it were not a fallacy, it does not deny that the opponent’s set of axiomatic assumptions is still possible, though unlikely. So the opponents can still go on believing what they want.


theevidence said...

One of the fundamentals of science is repeatability. You don't just pull out one ball. You do it over and over again, until you have statistical significance. After a bit, I'll easily realize there are more golf balls than ping pong balls, because the process of science uses repeated observations to increasingly make better predictions about the truth.

theevidence said...

One other thought, "Possibilities are infinite, probabilities are few". Just because you picked one particular "fact" that isn't repeatable, doesn't mean that there aren't other experiments that are repeatable that do give us statistically significant evidence. Like the multiple methods for determining the age of the universe and the distance of the stars. There are an infinite plurality of samples that make the probability of a 6000 year old god/religion highly improbable / approaching impossibility.

Randy Kirk said...

Interesting that you would choose to go to the issue of what is evidence as being a major underpinning of the entire issue. I was researching definitions of evidence with much the same purpose.

First, as to your Bernardo's approach: If we only have the three variables and only know two of them, then your assumptions are very sound, but eventually come down to never having a satisfactory answer at all.

On the other hand, if we have 10,000 variable or a 100,000 and we have strong evidence for a 5,000 or 60,000 of those, then we can certainly feel comfortable in making more than a guess. Obviously in the case of naturalism vs Supernaturalism, we have huge amounts of evidence going both ways. Which is why it becomes fun and instructive to debate, assuming neither side claims that the other sides "evidence" isn't.

So then what is evidence. Seems that it depends on the discipline. Evidence for an historian is quite different than for a scientist. However some of the scientist's evidence is much like the historians (not replicable in a lab as it is from the fossil record or geological record or ice core record.) The biggest problem for the current definition of evidence in science is that it assumes naturalism and therefore defines out God. This has not always been the case and makes debate impossible. (Or at least creates confusion on the part of some atheistic naturalists as to how to have an intellectually honest debate.) You can't just define away the issue. If I experience spiritual sensations, you can no more define that away than if you are color blind and you want to define away my sensation of color.

Legal evidence is altogether different again as it is broken down into various kinds (at least 10) such as circumstantial and hard.

Therefore, in order to have a debate, it would seem absolutely necessary to accept as evidence the truth claims of both sides. We can then debate the quality, quantity, applicability, etc of that evidence.
For example the Old Testament prophesies and their fulfillment in later OT books or in the NT. Clearly there is such a record. Clearly there is massive documentation. It can be debated as to when what was written, whether it was changed, and whether the later historical events were fulfillments or not. Even whether or not those later events took place, or took place in the way that they are described historically. Then one could argue whether if such events were literal fulfillments of prophesy, whether that is meaningful evidence of God or something else.

Randy Kirk said...

Just a quick tag. The current return of the Jews from the far reaches of the planet to reclaim Israel would be a great place to start the specific debate I mentioned. To this pea brain, it seems almost impossible to imagine that this one of a kind religious group has survived 2000 years of every possible attempt to destroy it, and has then resettled a land surrounded by hostiles and defended every attempt to crush it. And that such a return was clearly prophesied.

Bernardo said...


It's smart of you to bring up things like circumstancial evidence and hard evidence. You're right about how this is not a simple probabilistic calculation with 3 variables.

But I do still think that you're wrong about how "we can certainly feel comfortable in making more than a guess" and how "in the case of naturalism vs Supernaturalism, we have huge amounts of evidence going both ways".

I think that debating these ideas is fun and instructive, but I do insist that your "evidence" isn't "evidence". This debate is instructive because of what it reveals about the human mind, not because of what it reveals about the universe or God... because the debate doesn't reveal anything about the universe or God. The more I think about it, the clearer it becomes that we just assume the "truths" that work for us as far as helping us formulate satisfactory explanations.

Let me try to explain this using the imagery you have brought up: circumstancial evidence and hard evidence in a criminal investigation, and the evidence used by historians and scientists.

All those kinds of evidence have a certain property. All "evidence" in the God-versus-no-God debate lacks this property. Derrida mentioned this very briefly in passing, and the blog post I wrote aimed to prove this mathematically, but neither explanation hit the mark. Let me try again to explain it.

(Comment too long. Part 2 coming up).

Bernardo said...

(Part 2)

The property is as follows: Saying that "X is evidence for Y and against Z" means "We observe X. Explaining X as caused by Z requires a series of events that is less probable, more of a coincidence, more of a stretch of the imagination, more different from the kinds of events we have observed in the past, and less consistent with how we all understand the world to work (to the extent that we can agree on how the world works), than explaining X as caused by Y".

Evidence for theism does not have this property. When you say "X is evidence for theism", you are literally saying "X is more likely to have been caused by a miracle than by naturalistic or earhtly events". In other words, you are literally saying "We observe X. Explaining X as caused by naturalistic or earthly mechanisms requires a series of events that is less probable, more of a coincidence, more of a stretch of the imagination, more different from the kinds of events we have observed in the past, and less consistent with how we all understand the world to work (to the extent that we can agree on how the world works), than explaining X as caused by a MIRACLE".

But you see, ANYTHING is more probable than a miracle! This is true almost by definition of "miracle".

You say "We observe the records of some prophecy that came true. They are more likely to have been caused by a miracle than by editing-after-the-fact or by stretchy exaggerated interpretations and mistranslations", or "We observe the records of Jesus rising from the grave. They are more likely to have been caused by a miracle than by editing-after-the-fact, conspiracies, or mass delisions". What you are saying is "We observe the records for [blah]. Explaining the records for [blah] as caused by naturalistic or earthly mechanisms (editing-after-the-fact, stretchy exaggerated interpretations, mistranslations, conspiracies, or mass delisions) requires a series of events that is less probable, more of a coincidence, more of a stretch of the imagination, more different from the kinds of events we have observed in the past, and less consistent with how we all understand the world to work, than explaining X as caused by a miracle". And a naturalist will never agree with that. I, for one, wouldn't. The way that I understand the world to work, a miracle is less likely than any of those things. Editing after the fact? It's been known to happen. Exaggerated interpretations and mistranslations? I see them all the time. Conspiracies? Plausible, even if improbable. Miracles? Pretty much impossible, almost by definition.

If I can explain some event, observation, record, or claim, either by "God broke physical law and made the universe behave in a way that it has never been observed to behave before or since", or "Option B" (e.g. editing-after-the-fact, stretchy exaggerated interpretations, mistranslations, conspiracies, or mass delisions), then "Option B" will ALWAYS seem more probable than miracles. This is how naturalists understand the world to work. A miracle is much less likely, much more of a stretch of the imagination, than the most improbable (but naturalistic and earthly) coincidence.

Miracles! When someone proposes this with a straight face, I can't take them too seriously.

You mentioned how "evidence" has a different meaning in court, in science, in history, etc. In any of those settings, just try to propose "miracles" as an explanation for your observations, and tell me how it goes.

Bernardo said...

Oh yeah, and...

Then a miracle occurs


Randy Kirk said...

We now need to define miracle. In God's economy, the only miracle would be an action that was not in accord with His nature. A virgin birth might be called miraculous in terms of science and our normal expectation, but is no more miraculous than life from non-life, or a universe exploding out of nothing with all the attributes we see.

As you've well stated elsewhere, our worldview is what dictates our solutions. Your solution for life from non life seems to me to be more supernatural or less likely than God did it. And until you can replicate it or show that life is being formed from non life right now, it is unlikely that I will see your POV.

Then you have the matter of my experience of the spiritual things. You can explain it all day long with "neurons" firing, but you'd have about as much success with that as explaining the amazing sensation that occurs in our spirit when a rookie pitcher strikes out 14 in 7 innings his first time out as neurons firing. There is a spiritual realm, and it may be as natural as can be. If you've never experienced it, then it may just be that you are spiritually blind.

However, I know you feel pretty intensely about flight, so I'm guessing you have this sense.

Bernardo said...

When I've been writing, I've been assuming definitions that are different from yours. We can debate on which definitions are most useful, or at least agree on one set for now. Unless we use the same definitions we'll be talking past each other.

My definition of "natural" or "naturalistic" is something that is caused and governed by the mechanical and quasi-deterministic (to the extent that any macroscopic physical phenomenon is deterministic) processes relating to the motions of particles and their interactions. "Little billiard balls bouncing around" is a little oversimplified but it captures the idea.

The opposite of that is anything "supernatural" or "miraculous": Something caused and governed by non-material spiritual forces, by gods or demons or ghosts or other entities that are to some extent not knowable or measurable or material, that manifest themselves on some medium that can be completely outside our ability to detect them, that have intentions that we can only guess at, and that supposedly influence our material world once in a while.

Under those definitions, a virgin birth (or a divinely-inspired prophecy, or a burning bush, etc) is a miracle. It is more miraculous than life-from-non-life, than the Big Bang, than the normal fire in a candle, because these things are just particles interacting without the need for spirits to break the laws of physics.

Do I KNOW that the Big Bang was a naturally-caused (non-miraculous) event? No. But I assume it was, since I have no reason not to. I COULD assume that the Big Bang wasn't naturally-caused, but that would block me from understanding it if it was. History is littered with claims of "Spirits did it!" that have since been overturned by our sharpened understanding of natural processes.

I mean, do I KNOW that bird flight is a naturally-caused phenomenon? No. But I assume it is. I COULD assume that bird flight is miraculous and supernatural (as opposed to caused/governed by laws of physics that I can potentially learn and exploit), but I have no reason to. Assuming that bird flight is NOT miraculous/supernatural is what allowed folks like da Vinci, Cayley, Lillienthal, and the Wrights, to model its mechanisms well enough for us to exploit them. The same can be said for fire, the Big Bang, the mind, you name it.

My idea about life-from-non-life cannot be miraculous or supernatural, by (my) definition, since it is completely naturalistic and relies on no spirits, just chemistry and time.

My default view is that all things are naturalistic; I would need clear evidence before I ever say "Well, okay, maybe spirits are influencing this after all". Your default view on certain events and phenomena is supernatural: You would need clear evidence before you ever say "Well, okay, maybe life can arise out of non-life without spirits guiding the molecules".

(too long again. Here comes part 2)

Bernardo said...

(Part 2)

I have experiences like the ones you call "spiritual". I have been in love, something that has many facets each more intense and immersive and powerful than the last. I have been happy enough to cry, many times. I have listened to symphonies that made me forget who I am and enter a different state of mind. I have been exposed to overwhelming beauty. I have had experiences where the information provided by my senses blurred with sensory information generated by circuits in my brain that were not under my conscious control. So I don't deny that these things exist. I just think that attributing them to literal spirits is a mistake. I think that they can be explained by naturally-caused changes in brain chemistry and in the nature of neuron connections and signals. This does not mean that if I describe those neural changes to you, you'll immediately emulate the sensation that those neural changes cause. But this is a failure in our communication using language, and in your ability to emulate sensations, not in the model that says that those sensations are nothing more than neural changes. Just because you can't fully imagine some aerodynamic behavior I describe to you, doesn't mean that airplane flight is caused by spirits. Same thing for mental phenomena (even though mental phenomena are not understood and cannot be modeled as well as aerodynamics. At least not yet).

I am not blind to the phenomena you call "spiritual". It's just that I don't think those phenomena are caused by spirits. I think they are naturally caused. I agree that the conscious self, and the emotions and desires and sensations it feels (the "qualia" of experience), is just about the most amazing thing I have ever encountered; It practically defies logical explanation. But I find that when something seems to defy logical explanation, I can usually dig a little deeper and at least get a glimpse of the mechanisms behind the curtain. You, on the other hand, are happy to say "Spirits are doing it", which to me is not satisfactory.