Friday, September 28, 2012

Bayes’ Theorem and the LCA (p.2)

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Bayes’ Theorem and the LCA
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Some Preliminaries

The theist can say that the reason why something exists rather than nothing is that God is a necessary being he exists by the necessity of his own nature. An atheist could believe the reason why there is something rather than nothing by being a Platonist, thinking that abstract objects like numbers genuinely exist as nonphysical entities independently of the mind. Yet (in my humble opinion at least) it seems more likely that numbers are more like ideas in the mind and have no more real existence then Sherlock Holmes. If there is no explanation for why something exists (including the necessity of a thing’s nature), then the contingent universe is all there is, and the contingent universe just exists inexplicably without an external cause.

Yet “If there is no explanation for why there is something rather than nothing, then the contingent universe just exists inexplicably” is true then “If the contingent universe has an explanation of its existence, then there is an explanation for why there is something rather than nothing.” To simplify it then I’ll use “Why is there something rather than nothing?” and “Why does the contingent universe exist?” more or less interchangeably, since in all likelihood they probably have the same answer: either (1) it exists inexplicably; or (2) the explanation is a transcendent personal entity that is metaphysically necessary.

Bayes’ Theorem and the LCA

Maverick Atheism doesn’t give any reason to prefer “there is no explanation for why the universe would exist” over “there is an explanation for why the universe would exist.” He seems to argue that “there is an explanation” has only a very slight advantage from the evidence of the universe’s existence. Is he right? Here Bayes’ theorem can be of some help. Suppose we use the following symbols:
  • H is the hypothesis that the PSR holds true with respect to the universe (there is an explanation for why it exists; a sufficient reason for its existence).
  • ¬H symbolizes that it is not the case that the PSR holds true with respect to the universe.
  • E is the evidence of the universe existing.
Basically, ¬H says, “there is no reason for why the universe would exist” in the sense that if the universe does exist, it does so inexplicably with no external cause for its existence. One version of the Bayes’ theorem is the following:

Pr(H|E) = 
Pr(H) × Pr(E|H)
Pr(E)


What is the likelihood that the universe exists if there is a sufficient reason for its existence? That probability is 100%, so Pr(E|H) = 1. Suppose we have a true agnostic about whether there is an explanation for the universe’s existence (i.e. H and ¬H have equal prior probability) but hadn’t yet taken into account the evidential force E has for H. One might be tempted to set Pr(E) as 0.5, but then we’d get this:

Pr(H|E) = 
0.5 × 1
0.5
 = 1


As the old saying goes, if it sounds too good to be true, it probably is. There’s something wrong with the above equation; a little math shows that for Pr(E) to be 0.5 here Pr(¬H) must be 0, and that’s mathematically impossible if Pr(H) = 0.5. So what went wrong? Getting the right value for Pr(E) is trickier than one might think. We should be careful not to confuse Pr(E) with Pr(E|¬H); it may be true that in the absence of sufficient reason for the universe’s existence the universe is just as likely to exist as not, but that won’t necessarily hold true for Pr(E), because Pr(E) is equal to the following:
Pr(E) =  Pr(H) × Pr(E|H) + Pr(¬H) × Pr(E|¬H)
So by my lights, using the form of Bayes’ theorem where Pr(E) is on the bottom is not advisable here. With that in mind, another form of Bayes’ theorem is this:

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


What about Pr(E|¬H)? We’re looking for the likelihood of E given just ¬H. Without an external cause of the universe to make it exist (and the universe lacking necessary existence) we can estimate the likelihood of it existing given just ¬H as 50%. Suppose one starts out as truly agnostic as to whether there is an explanation for the universe’s existence (i.e. both H and ¬H have a prior probability of 0.5) but had not yet considered the evidential force of the universe’s existence; call this person Al. Agnostic Al is using these probabilities:
  • Pr(H) = 0.5
  • Pr(E|H) = 1
  • Pr(¬H) = 0.5
  • Pr(E|¬H) = 0.5
And plugging those values into above formulation of Bayes’ theorem gives us this:

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


So it seems that if one considered H and “there is no reason for why the universe would exist” to be equally likely but hadn’t yet taken into account the evidential force of the universe’s existence for H, one should assign H a probability of about 67% in the absence of further arguments. While that probability isn’t overwhelmingly huge, it’s not exactly negligible either. The existence of the universe would constitute some fairly significant evidence for H, in part because 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. So if in the absence of a sufficient reason “nothing at all exists” is just as likely as “something exists” then this provides a significant advantage for the hypothesis that there is some explanation for why there is something rather than nothing, at least for the aforementioned agnostic Al.

The mathematics is airtight; to reject it one would have to reject the input probabilities. I suspect Pr(E|H) would be pretty uncontroversial given the sense of “sufficient reason” being used here, but the other parts of the probability calculations are more vulnerable.

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