Sunday, September 30, 2012

Bayes Theorem and the LCA (p. 4)

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Bayes’ Theorem and the LCA
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Hypothesis H has some inferential virtues (things that make an inference good) that go beyond merely entailing E, and thus there are inferential virtues go beyond what is reflected Pr(E|H), and are instead reflected (albeit indirectly) in Pr(H).

   (1)  It provides an explanation. The big bang theory explains why this cosmic microwave background radiation exists but another hypothesis is that the background radiation exists inexplicably. Both hypotheses entail the existence of cosmic microwave background radiation, but that the big bang theory explains the radiation’s existence is an advantage the other hypothesis doesn’t have. Ceteris paribus, we prefer a hypothesis that explains the existence of a thing over one that merely entails the thing’s existence.
   (2)  Precision. Not only does H imply the existence of an explanation, we can actually think of a specific explanation for the contingent universe: an eternal, transcendent, metaphysically necessary personal entity that is the external cause of the universe. Granted, this level of precision is limited, but it’s a big improvement over “there is an explanation and but I can’t think of what it might be at all.” To give an illustration, suppose theory T implies the data but does not explain it. I say we should reject theory T because there might be some unknown hypothesis that explains the data, rather than merely implying it like theory T does. My case for rejecting theory T would be much stronger if I could think of a real example of such an explanation rather than merely asserting that some explanation exists (which would constitute some additional precision for my hypothesis of there being an explanation). Similarly, the fact that we can think of an explanation makes H more likely than it would be otherwise.


My claim here is modest: that items (1) and (2) make H more likely than it would have been without them. Some inferential virtues are also explanatory virtues (things that make an explanation a good one). Some inferential virtues are also explanatory virtues (things that make an explanation a good one). For the specific explanation for H (eternal, transcendent, metaphysically necessary, personal entity as the external cause of the universe), these explanatory virtues include:

   (3)  Plausibility. One factor going into plausibility is if it implies fewer falsehoods. I mentioned that an additional factor H has in its favor is that we can actually think of some explanation for the universe. But if the proposed hypothesis is the only known viable explanation and we have no evidence against it (as I claim), the fact that we have such a hypothesis known to us also makes H more likely than it would be otherwise. Think back to the situation of theory T. My case for rejecting theory T would be better if there were no evidence against my explanatory hypothesis.
   (4)  Tying in with background knowledge. Fulfilled to an albeit limited extent; we are intimately familiar with personal causes, and we experience personal causes being a reason for why things exist all the time.
   (5)  Simplicity. Fulfilled to an albeit limited extent. In my series on the Leibnizian cosmological argument, I posited only those attributes that were needed to explain the existence of the contingent universe: an eternal, transcendent, metaphysically necessary, personal entity that is the external cause of the universe. That these attributes derive so simply from there being an explanation for the existence of the universe makes the explanation more likely than it would be otherwise.
   (6) 
Explanatory scope. Fulfilled in multiple ways.
   (a)  Explaining the cosmos. The existence of a necessary personal being that exists by the necessity of its own nature is sufficient to explain why there is something rather than nothing; the explanation for the necessary being’s existence is the necessity of its own nature, and that necessary being entails the existence of something rather than nothing. In my series on the Leibnizian cosmological argument I noted that the eternal, transcendent, metaphysically necessary, personal entity as the external cause of the universe explains why there is something rather than nothing, why the contingent universe exists, and why the physical universe exists.
   (b)  Explaining morality. An eternal, transcendent, necessary personal being (with a few more attributes) also explains objective morality. For more on this, see my argument from ontological simplicity (which is part 1 in my argument from morality series).


These facets of the explanation make the explanation more likely than it would have been without them.[1] My claim is modest: the fact that we can think of an explanation for why there is something rather than nothing that meets criteria (3) through (5) to the extent that they do (e.g. the explanation implies no falsehoods) makes H more likely than it would be otherwise. Just as my case for rejecting theory T would be much improved if I could think of an explanation that met criteria (3) and (4), so my case for rejecting ¬H is improved via (3) and (4). By my lights, the entity explaining not only why there is something rather than nothing but also the existence of objective morality (6b) is a particularly significant item in favor of H, in part because of how simply the existence of an eternal, transcendent, metaphysically necessary entity is extrapolated from the existence of objective morality, but whether I’m right about that heavily depends upon that moral argument for God working.

Shoe On the Other Foot

In addition to assigning a lower prior probability for H to avoid believing it, one disputable point is how much additional evidential weight items (1) through (6) give to “there is an explanation for why there is something rather than nothing;” an atheist could say the evidential improvement is very small. Would such an atheist be right?

Here’s another way to look at it. Suppose the shoe were on the other foot and there being an explanation for why there is something rather than nothing were devastating to theism rather than atheism. Criteria (1) though (6) are met to the same level as the theist’s personal cause hypothesis (the atheist’s explanation is simply and straightforwardly derived from the data; the anti-theistic explanation is the only known viable explanation, etc.). The devastating-to-theism explanation implies no falsehoods and explains why there is something rather than nothing, why the contingent universe exists, and why the physical universe exists. Wouldn’t atheists be reluctant to assign a low prior probability to H in those circumstances? Wouldn’t they be right to do so? I also have a hard time believing atheists wouldn’t use this as fairly significant evidence against theism, particularly in light of the Bayes’ theorem equation that yields a 67% probability of there being an explanation for someone like Al.

Summary and Conclusion

The symbols were these:
  • H says there is an explanation for why the universe exists; a sufficient reason for its existence.
  • ¬H says is not the case that H is true; there is no reason for why the universe would exist.
  • E is the evidence of the universe existing.
The equation was this:

Pr(H|E) = 
Pr(H) × Pr(E|H)
Pr(H) × Pr(E|H) + Pr(¬H) × Pr(E|¬H)


One advantage H has over “there is no reason for why the universe would exist” is that on H it is more likely that the universe would exist, and for the agnostic (one who considered “there is an explanation for why the universe exists” to be equally as likely as “there is no explanation for why the universe would exist”), that’s enough for the universe’s existence to be fairly significant evidence for H, even if that evidence isn’t overwhelming. Using the above form of Bayes’ theorem to help show that (where Pr(H) = Pr(¬H) = 0.5, Pr(E|H) = 1, and Pr(E|¬H) = 0.5):

Pr(H|E) = 
0.5 × 1
0.5 × 1 + 0.5 × 0.5
 = 
0.5
0.75
 = 23 ≈ 0.67


With the mathematics being airtight, a person who disagrees with the probability result will have to dispute at least one of three things:
  1. Pr(E|H) = 1
  2. Pr(E|¬H) = 0.5
  3. Pr(H) = Pr(¬H) = 0.5
Pr(E|H) is unassailable; there existing an explanation for why the universe exists entails that the universe exists. Pr(E|¬H) is more vulnerable, but considering how special the number zero is to prior probabilities (as credibly illustrated in the case of “zero gods exist”), then abandoning all the background information we would otherwise have about things existing and focusing on just ¬H, it does seem that Pr(E|¬H) = 0.5 or something close to it.

Attacking Pr(H) = Pr(¬H) wouldn’t actually address this key point: a person who is truly agnostic but hadn’t yet considered the evidential force the universe’s existence has for H should re-assign a probability of about 67% for H upon considering such evidence.

In addition to Pr(E|H) = 1, there are various other factors that make H more likely than it would otherwise, and thus there are factors that increase the probability of H that aren’t reflected in Pr(E|H). For example, we can think of a specific explanation for why there is something rather than nothing that (a) implies no falsehoods; (b) is fairly simple (an entity with certain specified attributes that are derived simply and straightforwardly from the data); and (c) has an interesting connection to explaining the existence of objective morality. I’d venture to say that factors (1) though (6) I mentioned push the probability of “there is an explanation for why the universe exists” at least beyond the 70% range.

The degree to which all this counts as evidence for “there is an explanation for the existence of the universe” can be considered by imagining if the tables were turned in atheism’s favor. If in addition to H’s superior explanatory power for the existence of the universe, the devastating-to-theism explanation explained why there is something rather than nothing, why the contingent universe exists, why the physical universe exists, the explanation implied no falsehoods etc. it seems this would constitute fairly significant evidence in favor of atheism. But if that is true, rationality dictates we be consistent and recognize that these factors are equally as favorable to the pro-theism hypothesis of an eternal, transcendent, necessary personal being as the external cause of the universe.

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[1] I’m oversimplifying this, and some of this can be a bit tricky. Where B is our background knowledge, Pr(H) is really Pr(H|B) (the likelihood of H given some set of background information). For example, for (2), strictly speaking our own existence isn’t part of B. We could include in B something like, “If something were to exist in such a way that we would exist, then there would be personal entities like us who have a reason for why some things exist.” That said, we’re basically considering how one should adjust their probability of H when they hadn’t taken into account E’s evidential force for H.

1 comment:

  1. As in Pascal's Wager, the difficultly seems to be in ascertaining any probabilities. The agnostic H = .5, ~H =.5 may not hold, but the delineation of the rational factors here is interesting. A deck of cards is half red, half black; but some type of God is either there or not [1.0 / 0.0 or 0.0 / 1.0] ?

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