Thursday, December 09, 2010

Subjective probability assessments and the Bayesian method

I've been reading John Loftus for about 5 years now, a considerable sample size.  The chart below represents (rather facetiously) an assessment of topics, of which any one might come up as the main subject of a blog post. Now, let's assume my pie chart could have been created using one of two methods.  Both methods start with the same list of topics and then rank them, but each method suggests a different approach to ranking, and perhaps even a different interpretation of what the ranking means.  

1. How frequently the topic has been the main subject of a post
2. My degree of belief that the topic will be the main subject of the next post

Assuming we must rely on memory, an approach to #1 might be to rank each topic's probability in relation to the others on the list.  (Remember this is just a subjective ranking.)  Given this approach, we will  end up with a nice tidy list where the probabilities add up to 1, and each probability is relative to the list.

With #2, the approach could be different.  I think we're less inclined to make our probabilities coherent (add up to 1) when we think in terms of 'degree of belief', and perhaps for good reason.   When asked about degree of belief, we are apt to consider additional evidence beyond mere frequency.  For example, we might hold an additional belief that John gets more frustrated with Christians during the holidays.

But is this an unfair characterization?  My perception of the frequencies (in #1) might account for additional evidence (such as John's holiday disposition) whether or not I realize that it does.  For instance, the fact that his posts tend to gravitate towards "Christians are deluded!" during the holidays could very well be the reason I believe John is more frustrated with Christians during the holidays.  Should we characterize this "fact that his posts tend to.." as a property out there in the world, a degree of belief, or both?

How do we sort things like this out, assuming we have good reason to?  One promising approach to critical thinking about probability is called the Bayesian method.  This method starts with a process for ranking one's degrees of belief with respect to some set of propositions, and when new evidence comes to light those degrees of belief are updated (recalculated), and thus the change in our degree of belief in some proposition with respect to new evidence can be accounted for mathematically.  The hardest part about this method is simply figuring out where to start.  What is the probability of any given belief we already hold?  One way to answer this question is to ask another one, "Would you rather bet on belief x or bet on the chance that a fair coin toss will land heads?"  If you think your belief is more likely than heads, now you're a little closer to knowing your degree of belief (more than 50% sure).  After we get that nailed down, things fall into place as new evidence comes along.  Critics of the Bayesian method accuse it of being overly subjective, yet it isn't clear to me that other approaches to interpreting probability are exempt from subjectivity.  Regardless, each approach to probability might prove useful in different situations.

Bayesian theory can also lead to a humanistic vision for the future...an optimistic outlook that may have been previously marred by the sense that philosophy might never reach agreement on much of anything.

"The model of reasoning that I am inclined to accept is that everyone starts where they start, and then conditionalizes their belief systems on the evidence. And then, maybe, however many generations it takes, we can reach a consensus." - Dr. Victor Reppert, in a discussion about his essay on the probability of miracles.

This looks very similar to scientific optimism.  A scientist might say, "given enough observations, humans will eventually develop a coherent theory of the physical world."  A Bayesian might say (eg., Dr. Reppert), "given enough evidence, humans will eventually develop a coherent (shared) set of beliefs about the world."

Notice how the Bayesian account didn't put physical in front of world?  Of course, a materialist will define evidence as purely physical (remember, we're talking about a Bayesian definition of evidence here).  But it isn't clear to me that evidence qua evidence must be physical in a Bayesian model.  Perhaps I'm wrong about that?  I'm not sure.  Dr. Reppert is a theist, so I can only assume that he believes humans will ultimately converge on a belief system that is...at minimum...not eliminatively [sic?] materialistic?  But yet science, by it's own lights, is headed towards reducing the world down to some ultimate materialistic explanation.  So why is Bayesianism headed in another direction?  Perhaps there is a catch regarding one's view of the mind here?

What if science eventually reduces the mind down to some strictly materialistic explanation?  Could a Bayesian continue to hope that humans might eventually converge on a system of beliefs that is compatible with (bare bones) Christianity or theism?  At face value, I don't see any reason why one can't include religious experience in their probability model.  However, I don't see the optimistic side of all this (as a Christian).  To me, it seems that one's belief/non-belief in God will determine which direction Bayesianism takes you.  But then again, how does one account for the man that doesn't believe in God and then encounters some set of experiences that persuade him otherwise (and vice versa for apostates)?  How does one account for John Loftus' Outsider Test for Faith, which seems to suggest that we should provisionally "reset" our prior probabilities (with respect to one's religion) back to a neutral position, and then have a look at the evidence relative to that.  Looks like I have much more thinking to do about this.

*Update*
Dr. Reppert has posted his reply over at dangerous idea.  And in case any readers are (inexcusably!) unaware of the Monty Python reference:

4 comments:

Victor Reppert said...

David: I should say that I am unpersuaded of "trajectory of science" arguments which suggest that as we investigate further we will find greater and greater support for reductionism. Two aspects of the materialistic vision of the world as it has been historically understood are the following:

1) The universe had no beginning, and has always existed.

2) The universe is deterministic, and as we do science we will come closer and closer to finding determining causes for everything.

Now, thanks to the development of the Big Bang theory in the first instance, and quantum mechanics in the second instance, confidence in both of these theses has eroded in comparison to what might have been thought in the early days of the 20th Century.

Now, of course, naturalists have revised their conception of what is naturalistically acceptable to accommodate a universe with a temporal beginning, and a universe with quantum level indeterminism. But the point is that science frustrated the expectations of what at the time were the expected results of the naturalistic thrust of science.

With respect to the analysis of mind, I see a lot of bravado about reductive analyses but no real hard evidence that reductions are going to be successful. In fact, given the fact that "the material" or "the natural" has to be defined in terms of the absence of the mental, it looks to me as if reduction of the mental to the physical is logically impossible, and that the more we study things scientifically the more evident this will become.

David Parker said...

Dr.Reppert,

Thanks for stopping by and taking the time to comment. I am in agreement with the points that you've made.

Being fairly new to these areas (philosophy of the mind and epistemic probability), I seem to be having a bit too much fun probing possible worlds. I am intrigued by your view that beliefs will self-correct in the long run. Any further reading recommendations are much appreciated.

I also have an old backissue of Philosophia Christi headed my way (volume 5). I am more familiar with Plantinga's EAAN, but want to get more acquainted with your argument from reason.

Cheers,
David

Brenda said...

"given enough observations, humans will eventually develop a coherent theory of the physical world."

What if we don't? What if the universe is such that there is no single coherent perspective under which all others may be ordered?

"But yet science, by it's own lights, is headed towards reducing the world down to some ultimate materialistic explanation."

This seems doubtful to me. At this time quantum mechanics and relativity are in conflict. String theory, the proposed solution that would unite them both, is (I believe) a house of cards. See "Not Even Wrong" by Peter Woit.

"What if science eventually reduces the mind down to some strictly materialistic explanation?"

I don't think that consciousness is reducible to neural activity in the brain and yet I think that it is entirely the product of that activity.

David Parker said...

Brenda,

I am skeptical about all of the theories coming out of quantum physics right now because of how confusing the language is. What they believe to be 'real' and 'true' needs to be held in mind when evaluating their claims.

For instance, Hawking has said that it doesn't make sense to speak of truth as 'what corresponds to reality' because we don't know what reality is. So, he defines reality as a useful model. The fish in a round bowl doesn't have a distorted view of reality, since there could be a model fashioned to accurately predict outcomes (relative to an observer in a fishbowl).

Since I disagree with that view of truth and reality, it's difficult to say what I make of those theories right now.

This Peter Woit book looks great, thanks for recommending!

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