The Parable of the Sheep

The young girl awoke with a start to the sound of desperate bleating. She quickly dressed and ran outside, fearing the worst. The pen which held her sheep—her precious sheep that she cared for daily—was open. Carcasses and blood were everywhere. She had to find her father; he was the only one who could help. …

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New Contents Page

I’ve added a new page to the site: http://foxholeatheism.com/contents/. I think this will make browsing the archives of this site much easier. I’ve arranged posts by category and sub-category, where appropriate. I haven’t included every post I’ve ever written, but I have included many of them. Next up, I’ll be doing something similar for my …

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Absence of Evidence is Evidence of Absence

Recently, I’ve written about some introductory topics in Bayes’ Theorem. If you did not read these earlier pieces, you may want to go here and here before reading this post. The initial impetus was to use the theorem to defend a famous maxim often attributed to Carl Sagan—extraordinary claims require extraordinary evidence. This time, I’m …

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God and Intrinsic Value

I have a new post at An American Atheist. It argues that intrinsic value cannot come from God. If interested, you can read it here: God and Intrinsic Value Bookmark on Delicious Digg this post Recommend on Facebook share via Reddit Share with Stumblers Tweet about it Subscribe to the comments on this post

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Are the Ten Commandments just?

Here is an argument with premises that are fairly easy to defend, but which leads to powerful conclusions where traditional theism is concerned: 1. Justice means to give people what they deserve. 2. People do not deserve to be punished for acts in which they had no role. 3. Descendants who are not yet born …

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The Parable of the SheepNew Contents PageAbsence of Evidence is Evidence of AbsenceGod and Intrinsic ValueAre the Ten Commandments just?

Jan
05

The Parable of the Sheep

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The young girl awoke with a start to the sound of desperate bleating. She quickly dressed and ran outside, fearing the worst. The pen which held her sheep—her precious sheep that she cared for daily—was open. Carcasses and blood were everywhere. She had to find her father; he was the only one who could help.

“Daddy!” she cried, running up to him, “a wolf’s gotten in the pen.”

“I know,” the father said.

“Please do something,” she sobbed. “They’re being killed and suffering terribly.”

“But the sheep let the wolf into the pen,” said the father as he turned away to resume his work. “They were curious about the animal. They had never seen a wolf before, so they knocked the gate open.”

“But they’re sheep!” the girl practically screamed at him. “They couldn’t possibly understand the consequences.”

“I’m sorry, dear,” said the father. “I warned them what would happen. They did not listen.”

“But they’re sheep!” she said again—this time even louder. “You would let them suffer and die over this stupid choice? They can’t see and think like you.”

“Even so,” said the father solemnly, “it is better for them if I do not interfere.”

Outside the slaughter continued as the wolf savagely ripped out the throats of its victims. The father did nothing.

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Jan
04

New Contents Page

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I’ve added a new page to the site: http://foxholeatheism.com/contents/.

I think this will make browsing the archives of this site much easier. I’ve arranged posts by category and sub-category, where appropriate. I haven’t included every post I’ve ever written, but I have included many of them.

Next up, I’ll be doing something similar for my recommended reading page. My current widgets from Amazon are too limited and don’t show up on mobile devices without flash. I’ll recommend books based on subject matter and level of difficulty.

Feel free to leave feedback about the new page.

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Dec
30

Absence of Evidence is Evidence of Absence

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Recently, I’ve written about some introductory topics in Bayes’ Theorem. If you did not read these earlier pieces, you may want to go here and here before reading this post.

The initial impetus was to use the theorem to defend a famous maxim often attributed to Carl Sagan—extraordinary claims require extraordinary evidence. This time, I’m going to use the theorem to argue against another maxim associated with Sagan.

Sagan was an outspoken supporter of the Search for Extra Terrestrial Intelligence (SETI). In the following embedded video, he introduces the Drake Equation on his show Cosmos.


This equation leads many to believe there is almost certainly intelligent life elsewhere, even in our neighborhood of the galaxy. However, we face a problem. So far our search has been fruitless. In response to this, Sagan noted that absence of evidence is not evidence of absence.

What Sagan should have said, though it wouldn’t have been as catchy, is that absence of evidence is not proof of absence. There is a difference between something being a proof of a claim and being evidence for or against a claim. If we go back to thinking in Bayesian terms, we can put it like this: If something is evidence against a hypothesis, then the posterior probability will be lower than the prior probability after taking said evidence into account.

Let’s run some numbers and see whether absence of evidence leads to this lower posterior probability. It might be useful to think of this like a function. A measure of probability goes in, stuff happens, and another measure of probability comes out the other end. It doesn’t really matter for the purposes of this post what the prior probability is; rather, we are just concerned with how the output compares to the input. In terms of the theorem, that means we’ll want to focus our discussion on the two figures assessing the likelihood of observed evidence.

Let’s consider an example with a prior probability of H set at 0.5:

  • 0.5 * Pr(E│H) / [0.5 * Pr(E│H)] + [0.5 * Pr(E│¬H)] = ?

We are concerned with the likelihood of our observed evidence given a true hypothesis, Pr(E│H),and the likelihood of the same evidence given a false hypothesis, Pr(E│¬H).

First, we should observe how the equation will react based on what we plug in for these numbers. If we plug in the exact same number for both figures, then our outcome will not change. The posterior probability will be 0.5, which will mean our evidence did not specifically favor either H or ¬H. Plugging in the same number for both essentially means the observed evidence was equally expected by both hypotheses.

But what happens if one is higher or lower, meaning the evidence is expected under one hypothesis more than the other? Let’s try plugging in Pr(E│H) = 0.7 and Pr(E│¬H) = 0.3. Our output is 0.7. Compared to the prior probability of 0.5, this is an increase, so this was evidence in favor of H. How about if we switch the figures so that Pr(E│H) = 0.3 and Pr(E│¬H) = 0.7? This time, the output was 0.3, a decrease, so this was evidence against H (against H and in favor of ¬H is really the same thing).

Now on to the big question of whether absence of evidence for some hypothesis (H) will mean a higher number in Pr(E│H) or Pr(E│¬H) or whether they will be the same. Let’s first eliminate one irrelevant possibility. In these cases, Pr(E│¬H) will always be =1. That is because in cases where someone claims something does not exist, like God or ghosts or aliens, there should always be an absence of evidence. That is expected 100% of the time. This means Pr(E│H) will never be higher than Pr(E│¬H); it can only be ≤1.

Whether or not Pr(E│H) will be lower than or equal to Pr(E│¬H) will depend on what H predicts. For example, say you specifically predict life on Titan, a moon of Saturn. If someone observes that there is no evidence of life on Mars, that doesn’t affect your hypothesis. So, it certainly is possible in cases of irrelevant evidence to achieve a neutral outcome. You can try plugging in some numbers yourself to see. In the following cases, the posterior probability shows no change from the prior probability because both likelihood measurements are =1:

  • 0.5 * 1 / [0.5 * 1] + [0.5 * 1] = 0.5
  • 0.9 * 1 / [0.9 * 1] + [0.1 * 1] = 0.9
  • 0.1 * 1 / [0.1 * 1] + [0.9 * 1] = 0.1

Many hypotheses, however, will not be so lucky. That is because the search for evidence is often quite relevant to the hypothesis (otherwise it would be a pretty fruitless search). So, in most cases where the evidence is relevant to the hypothesis, Pr(E│H) will be lower than Pr(E│¬H), which leads to a lower posterior probability, as shown in the following examples:

  • 0.5 * 0.9 / [0.5 * 0.9] + [0.5 * 1] = 0.47
  • 0.5 * 0.75 / [0.5 * 0.75] + [0.5 * 1] = 0.43
  • 0.5 * 0.5 / [0.5 * 0.5] + [0.5 * 1] = 0.33

In review, as long as the lack of evidence is relevant to the hypothesis, this lack of evidence is indeed evidence against that hypothesis being true. The degree to which that is the case will depend specifically on the initial predictions of H, as shown in the last set of examples.

 

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Dec
29

God and Intrinsic Value

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I have a new post at An American Atheist. It argues that intrinsic value cannot come from God. If interested, you can read it here: God and Intrinsic Value

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Dec
28

Are the Ten Commandments just?

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Here is an argument with premises that are fairly easy to defend, but which leads to powerful conclusions where traditional theism is concerned:

1. Justice means to give people what they deserve.

2. People do not deserve to be punished for acts in which they had no role.

3. Descendants who are not yet born (or are very young) can play no role in the acts of their ancestors.

4. Therefore, punishing said descendants is not just.

You might think this is an obvious conclusion to draw. Good; I hope you do think that. To be safe, though, Ill defend the premises briefly.

I almost consider (1) to be a tautology. I have a hard time separating the notions of justice and desert. If necessary, we can fall back on discussions by Aristotle and many others defending this concept of justice. If someone wants to propose a different definition, I’m willing to entertain it, but I can’t imagine any definition of justice that would escape the problem of this argument.

I also think (2) should be fairly obvious, but I want to bring in support from a theist here before we discuss possible further conclusions. William Lane Craig had the following to say in his debate with Sam Harris:

“His [Harris's] thoroughgoing determinism spells the end of any hope or possibility of objective moral duties because, on his world view, we have no control over what we do.”

Craig clearly thinks that control over the act is required to create any sort of duty or obligation. If no obligation is violated, it is not clear how any reciprocal punishment for the act can be deserved. I welcome some argument to the contrary.

I can’t imagine anyone denying (3) without invoking some kind of very strange backward causation. Time travel could potentially be trouble (I don’t actually think it is), but I’m going to set that concern aside for this discussion.

Then, the conclusion simply follows from the premises.

So, why does this matter? Well, it creates a tension between certain theistic claims: (i) God is completely just; (ii) The Ten Commandments were given by God.

There are actually multiple versions of the Ten Commandments, but I will be specifically quoting from Exodus 20:4-6 (NIV):

“You shall not make for yourself an image in the form of anything in heaven above or on the earth beneath or in the waters below. You shall not bow down to them or worship them; for I, the LORD your God, am a jealous God, punishing the children for the sin of the parents to the third and fourth generation of those who hate me, but showing love to a thousand generations of those who love me and keep my commandments.”

Here, God is promising to punish descendants to the third and fourth generation specifically for the acts of their ancestors. Given the argument above, this means we should at the very least reject either (i) or (ii), both of which are central components of Judeo-Christian theism.

 

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Dec
16

Creationist Math

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A common misconception is that evolution relies entirely on chance. For example, I once read part of a science textbook intended for homeschooling parents that included this analogy:

Imagine a yard containing all of the parts of a working computer that has been disassembled and the parts have been strewn all over the yard. How likely does it seem that a tornado could blow through the yard and randomly reassemble the parts to once again form a working computer?

This is an attempt to update the old Boeing 747 analogy from Fred Hoyle. He was attempting to illustrate the improbability of certain elements of life originating by chance.

Since such abuses of probability estimation seem to thrive still today, particularly in forums that are dominated by amateur commentators (I’m talking about you, Facebook), I thought it might be worthwhile to discuss a one of the many problems with analogies like this.

The modern theory of evolution—or you may hear it referred to as the modern synthesis—does not suggest that evolutionary events are the products of mere chance. While chance is a factor, such as in random mutations, it is not the only one. Consider 10 types of birds living on an island with 10 different beak shapes. Let’s suppose the available food source for these birds is only reachable by one of the beak shapes (perhaps it’s in a narrow hole or something). A naïve treatment of the probability of survival here would assign equal weight to every type of bird. However, we should easily recognize that survival is not random here. It will specifically favor the bird type that is able to reach the food source. So, there is a non-random factor at work. Specifically, reaching the food is needed for reproduction and survival and not all bird types can reach the food. An entirely natural process is performing selection.

Let’s also look at an example that does not involve living things. If you’re on a rocky beach, you might notice the distribution of rocks and pebbles has a specific pattern. Rocks will be sorted according to their size. There will be fairly uniform layers running parallel to the water. Let’s approach the problem like a creationist and see how we incorrectly determine the probability by thinking it’s random. To keep it simple, we’ll assume a small sample space of 16 rocks. Each letter group means the rocks are roughly the same size.

 

A A A A
B B B B
C C C C
D D D D

 

If I were to randomly pull rocks out of a bag and place them into the 16 squares, I calculate the chance as only 0.0000000159 that this pattern would appear. This is being pretty generous in that we have only 16 squares to fill and any of the A rocks can be in the first row, B rocks in the second row, etc. Even given these concessions, random chance is an unlikely explanation. So, should we conclude that there must have been intelligent involvement? Of course not. We know there are natural processes selecting for rock placement just as natural processes select for survival.

Any argument that calculates a probability based on random chance alone ignores this known feature, thus, is arguing against a straw man.

[Cross-posted at An American Atheist]

 

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Dec
08

Repeating Nonsense

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I began listening to the debate on Naturalism vs. Theism between Jeffrey Jay Lowder and Phil Fernandes the other day. I’m about halfway through (it’s over two hours long) and, thus far, Lowder is “winning” decisively. There are several claims made by Fernandes that inspire a facepalm, but I found one claim especially annoying.

You will often hear creationists talk about the Earth residing in a Goldilocks Zone. Here was a claim along these lines made by Fernandes:

If the distance between the Earth and the Sun was to differ by just 2% in either direction, no life on Earth would be possible.

This, like nearly every creationist claim, is demonstrably false. Yet, such things are repeated ad nauseum. Now, we could say why this claim being used as proof of a designer is problematic in terms of philosophy, but I think some fairly simple science will be our best method of debunking here.

The Earth’s orbit around the sun looks something like this (not to scale):

 

 

You’ll notice that it is an ellipsis. Why does this matter? Well, we can see that the existing orbit does differ. At the Earth’s greatest distance from the Sun, the distance between them is about 152 million km or 1.0167 AU. This is called the aphelion. When the Earth is closest to the Sun, the distance between them is 147 million km or 0.9833 AU. This is called the perihelion. At this point, you’re probably tempted to take out your calculator. If so, you would find that the distance between the Earth and Sun actually differs by 3.3% or 3.4%, depending on whether you use the aphelion or the perihelion. Either way, we don’t have to wonder what would happen to the Earth if its distance from the Sun differed by more than 2% because it already does. Yet, here we are.

At best, the creationist claim here is poorly phrased. At worst, it’s plainly false.

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Dec
02

How to use Bayes’ Theorem

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Recently, I wrote a Bayesian formulation of Carl Sagan’s famous maxim, ‘Extraordinary claims require extraordinary evidence.’ However, since the aim of that post was not to teach Bayes’ Theorem, but to reply to a criticism of the maxim, I may have left readers unprepared to actually use this theorem. I thought it might be helpful to provide an explanation. The equation looks difficult, but it’s actually quite simple once you understand the symbols. It’s just a matter of figuring out a few numbers.

 

Bayes’ Theorem

Here is one way to formulate Bayes’ Theorem when you want to know the probability of A given B:

Since A and B might sound a little abstract, let’s provide a concrete example and we’ll walk through the equation step-by-step.

 

Breast Cancer

Eliezer Yudkowsky uses the following example[i] that most doctors get wrong when polled:

1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get a positive mammography. 9.6% of women without breast cancer will also get a positive mammography. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

So, we want to find the probability of breast cancer (A) given a positive mammography (B). We will just need to concern ourselves with four figures—the two in the red box (the numerator is repeated exactly in the denominator) and the two in the green box. These four figures are:

  • P(B│A) = the probability that you will have a positive mammography if you have breast cancer
  • P(A) = the probability that you have breast cancer prior to considering the evidence, which is why it is often called the ‘prior probability’
  • P(B│¬A) = the probability that you will have a positive mammography if you do not have breast cancer
  • P(¬A) = the prior probability that you do not have breast cancer

So, now we should be able to figure out where to plug in the numbers from the initial problem:

  • P(B│A) = 0.80
  • P(A) = 0.01
  • P(B│¬A) = 0.096
  • P(¬A) = 0.99

The last figure of P(¬A) was not explicitly given above, but we can always figure it out if we know P(A). That’s because P(A) + P(¬A) = 1. Think of it this way: 1 is the total amount of possible probability to be divided up among options because 1 is really the same as 100%. So, whatever the probability is that A is true, it is necessarily the case that the probability that A is not true completes whatever is left to add up to 1. If there is a 70% chance that A is true (0.7) then there is a 30% chance that A is not true (0.3). In this particular case, the prior probability of A was 1%, so the prior probability of ¬A is necessarily 99%.

Now that we have our numbers matched to the terms, we can return to the original formula:

P(A│B) =   0.80*0.01_____________

(0.80*0.01) + (0.096*0.99)

Simplified further, it reads:

P(A│B) =   0.008____________

0.008 + 0.09504

P(A│B) = 0.0776

This means that if you are a woman at age 40 and receive a positive mammography, there is actually only a 7.76% chance that you have breast cancer. Only 15% of doctors across several surveys gave the correct answer. Most of them replied that there was an 80% chance that you have breast cancer in this scenario. That’s quite a difference, especially to the woman wondering whether she has cancer!

 

Belle and Gaston

That problem is relatively easy, since you are given the numbers. What happens, though, in a fairly ambiguous situation where estimation is required?

Let’s say that Gaston has approached Belle for a date and been rejected on multiple occasions. He is trying to determine whether Belle really likes him and is playing hard to get or whether she genuinely does not like him. So, we are looking for the probability that Belle likes Gaston (A) given the evidence of her repeated rejections (B). Again, we will just need to concern ourselves with four figures, tackled one-by-one:

  • P(B│A) = the probability of rejections if Belle likes Gaston

We can recognize that the first figure should be quite low, given that Belle is known to be a very honest and straightforward person. If Belle actually likes Gaston, we would expect an honest admission of this, rather than repeated rejections. However, women can be funny in showing their affection, so I would say it’s still greater than 0. Let’s place the probability at 0.05. It’s very low, but leaves a little room for error in case our perception of Belle has been mistaken.

  • P(A) = the prior probability that Belle likes Gaston

The second figure should be high, given Gaston’s popularity with women in general. Remember that in factoring this number, the evidence, such as the rejections, should not count. In other words, there is no B in this part of the equation. So, let’s err on the side of Gaston’s prominent features—good looks, size, fighting ability, egg consumption—and place the probability at 0.9.

  • P(B│¬A) = the probability of rejections if Belle does not like Gaston

The third figure should be very high for the reasons previously discussed about Belle’s honest manner. If Belle does not like Gaston, then it is almost certain she would reject him. Let’s place that probability at 0.99.

  • P(¬A) = the prior probability that Belle does not like Gaston

Finally, the last figure is 0.1. Remember, it is simply filling out the remainder from P(A).

So, now we can plug in the numbers:

P(A│B) =   0.05*0.9____________

(0.05*0.9) + (0.99*0.1)

Simplified further, it reads:

P(A│B) =   0.045____________

0.045 + 0.099

P(A│B) = 0.3125

Even though Gaston finds himself desired by women in the overwhelming majority of cases, the evidence here is such that he can be confident that Belle does not like him. There is only a 31.25% chance that she does in fact like him (A) given her rejections (B).

 

Conclusion

If you came into this not knowing much about Bayes’ Theorem, hopefully you now understand it a bit more. The math involved here is really not that difficult. You simply have to determine four numbers and then just let the formula do the heavy lifting. I would recommend plugging in some numbers to see what happens to the outcome. For example, when you have really high or low prior probabilities, it’s hard to overcome them with evidence. This was the essential point in my earlier post about extraordinary evidence.

 

[i] If you want to dig into the details more, he gives a really nice, in-depth explanation with lots of examples here: http://yudkowsky.net/rational/bayes.

 

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Nov
29

The hiddenness argument revisited (II) by J.L. Schellenberg

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In an earlier post, I outlined the argument from hiddenness by J.L. Schellenberg along with his responses to several criticisms of the argument. These criticisms were grouped together in virtue of being irrelevant, according to Schellenberg. They generally were either already covered by one of his premises or could be explained away by further clarification.

In this post, I’ll explain Schellenberg’s second article that covers criticisms he does find relevant. This should be quite simple to understand since he recommends using the same general approach to every such criticism. He calls this approach the Accommodationist Strategy (hereafter, AS). AS may need slight tailoring in each case, but the overall structure will be the same.

So, what is the AS and how does it work? Essentially, the AS works by placing an enormous burden of proof on the opponent who claims to have a defeater for the hiddenness argument. Such defeaters include offering a reason for the hiddenness (we’ll see one example). When presented with a reason for hiddenness, ask yourself whether the proposed good brought about by it can be achieved by any other means that does not result in reasonable non-belief. Remember that God is all-knowing and all-powerful, so can really use any means for achieving his ends. As Schellenberg says, “It comes up against the unsurpassable immensity of divine resourcefulness.”[i]

Let’s look at one quick example. Swinburne, in Providence and the Problem of Evil, argues that certain goods, like responsibility, flow from not having a constant or immediate knowledge of God. But can such goods, Schellenberg asks, really not be derived by any other means at this immensely resourceful God’s disposal? The fact that we can come up with ways to do it ourselves with all of our limitations would strongly suggest that God could indeed derive them in other ways. Schellenberg says these are just tokens of certain types of good. There are other ways to actualize the types of good without hindering a relationship with God. A perfectly loving and relationship-seeking God would necessarily prefer these other tokens, if available.

Similarly, the AS can be applied to soul-making and other relevant criticisms that posit some type of good achieved by hiddenness.

 

Further Thoughts

Since that strategy should be grasped pretty easily, I thought I’d include some bonus material for responding to reasons given for hiddenness. Atheist philosopher Stephen Maitzen, whose arguments I highly recommend, poses another difficulty for such objections. While Schellenberg’s AS makes it difficult to establish said criticisms as defeaters, we still might wonder how well they work as undercutters.[ii] I think Maitzen’s response to such criticisms gives us good reason to think they don’t even work well on that front. Maitzen argues that any explanation for hiddenness will have to also account for the geographic distribution of theistic belief and non-belief. For example, Afghanistan is almost uniformly theistic and Cambodia the opposite. If we assume that the reasons given for hiddenness are at work, then it’s hard to see why religion should be geographically distributed in a way that makes more sense under naturalistic explanations.


[i] “The hiddenness argument revisited (II)” p. 288.

[ii] A defeater renders a premise false outright where an undercutter renders it less probable.

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Nov
23

Extraordinary Claims Really Do Require Extraordinary Evidence

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Yes, it’s still true. The basic principles of mathematics and probability have not changed. Thus, it is surprising to read an article saying that requiring extraordinary evidence for extraordinary claims doesn’t make sense. At least, it’s surprising until you realize the person making the claim is probably an evangelical Christian with an extraordinary claim to promote. So, how should we respond? I’ll provide a short answer and a long answer.

 

The Short Answer

Bayes’ Theorem, which has been proven to be formally valid, tells us that extraordinary claims (things with very low prior probabilities) do indeed require extraordinary evidence. For example, imagine my friend tells me he won the World Series of Poker. I would first be struck by the improbability of such a claim considering the Main Event has over 6,000 entrants and this particular friend isn’t that good at poker. Even so, it’s not impossible and there is evidence which would overcome that initially low probability. Examples of significant probability-raising evidence would include if he had millions of dollars suddenly and if he had a WSOP bracelet. If you want to see this idea demonstrated mathematically, keep reading. 

 

The Long Answer

This is Bayes’ Theorem:

P(h|e.b) = P(h|b) x P(e|h.b)  /  [ P(h|b) x P(e|h.b) ] + [ P(~h|b) x P(e|~h.b) ]

 

Symbol Meaning
P Probability
H Hypothesis
E Evidence
B Background Knowledge
| Given
~ Not (or the negation of)
. And

 

In English, this says, “The probability that a hypothesis is true, given available evidence and background knowledge is equal to [and then you have the equation].” To understand the equation, let’s dig into the details just a bit.

The first term you’ll notice once you read past the = symbol is P(h|b); this is the prior probability. This term is concerned with the probability that a hypothesis is true given your background knowledge. So, when we say that the virgin birth has an incredibly low prior probability, that means that based on everything we know about the world through science, history, etc. this sort of thing doesn’t generally happen. Quite simply, we understand how babies are made, and this isn’t it. Further, in cases where parthenogenesis does actually occur in other animals, the resulting offspring are always female. So, if true, this would seem to be a one-time thing. I have no idea how many humans have ever existed, but let’s say there have been 100 billion. This would make the prior probability 1/100,000,000,000. Note that this is prior to considering any evidence for the case in question, hence the term prior probability.

The second term above is P(e|h.b), meaning the probability that you would have the available evidence given your hypothesis and background knowledge. In other words, would the available evidence be expected under the hypothesis? For example, say someone claims they’ve been to the beach for the past few hours. You notice they are sunburned and in their car is a beach towel. These things fit the claim well and would merit a high number. On the other hand, the lack of sun kissed skin and a movie ticket stub from two hours earlier would not be the sort of evidence you would expect.

This covers the numerator. It is the prior probability multiplied by the likelihood of the evidence.

Now, as you keep moving from left to right, you’ll notice that the first term in the denominator is the entire numerator repeated. That’s why the theorem is sometimes shortened to P = A / A + B. So, what is B? It is basically the same as discussed above, only for ~H, rather than H. It is the probability that your hypothesis is false and the likelihood of the available evidence given a false hypothesis.

Let’s plug in some numbers using a very low prior probability and see the earlier claim in action.

  • P = 0.01 x .9 / (0.01 x .9) + (0.99 x .75) = 1.2%

Here we see a low probability event where the evidence is almost nearly as well explained by negating hypotheses and the probability is slightly raised, but remains very low. Now, let’s slowly decrease the likelihood that the evidence can be explained by alternative hypotheses and watch what happens to the outcome.

  • P = 0.01 x .9 / (0.01 x .9) + (0.99 x .50) = 1.8%
  • P = 0.01 x .9 / (0.01 x .9) + (0.99 x .25) = 3.5%
  • P = 0.01 x .9 / (0.01 x .9) + (0.99 x .10) = 8.3%
  • P = 0.01 x .9 / (0.01 x .9) + (0.99 x .01) = 48%

We don’t see a substantial increase in probability until we get into very low ranges of likelihood for the same evidence to be observed on alternative hypotheses. In the last example, there is only a 1% chance that the evidence can be explained by alternative hypotheses. In cases of low prior probability, the evidence must be such that it basically rules out alternative hypotheses to a very high degree.

In other words, extraordinary claims really do require extraordinary evidence.

 

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