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.

Saturday, September 29, 2012

Bayes’ Theorem and the LCA (p. 3)

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Bayes’ Theorem and the LCA
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Pr(E|¬H)

One could make this objection: there are infinitely many ways for there to be something, and only one way for there to be nothing. Pr(E|¬H) should therefore be extremely high, somewhere near 1.

I don’t think this objection is quite as good as it first appears. In some respects simplicity is evidence of truth, and we tend to have a special place in our intellectual hearts for the number zero. To illustrate, consider the case of “How many gods are there?”

Suppose one has no background information regarding the existence of “How many gods are there?” and little to no background information about what sorts of gods, if any exist, are like (e.g. whether they are friendly or hostile, whether they are beautiful or ugly, and whether any have six arms). Given the lack of such background information, what should one’s default view be about whether there are any gods? The default view, it seems to me, is to be truly agnostic, i.e. award “there are no gods” the probability of 50%. Some atheists would award “there are no gods” a default probability greater than 50%, so to broaden the agreement let’s say that the default probability of “there are no gods” is greater than or equal to 50%. But provided we would also award nonzero (even if low) probabilities for “there is exactly one god” and “there are exactly two gods,” we would be giving especially high probability status to there being zero gods, as in “the probability of there being zero gods is greater than or equal to 50%,” thereby awarding the “zero” value to be more probable than any other number of gods and it being no less probable than any other proposition that entails the existence of gods (e.g. “there is a god with six arms”). Notice how significant this is: we are awarding “there are zero gods” a probability value that is no less than all other possibilities combined even when there are infinitely many ways for gods to exist. This strongly suggests, I think, the value of ontological simplicity when coming up with prior probabilities especially as they relate to “there exist things of type X” when there is no background information.

The same principle holds, I believe, for “Why is there something rather than nothing?” Nothing is the ultimate zero, and abandoning all the background information we would otherwise have about things existing and focusing on just ¬H, it seems to me that Pr(E|¬H) is 0.5 or something close to it. The question, “Why is there something rather than nothing?” is interesting precisely because in the absence of a sufficient reason, there being nothing at all to exist seems like a very real possibility.

Prior Probabilities: Pr(H) and Pr(¬H)

Disputing the prior probabilities doesn’t actually affect my main point: a person who is truly agnostic (thinks “there is an explanation for the universe’s existence” is equally likely as “there is no reason for why the universe would exist”) 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 in the absence of further arguments. That said, one could say that a person who didn’t consider the evidence for E has for H should have Pr(H) be less than Pr(¬H).

One could say that, but in that case the objector would have to argue that in this situation “there is an explanation for the universe’s existence” is less probable than “there is no reason for why the universe would exist.” In abandoning the agnostic’s “neutral probability” default position, the objector would need to give some argument for why the background information would favor ¬H over H. Furthermore, one could argue that if anything the opposite should be done. Hypothesis H has some inferential virtues (things that make an inference good) that go beyond merely entailing E, and thus there are inferential virtues that aren’t captured in Pr(E|H).

One reason to rank H higher than we would otherwise rank it is that it’s part of the nature of rational inquiry to look for explanations for why things exist. To quote what I said in a previous entry:
Here I’ll borrow a bit from philosopher Richard Taylor’s illustration of finding a translucent ball in the woods. “How did it get there?” you ask. I reply, “There is no explanation for it being in the woods; the ball just exists inexplicably.” My response seems less plausible than the idea that there is some explanation for the ball’s existence. What if we enlarged the ball to the size of a car? Same problem: some explanation seems to be needed. How about a city? Same problem. A planet? Same problem. A galaxy? Same problem; increasing the size does nothing to remove the need for an explanation. How about if the ball were as big as the universe? Same problem. All things considered, it seems intuitively plausible that if a contingent thing exists, there is some reason why it exists, since it could have failed to exist.
The universe is contingent, and our default rational preference should be to accept that there is an explanation for its existence. Regarding the translucent ball illustration, Maverick Atheism says:
True enough, increasing the size of the translucent ball does nothing to remove the need for an explanation (let us also assume arguendo that all translucent balls are contingent). Size doesn’t matter, but what does matter is whether the translucent ball existed eternally. It is quite conceivable that there are possible worlds where a translucent orb has existed for all eternity without an external cause. If we had no evidence that the translucent ball began to exist, it would seem at least premature to simply assume it had an external cause
This may be a situation where reasonable people can disagree, but I’m not quite convinced that knowing the translucent ball to be eternal is sufficient to remove the need for an explanation (recall that it wasn’t sufficient in the case of the eternal monument scenario). In any case, suppose also in our eternal translucent ball scenario we had an explanation for its existence that was readily available, fairly straightforward, is the only known viable explanation, and we have no reason to believe the explanation is false. In that case I think the prior probability of there being a sufficient reason for the translucent ball’s existence was fairly good, at least more than 50%.

Similarly, perhaps it is logically possible for the contingent universe to exist eternally and uncaused, and for there to be no explanation for there being something rather than nothing. But it seems more intellectually satisfying to accept that there is an explanation for why there is something rather than nothing, especially if we have an explanation readily available, is the only known viable explanation, and no reason to think the explanation is false. The atheist could argue that we do have reason to think that the theist’s explanation (a transcendent personal cause) is false, but again that would require some kind of argument. In the absence of such an argument, the background information (the nature of rational inquiry, a readily available explanation etc.) if anything favors H over ¬H.

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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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Thursday, September 27, 2012

Bayes’ Theorem and the LCA

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Bayes’ Theorem and the LCA
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A while back the Maverick Atheism blog wrote a rebuttal to the Leibnizian Cosmological Argument (LCA), that was in part a rebuttal to my own series on the Leibnizian cosmological argument (though it was also largely a rebuttal to Christian philosopher William Lane Craig, whose form of the LCA I borrowed).

A Brief Recap

For those who aren’t familiar with Bayes’ theorem, I recommend reading this quick introduction to Bayes’ theorem. To recap my series on the LCA a bit, here’s one of the versions I used (one I labeled LCA 1A):
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The universe exists.
  3. If the universe does have an explanation for its existence, that explanation is God.
  4. Therefore, the universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation for the existence of the universe is God (from 3 and 4).
Premise 1 then is a version of the principle of sufficient reason. Another version of the LCA, one I labeled LCA 3, goes like this, where the contingent universe is the totality of all contingent things:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is God.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation of the contingent universe is God (from 3 and 4).
I argued that the contingent universe could have failed to exist, and I also argued that the external cause of the universe would have to be an eternal, transcendent, metaphysically necessary, personal entity. Here’s how it worked: The external cause of the contingent universe could not itself be contingent (since then we wouldn’t have an explanation for the contingent universe) and so must be necessary. What is necessary is also eternal, since at no time and in no circumstances can metaphysically necessary entities fail to exist. Since the physical universe itself is contingent, the external cause would have to be nonphysical, and there are only two candidates in the metaphysical literature for nonphysical entities: abstract objects (like numbers) and unembodied minds (like God). But abstract objects can’t cause anything, so the only known viable explanation would be a nonphysical mind. We thus wind up with an eternal, transcendent, metaphysically necessary, personal entity that is the external cause of the universe. I also argued for a more modest conclusion: that the only known viable explanation for the contingent universe is eternal, transcendent, metaphysically necessary personal entity as the external cause of the universe—and that of course sounds suspiciously like theism, enough to make atheism rather implausible if we knew such an entity exists. Hence this toned down version:
  1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
  2. The contingent universe exists.
  3. If the contingent universe has an explanation for its existence, that explanation is is an eternal, transcendent, metaphysically necessary, personal entity.
  4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
  5. Therefore, the explanation of the contingent universe is an eternal, transcendent, metaphysically necessary, personal entity (from 3 and 4).
God is classically a metaphysically necessary being (i.e. he exists in all possible worlds) and the external cause of the physical universe. But if God is metaphysically necessary, his existence would also explain why there is something rather than nothing.

Maverick Atheism’s Rebuttal

Maverick Atheism attacked the first premise and argued that perhaps the universe does not have an explanation for its existence. The position he put forward:
So we can accept that ceteris paribus a worldview that explains e.g. why there is something rather than nothing is better than one that doesn’t, but given the plausibility of physical reality existing eternally without an external cause, the degree of evidential support this provides is rather small.
There is an element of plausibility in this; most of us can at least envisage the physical universe existing eternally with no external cause. But now consider the following scenario, borrowing largely from my first article on the Leibnizian cosmological argument.
Suppose we humans learned of an eternally existing monument at the center of the universe that says, “I, the Lord thy God, am the sustainer of the universe and have sustained it throughout all eternity” (if questions of different languages bother you, imagine further that it displays this message through a kind of mechanical telepathy such that anybody who looks at the monument sees the message in her own language).
Most of us can at least envisage the monument existing eternally and without an external cause for its existence. Still, somehow the monument existing eternally with no explanation for its existence sounds rather implausible here. Why?

One factor is the likelihood of there being a monument like this in the absence of a sufficient reason for its existence (like some intelligent entity being the external cause of the monument’s existence), and intuitively that likelihood is low. Where H is the hypothesis (of there being a sufficient reason) and E is the evidence (the monument’s existence), we might say that Pr(E|¬H) is very low. If you’ll recall my quick introduction to Bayes’ theorem, you’ll know that a low Pr(E|¬H) is a factor in making Pr(H|E) high.

Is the likelihood of there being something rather than nothing extremely low in the absence of a sufficient reason, an improbability akin to the hypothetical monument existing without a sufficient reason? Maybe not. Still, the question, “Why is there something rather than nothing?” is interesting precisely because (in the absence of a sufficient reason) it seems like it could have been the case that there existed nothing rather than something. Something very similar holds with the contingent universe, since the contingent universe is the totality of contingent things and thus “no contingent thing exists” is no less a real possibility than “nothing at all exists.” Similarly, “no physical thing exists” is no less a real possibility than “nothing at all exists.” In this article, I’ll examine the insights Bayes’ theorem has on this possibility of there being nothing rather than something.

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Wednesday, September 26, 2012

Internet Dating

Like many nerds, I’m a bit like the last dodo bird: very single with approximately equal probability of finding a suitable mate. Fortunately I have access to the internet, but I find it to be of limited use. Some online match-up sites I’ve used:
  1. eHarmony.com
  2. match.com
  3. falseHope.com
  4. tooDifficultToFindTheRightOne.com
  5. bangYourHeadAgainstTheKeyboard.com

Saturday, September 8, 2012

Bayes’ Theorem and the Moral Argument

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In my previous blog entry I gave a quick introduction to Bayes’ theorem. In this blog entry I’ll show how Bayes’ theorem can be used in the service of theism.

When I put forth for the moral argument, I that argued that “If God does not exist, then objective morality does not exist” was likely true, and that it seems unlikely on atheism that objective morality exists. Another way to put it: “The probability that objective morality exists given that atheism is true is low.” With the conception of God I’m using, the existence of God entails that objective morality exists (God is morally good independently of human opinion), such that “The probability that objective morality exists given that God exists” is 100%. So we can construct a form of the moral argument that uses Bayes’ theorem and the following symbols:
  • H is the hypothesis that God exists.
  • ~H is the proposition that God does not exist (and thus that atheism is true).
  • E is the evidence of objective morality.
  • Pr(E|H) is the probability that objective morality exists given that God exists.
  • Pr(E|~H) is the probability that objective morality exists given that atheism is true.
That Pr(E|H) = 1 is pretty straightforward given the conception of God we’re using here, but the other probabilities aren’t so straightforward and will perhaps vary depending on the individual. Still, suppose we have someone who is truly agnostic (and thus Pr(H) and Pr(~H) are both 50%) but comes to believe in objective morality, and assigns the following probabilities:
  • Pr(E|~H) = 0.25
  • Pr(H) = 0.5
  • Pr(~H) = 0.5
  • Pr(E|H) = 1
One limitation of Bayes’ theorem is that the probabilities aren’t always clear, e.g. not everyone will agree on the correct value of Pr(E|~H). That said, if we had an agnostic that uses the above probabilities, the rules of probability suggest the agnostic should adjust his probability belief in theism using this version of Bayes’ theorem:

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

Plugging in the values into Bayes’ theorem...

Pr(H|E) = 
0.5 × 1
0.5 × 1 + 0.5 × 0.25
 = 
0.5
0.625
 = 0.8

Thus leaving the probability of theism given objective morality at 80%. Your own beliefs about the probabilities of e.g. Pr(E|~H) may differ than those of our hypothetical agnostic. If so, experiment and see what you get!

Bayes’ Theorem, A Quick Introduction

We all know that the probability of a hypothesis being true often changes in light of the evidence. Wouldn’t it be cool if math could help us show how it works? Fortunately, math is cool enough to help out here thanks to something called Bayes’ theorem. In this article I’ll introduce Bayes’ theorem and the insights it gives about how evidence works. In my next blog entry I’ll show how Bayes’ theorem can be applied in the service of theism.

One Form of Bayes’ Theorem

Bayes’ theorem is often used to mathematically show the probability of some hypothesis changes in light of new evidence. Bayes’ theorem is named after Reverend Thomas Bayes, an ordained Christian minister and mathematician, who presented the theorem in 1764 in his Essay towards solving a problem in the doctrine of chances. Before showing what the theorem is, I’ll recap some basic probability symbolism.

Pr(A) = The probability of A being true; e.g. Pr(A) = 0.5 means “The probability of A being true is 50%.”
Pr(A|B) = The probability of A being true given that B is true. For example:
Pr(I am wet|It is raining) = 0.8
This means “The probability that I am wet given that it is raining is 80%.”
Pr(¬A) = The probability of A being being false (¬A is read as “not-A”); e.g. Pr(¬A) = 0.5 means “The probability of A being false is 50%.”
Pr(B ∪ C) = The probability that B or C (or both) are true.
Pr(B ∩ C) = The probability that B and C are both true.
Pr(A|B ∩ C) = The probability of A given that both B and C are true.


Some alternate forms:

One VersionAlternate Forms
Pr(A) P(A)
Pr(¬A)  Pr(~A), Pr(−A), Pr(AC)
Pr(B ∪ C) Pr(A ∨ B)
Pr(B ∩ C) Pr(B ∧ C), Pr(B&C)
Pr(A|B)Pr(A/B)


The alternate forms can be combined, e.g. an alternate form of Pr(H|E) is P(H/E).

Bayes’ theorem comes in a number of varieties, but here’s one of the simpler ones where H is the hypothesis and E is the evidence:

Pr(H|E) = 
Pr(H) × Pr(E|H)
Pr(E)


In the situation where hypothesis H explains evidence E, Pr(E|H) basically becomes a measure of the hypothesis’s explanatory power. Pr(H|E) is called the posterior probability of H. Pr(H) is the prior probability of H, and Pr(E) is the prior probability of the evidence (very roughly, a measure of how surprising it is that we’d find the evidence). Prior probabilities are probabilities relative to background knowledge, e.g. Pr(E) is the likelihood that we’d find evidence E relative to our background knowledge. Background knowledge is actually used throughout Bayes’ theorem however, so we could view the theorem this way where B is our background knowledge:

Pr(H|E&B) = 
Pr(H|B) × Pr(E|H&B)
Pr(E|B)


To simplify it though I’ll leave the background knowledge in Bayes’ theorem implicit.

An Example

Here’s an example of Bayes’ theorem in action. Suppose we have a lottery and the odds are 1 in 5,461,512 that the following lottery numbers are chosen:
(4) (19) (26) (42) (51)
Let H be the hypothesis that the above lottery numbers were chosen. Let E be a newspaper called The Likely Truth reporting those numbers. The Likely Truth reports the lottery numbers with 99% accuracy (though it never fails to report some series of five lottery numbers of the sort that the lottery can result in, accurate or not), thereby making, making Pr(E|H) = 0.99. The odds that any particular series of five lottery numbers will be reported is likewise 1 in 5,461,512, making Pr(E) = 1 in 5,461,512. With that, we have the following probabilities:

Pr(H) = 
1
5,461,512

 ≈ 0.0000002
Pr(E) = 
1
5,461,512

 ≈ 0.0000002
Pr(E|H) = 0.99 

Plugging them into this version of Bayes’ theorem:

Pr(H|E) = 
Pr(H) × Pr(E|H)
Pr(E)

Gives us this:

Pr(H|E) = 
1
5,461,512
 × 0.99
1
5,461,512
 = 0.99

What’s interesting about this is that even though the prior probability of the hypothesis (≈0.0000002) is much lower than the probability that the newspaper made a mistake (0.01), the newspaper’s report still makes it rational to believe that the lottery numbers are probably accurate.

Another Form of Bayes’ Theorem

Keeping in mind I’ll leave the background knowledge in Bayes’ theorem implicit, another form of is Bayes’ theorem is this:

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

One insight the above formula gives us is that ceteris paribus the more unlikely it is we’d find the evidence if the hypothesis were false (i.e. a lower Pr(E|~H)), the stronger the evidence becomes for the hypothesis. Another insight is that ceteris paribus the more likely it is we’d find the evidence if the hypothesis were true (i.e. a higher Pr(E|H)), the stronger the evidence is for the hypothesis.

In my next blog entry I’ll show how Bayes’ theorem can be used for theism.