Mike Dunford has drawn my attention to a situation in Piasa, Illinois where high school English teacher Dan DeLong assigned extra-credit reading about gay animals. The situation seems to be a bit complicated. The assigned article discusses how homosexuality is common in the animal kingdom. DeLong has been suspended from his job and faces a disciplinary hearing.
If a teacher simply assigned reading from areas of study outside the teacher's area that would be a legitimate concern. However, it appears that the assignment was to read the essay and examine the essay's structure and argumentative form. Therefore, this is simply a normal high school assignment. Teachers assign all sorts of different reading about different subjects. That's part of the normal school curriculum. One might be able to describe a problem if one had evidence that DeLong had assigned the essay to make some sort of political, moral, theological or other point. However, I haven't seen any claim that that was the case. The objection simply seems to be that some parents were uncomfortable with the reading.
Unless the school had some sort of blanket policy about the teaching sexual material requiring parental permission I don't see anything that DeLong has done. While news reports are still sparse with details it seems like this is more about fear of gays and homosexuality extended to such a point that even discussing gay sex in animals triggers a backlash. That's stupid. There's simply no other way to describe it. DeLong should be restored to his position as soon as possible.
Update: DeLong has been reinstated.
Saturday, October 31, 2009
Thursday, October 29, 2009
Benford’s Law: Human Intuition, Randomness and Fraud
Suppose we look at some set of fairly natural data, say the populations of various countries. Let’s look at the leading digits of their populations. For example, the United States has a population slightly over 300 million. So, for the United States, the leading digit would be 3. What fraction of countries would you expect to have a leading digit of 1? Most people would guess 1/9th since there are 9 possibilities (1,2,3,4,5,6,7,8,9) and no obvious reasons to prefer any digit over any other digit. However, you’d be wrong:The actual percentage is slightly over 25%. In fact, it turns out that we should expect about 30% of countries to have a leading digit of 1. This strangely large number of 1s shows up frequently in natural data and is known as Benford’s law. It turns out that for many natural data sets we expect about 30% to have a leading digit of 1, about 18% to have a leading digit of 2 and so on. Benford’s Law is an important statistical rule that has a variety of generalizations and has practical applications (such as in detecting fraudulent or manipulated data).
Where is this strange pattern coming from? I didn’t have a good understanding of this until it was explained to me by Steven Miller. Consider a finite list of numbers that is reasonably well-behaved. For each member of our list, we can write it in scientific notation. So, for example. if we had 236 on our list, we would write it as 2.36 * 10^2. Note that the lead digit is then determined by what the lead digit is in the part of the scientific notation that isn’t the exponent. (This part is sometimes called the mantissa when one wants to be fancy) Now, for each number on our list, instead of looking at the number x, we can look at log10 x. What does this do to the scientific notation? Well, scientific notation then corresponds to the log of the mantissa + an integer that is the power of the exponent. So, for example, log10 (2.36 * 10^2)= .3729...+ 2.
Let’s examine the non-integer part of log10 x (call this f(x)) What distribution do we expect for f(x)?, It is a fixed value between 0 and 1 with no clear cut offs or biases in any direction so the most obvious thing to do is to make it a uniform distribution. That means that there’s about a 5% chance that f(x) falls below .05, about a 20% chance that f(x) falls below .2, about a 50% chance that f(x) falls below .5 and so on.
So what does this tell us about the leading digit? If the mantissa is below 2, then we have lead digit 1. If the mantissa is between 2 and 3, then we have lead digit 2 and so on. The mantissa is below 2 if f(x) is less than log10 2 = .301. Accordingly, we should expect that the mantissa is below 2 about 30.1% of the time. Thus, a number should have a lead digit of 1 about 30.1% of the time. Similar logic works for how frequently we should expect the lead digit to be 2, or 3 or so on.
What is wrong with the intuition that every lead digit is just as common? When we calculate probabilities, we are used to using the simplest probability distribution we can imagine, something like picking a positive integer from 1 to 10^n for some fixed n. We are used to this approach primarily because it is easy to calculate. Consequently, most probability problems in high school and college assume that we have such a uniform distribution since that assumption makes the math much easier. But actual distributions in real life don’t often look like this. For example, we might have a Bell curve or some other distribution. For almost any distribution that arises in nature, Benford’s law will apply due to the logic we used earlier.
So what does this do for us? Perhaps most importantly, we can use this insight to detect fraud. When humans try to make up data, it often fails to fit Benford’s law. In general, humans are bad at constructing data that passes any minimal test for randomness. Failure to obey a generalized version of Benford’s law was one of the major pieces of evidence for election fraud in the last Iranian election. The recent questions regarding whether Strategic Visions Polling was falsifying poll data arose when Nate Silver noticed that its results diverged substantially from Benford’s law.
For more information on Benford’s Law and related patterns in data, as well as more mathematical discussions of that data, see Terry Tao’s blog post from which I shamelessly stole the hard data about populations of nations.
Where is this strange pattern coming from? I didn’t have a good understanding of this until it was explained to me by Steven Miller. Consider a finite list of numbers that is reasonably well-behaved. For each member of our list, we can write it in scientific notation. So, for example. if we had 236 on our list, we would write it as 2.36 * 10^2. Note that the lead digit is then determined by what the lead digit is in the part of the scientific notation that isn’t the exponent. (This part is sometimes called the mantissa when one wants to be fancy) Now, for each number on our list, instead of looking at the number x, we can look at log10 x. What does this do to the scientific notation? Well, scientific notation then corresponds to the log of the mantissa + an integer that is the power of the exponent. So, for example, log10 (2.36 * 10^2)= .3729...+ 2.
Let’s examine the non-integer part of log10 x (call this f(x)) What distribution do we expect for f(x)?, It is a fixed value between 0 and 1 with no clear cut offs or biases in any direction so the most obvious thing to do is to make it a uniform distribution. That means that there’s about a 5% chance that f(x) falls below .05, about a 20% chance that f(x) falls below .2, about a 50% chance that f(x) falls below .5 and so on.
So what does this tell us about the leading digit? If the mantissa is below 2, then we have lead digit 1. If the mantissa is between 2 and 3, then we have lead digit 2 and so on. The mantissa is below 2 if f(x) is less than log10 2 = .301. Accordingly, we should expect that the mantissa is below 2 about 30.1% of the time. Thus, a number should have a lead digit of 1 about 30.1% of the time. Similar logic works for how frequently we should expect the lead digit to be 2, or 3 or so on.
What is wrong with the intuition that every lead digit is just as common? When we calculate probabilities, we are used to using the simplest probability distribution we can imagine, something like picking a positive integer from 1 to 10^n for some fixed n. We are used to this approach primarily because it is easy to calculate. Consequently, most probability problems in high school and college assume that we have such a uniform distribution since that assumption makes the math much easier. But actual distributions in real life don’t often look like this. For example, we might have a Bell curve or some other distribution. For almost any distribution that arises in nature, Benford’s law will apply due to the logic we used earlier.
So what does this do for us? Perhaps most importantly, we can use this insight to detect fraud. When humans try to make up data, it often fails to fit Benford’s law. In general, humans are bad at constructing data that passes any minimal test for randomness. Failure to obey a generalized version of Benford’s law was one of the major pieces of evidence for election fraud in the last Iranian election. The recent questions regarding whether Strategic Visions Polling was falsifying poll data arose when Nate Silver noticed that its results diverged substantially from Benford’s law.
For more information on Benford’s Law and related patterns in data, as well as more mathematical discussions of that data, see Terry Tao’s blog post from which I shamelessly stole the hard data about populations of nations.
Sunday, October 18, 2009
HPV, Cancer and Friends
Stephanie Zvan who blogs at Almost Diamonds has cervical cancer. A recent Pap smear came back with abnormal results and follow-up work found the cancer. Stephanie has talked about her experience and given follow-up information about her prognosis. Pieces responding to Stephanie's situation are at ERV and PalMD that are both worth reading.
I'm going to use this as an opportunity to go over some very basic issues:
First, for many cancers, getting them caught early matters. This is the case for many cancers. If you are a woman make sure you get regular Pap smears. If you are a man make sure to get regular prostate exams. No matter who you are, if you are older, make sure to get a regular colonoscopy. Etc. Etc. These exams save lives. When cancer is found doesn't just impact what treatment options there are but can be the difference between life and death.
Second, get the HPV vaccine. Unfortunately, this vaccine is not yet available for men but is available for women. HPV is one of the leading causes of cervical cancer. HPV is also associated with penile cancer in males. Once one has a given strain of HPV, getting a vaccine later isn't going to help matters. Moreover, HPV transmission is not prevented by condom use. HPV is a nasty little bugger and is much more easily transmitted than most STDs.
Stephanie grew up before there was a vaccine. However, in another respect, she is lucky. Stephanie and her husband don't intend to have children. However, for others they aren't so lucky. If you are a woman, get vaccinated. Men, when the vaccine is finally approved for males, please get vaccinated. Not only will this protect you, it will prevent you from getting a nasty disease which could then be given to those you love.
Stay informed. Stay protected. Stay safe.
Edit: Apparently the vaccine has now been approved for males. Good timing.
I'm going to use this as an opportunity to go over some very basic issues:
First, for many cancers, getting them caught early matters. This is the case for many cancers. If you are a woman make sure you get regular Pap smears. If you are a man make sure to get regular prostate exams. No matter who you are, if you are older, make sure to get a regular colonoscopy. Etc. Etc. These exams save lives. When cancer is found doesn't just impact what treatment options there are but can be the difference between life and death.
Second, get the HPV vaccine. Unfortunately, this vaccine is not yet available for men but is available for women. HPV is one of the leading causes of cervical cancer. HPV is also associated with penile cancer in males. Once one has a given strain of HPV, getting a vaccine later isn't going to help matters. Moreover, HPV transmission is not prevented by condom use. HPV is a nasty little bugger and is much more easily transmitted than most STDs.
Stephanie grew up before there was a vaccine. However, in another respect, she is lucky. Stephanie and her husband don't intend to have children. However, for others they aren't so lucky. If you are a woman, get vaccinated. Men, when the vaccine is finally approved for males, please get vaccinated. Not only will this protect you, it will prevent you from getting a nasty disease which could then be given to those you love.
Stay informed. Stay protected. Stay safe.
Edit: Apparently the vaccine has now been approved for males. Good timing.
Thursday, October 15, 2009
Where Are All the Smart Apologists?
Recently I read The Screw Tape Letters, a series of letters written by C.S. Lewis. The letters purport to be from an old demon giving advice to a young demon. Lewis is a smart, funny and talented writer. What the demon Screwtape says is as revealing and clever as what he does not say. This is not the only great work of apologetics by Lewis. Why are there no great Christian apologists like Lewis today?
Lewis is far from perfect. Lewis gave birth to one of the most annoying apologetic arguments, the Trilemma. However, even there his intelligence and originality shine through. I have not seen any contemporary apologist produce any argument that isn’t a tired repackaging of pre-existing arguments.
Who are the major apologists today? There really aren’t any in the influential way that Lewis was. But if one had to identify those who today continue the tradition of Christian apologetics, one would probably list Ray Comfort, William Demsbki, and Alister McGrath.
Do any of these people measure up to C. S. Lewis? No. Consider these writers individually:
Does Ray Comfort stack up to Lewis? No way. Ray Comfort is an idiot and an ignoramus. He’s the man who most famously tried to claim that the modern shape of the banana was evidence for a divine creator. Yes, the banana, a fruit that has been heavily modified by extensive breeding by humans, a fruit whose wild form is a nasty hard thing full of seeds.
Does William Dembski stack up to Lewis? Wililam Dembski isn’t an idiot like Comfort. He has a real PhD in mathematics. But this also is a man who, after intelligent design failed in the courts, was reduced to teaching apologetics at a second rate seminary while giving course credit to students for trolling pro-evolution websites. I can’t see C.S. Lewis doing that. Moreover, Dembski’s writing ability resembles that of a 7th grader trying to sound like he’s really bright and well read. I should know. I used to write like Dembski when I was in 7th grade. Demsbki also seems to spend most of his time fighting with other Christians. (He really, really doesn’t like theistic evolution.)
Does Alister McGrath stack up to C. S. Lewis? Now we are getting closer. McGrath is a respected theologian who also has a degree in biophysics. He’s bright. He’s willing to accept both science and religion. He has on occasion made cogent arguments. But there are two problems: First, he’s a dreadfully boring writer. I have trouble staying awake when I read anything he writes. Someone needs to get Ben Stein to do a book on tape of one of McGrath’s books. It would be the ultimate sleep aid. Or maybe it would be a weapon of mass destruction as just playing it nearby would cause individuals within a hundred mile radius to fall into irreversible comas. This brings us to the other issue with McGrath: The subjects and titles of his books are equally dreadful. His two most well known books are "The Dawkins Delusion?" and "Dawkins' God." Ok, Alister. We get the point. You don’t like Richard Dawkins.
So why are there no great apologists for Christianity today? Here are four possible explanations:
First, perhaps great apologists are simply rare and C.S. Lewis is a great outlier. This isn’t a satisfactory explanation. I could compare the modern stock of apologists with G. K. Chesterton and they would still not match up.
A second argument is that Christianity is not the common belief among intellectuals that it was fifty or sixty years ago. Since a smaller fraction of intellectuals today are deeply Christian and since apologetics is valued less today as it has been in the past, intellectuals are much less likely to go into apologetics.
Third, the state of the evidence has changed over time to make belief in Christianity less probable. This argument is almost certainly wrong. The major modern controversies implicating Christianity and Judeo-Christian religions in general have existed for a very long time. The Documentary Hypothesis and similar theories about other Biblical texts have been around for more than a century. So has evolution. Thus, the need to address these issues (either by reconciling Christianity with them, or by refuting them) has existed for a long time.
Fourth, the modern focus of apologetics has been the watchmaker analogy and variations thereof. The watchmaker analogy is an argument for the existence of God based on an analogy to a watch found in the desert which one would immediately realize had a designer. It is no coincidence that the three apologists listed above, all have arguments that revolve around the watchmaker. Ray Comfort uses a particularly stupid form of the watchmaker argument. William Dembski uses a particularly obfuscated form of the watchmaker argument. And Alister McGrath doesn’t really use the argument itself but rather spends most of his time arguing that Richard Dawkins hasn’t sufficiently refuted the watchmaker argument and that if Dawkins is fallible God must then exist.
This focus is understandable: The watchmaker argument and other teleological arguments for the existence of God are some of the hardest to refute. However, the focus of all contemporary apologetics on a single argument has left the industry stagnant and uncreative. In such circumstances, it isn’t surprising that apologetics fails to attract many intellectuals. Moreover, the focus on the watchmaker argument has caused much of modern apologetics (and thus many of modern apologists) to go head to head with much well-established science. C. S. Lewis in contrast was open to the possibility that evolution was correct. If the entire apologetic system revolves around attacking basic science, one shouldn't be surprised that not many bright, educated people are willing to lead it.
I’m not completely satisfied with any of these explanations. However, the decline of contemporary Christian apologetics needs explanation.
Lewis is far from perfect. Lewis gave birth to one of the most annoying apologetic arguments, the Trilemma. However, even there his intelligence and originality shine through. I have not seen any contemporary apologist produce any argument that isn’t a tired repackaging of pre-existing arguments.
Who are the major apologists today? There really aren’t any in the influential way that Lewis was. But if one had to identify those who today continue the tradition of Christian apologetics, one would probably list Ray Comfort, William Demsbki, and Alister McGrath.
Do any of these people measure up to C. S. Lewis? No. Consider these writers individually:
Does Ray Comfort stack up to Lewis? No way. Ray Comfort is an idiot and an ignoramus. He’s the man who most famously tried to claim that the modern shape of the banana was evidence for a divine creator. Yes, the banana, a fruit that has been heavily modified by extensive breeding by humans, a fruit whose wild form is a nasty hard thing full of seeds.
Does William Dembski stack up to Lewis? Wililam Dembski isn’t an idiot like Comfort. He has a real PhD in mathematics. But this also is a man who, after intelligent design failed in the courts, was reduced to teaching apologetics at a second rate seminary while giving course credit to students for trolling pro-evolution websites. I can’t see C.S. Lewis doing that. Moreover, Dembski’s writing ability resembles that of a 7th grader trying to sound like he’s really bright and well read. I should know. I used to write like Dembski when I was in 7th grade. Demsbki also seems to spend most of his time fighting with other Christians. (He really, really doesn’t like theistic evolution.)
Does Alister McGrath stack up to C. S. Lewis? Now we are getting closer. McGrath is a respected theologian who also has a degree in biophysics. He’s bright. He’s willing to accept both science and religion. He has on occasion made cogent arguments. But there are two problems: First, he’s a dreadfully boring writer. I have trouble staying awake when I read anything he writes. Someone needs to get Ben Stein to do a book on tape of one of McGrath’s books. It would be the ultimate sleep aid. Or maybe it would be a weapon of mass destruction as just playing it nearby would cause individuals within a hundred mile radius to fall into irreversible comas. This brings us to the other issue with McGrath: The subjects and titles of his books are equally dreadful. His two most well known books are "The Dawkins Delusion?" and "Dawkins' God." Ok, Alister. We get the point. You don’t like Richard Dawkins.
So why are there no great apologists for Christianity today? Here are four possible explanations:
First, perhaps great apologists are simply rare and C.S. Lewis is a great outlier. This isn’t a satisfactory explanation. I could compare the modern stock of apologists with G. K. Chesterton and they would still not match up.
A second argument is that Christianity is not the common belief among intellectuals that it was fifty or sixty years ago. Since a smaller fraction of intellectuals today are deeply Christian and since apologetics is valued less today as it has been in the past, intellectuals are much less likely to go into apologetics.
Third, the state of the evidence has changed over time to make belief in Christianity less probable. This argument is almost certainly wrong. The major modern controversies implicating Christianity and Judeo-Christian religions in general have existed for a very long time. The Documentary Hypothesis and similar theories about other Biblical texts have been around for more than a century. So has evolution. Thus, the need to address these issues (either by reconciling Christianity with them, or by refuting them) has existed for a long time.
Fourth, the modern focus of apologetics has been the watchmaker analogy and variations thereof. The watchmaker analogy is an argument for the existence of God based on an analogy to a watch found in the desert which one would immediately realize had a designer. It is no coincidence that the three apologists listed above, all have arguments that revolve around the watchmaker. Ray Comfort uses a particularly stupid form of the watchmaker argument. William Dembski uses a particularly obfuscated form of the watchmaker argument. And Alister McGrath doesn’t really use the argument itself but rather spends most of his time arguing that Richard Dawkins hasn’t sufficiently refuted the watchmaker argument and that if Dawkins is fallible God must then exist.
This focus is understandable: The watchmaker argument and other teleological arguments for the existence of God are some of the hardest to refute. However, the focus of all contemporary apologetics on a single argument has left the industry stagnant and uncreative. In such circumstances, it isn’t surprising that apologetics fails to attract many intellectuals. Moreover, the focus on the watchmaker argument has caused much of modern apologetics (and thus many of modern apologists) to go head to head with much well-established science. C. S. Lewis in contrast was open to the possibility that evolution was correct. If the entire apologetic system revolves around attacking basic science, one shouldn't be surprised that not many bright, educated people are willing to lead it.
I’m not completely satisfied with any of these explanations. However, the decline of contemporary Christian apologetics needs explanation.
New Largest Prime Number Found
The largest known prime has now been bumped up. The recently discovered prime is 2^43112609 − 1 which in base 10 has around twelve million digits. As with all the largest primes discovered for most of the last hundred years, the prime is a Mersenne prime, that is it is one less than a power of 2. As with the last few Mersenne primes discovered, this was discovered by the Great Internet Mersenne Prime Search which uses distributed computing to search for Mersenne primes. I've blogged before about Mersenne primes. For more details on why we care about Mersenne primes and their history dating back to the ancient Greeks see this post and this post.
There are reasons to suspect that if 2^n-1 is prime that n-1 should be likely to have a lot of small prime factors. In this case, 43112609 -1= 2^5 * 7 * 11 * 17497. So while some of the prime divisors look very small, it has at least one prime very large prime divisor and so doesn't seem to fit this pattern well. This is in contrast to the last discovered Mersenne prime 2^42643801 - 1. In that case one has 42643800 = 2^3 * 3^3 * 5^2 * 53 * 149. It remains to be seen if this new example is simply an outlier or if we need to reevaluate what we expect the exponents of Mersenne primes to look like.
There are reasons to suspect that if 2^n-1 is prime that n-1 should be likely to have a lot of small prime factors. In this case, 43112609 -1= 2^5 * 7 * 11 * 17497. So while some of the prime divisors look very small, it has at least one prime very large prime divisor and so doesn't seem to fit this pattern well. This is in contrast to the last discovered Mersenne prime 2^42643801 - 1. In that case one has 42643800 = 2^3 * 3^3 * 5^2 * 53 * 149. It remains to be seen if this new example is simply an outlier or if we need to reevaluate what we expect the exponents of Mersenne primes to look like.
Monday, October 5, 2009
Glenn Beck, International Law, and Hypocrisy
The last time I discussed Glenn Beck on this blog, we discussed Beck's general ignorance and contempt regarding basic legal history. Most astounding of Beck's claims was the claim that the notion that laws change over time was an idea that arose in the 1920s as a response to "Darwinian evolution."
After remarks like that one might think that Beck isn't a bad person but just incredibly ignorant. However, Ed Brayton has drawn attention to Beck's latest interaction with international law demonstrates his true hypocrisy.
Beck frequently use the "I'm just asking questions" gambit to insinuate hurtful and borderline libelous claims. In response, One enterprising individual started a satirical website "Did Glenn Beck Rape and Murder a Young Girl in 1990?"Frankly, this seems to me to be over the line of reasonable civil discourse (even if it is amusing).
How has Beck responded to this website? Beck has petitioned World Intellectual Property Organization to remove control of the domain from the satirist. Beck's argument is that the domain creates confusion with his brand name and therefore is a violation of international conventions on trademarks. That claim is so profoundly stupid that I'm not going to bother addressing it.
The lawyers for the website responded by explaining in detail what was wrong with Beck's claim and noting that his appeal to WIPO seemed to be an attempted run-around of the First Amendment of the Constitution of the United States. What makes Beck's behavior noteworthy is that, as one of the commentators to Brayton's blog observed, Beck has a history not just of complaining about transnationalism but of any intervention of international law into U.S. affairs. Indeed, Beck wrote in a March 30, 2009 column:
After remarks like that one might think that Beck isn't a bad person but just incredibly ignorant. However, Ed Brayton has drawn attention to Beck's latest interaction with international law demonstrates his true hypocrisy.
Beck frequently use the "I'm just asking questions" gambit to insinuate hurtful and borderline libelous claims. In response, One enterprising individual started a satirical website "Did Glenn Beck Rape and Murder a Young Girl in 1990?"Frankly, this seems to me to be over the line of reasonable civil discourse (even if it is amusing).
How has Beck responded to this website? Beck has petitioned World Intellectual Property Organization to remove control of the domain from the satirist. Beck's argument is that the domain creates confusion with his brand name and therefore is a violation of international conventions on trademarks. That claim is so profoundly stupid that I'm not going to bother addressing it.
The lawyers for the website responded by explaining in detail what was wrong with Beck's claim and noting that his appeal to WIPO seemed to be an attempted run-around of the First Amendment of the Constitution of the United States. What makes Beck's behavior noteworthy is that, as one of the commentators to Brayton's blog observed, Beck has a history not just of complaining about transnationalism but of any intervention of international law into U.S. affairs. Indeed, Beck wrote in a March 30, 2009 column:
"Once we sign our rights over to international law, the Constitution is officially dead. When you say things like, 'We are not going to put the Constitution behind international law,' you say that in the international court, if you say that on the floor of the United Nations, you are a freak show."In the same column Beck also wrote this gem:
"Let me tell you something. When you can't win with the people, you bump it up to the courts. When you can't win with the courts, you bump it up to the international level."I'll let those comments speak for themselves.
Subscribe to:
Posts (Atom)
