The law of large WTF?

Andrew Gelman recently pointed out that some financial folks have some very strange ideas about the law of large numbers. Apparently, those strange ideas are deeply ingrained. Here are some excerpts from Investopedia's entry on the law of large numbers:

Definition of 'Law of Large Numbers.' In statistical terms, a rule that assumes that as the number of samples increases, the average of these samples is likely to reach the mean of the whole population.

So far so good; it's an informal definition of the same concept that statisticians call the law of large numbers. But then when you read,

When relating this concept to finance, it suggests that as a company grows, its chances of sustaining a large percentage in growth diminish. This is because as a company continues to expand, it must grow more and more just to maintain a constant percentage of growth.

...you start scratching your head. The above statement, while definitely true, has precisely fuck-all to do with the law of large numbers. As does the next one:

As an example, assume that company X has a market capitalization of $400 billion and company Y has a market capitalization of $5 billion. In order for company X to grow by 50%, it must increase its market capitalization by $200 billion, while company Y would only have to increase its market capitalization by $2.5 billion. The law of large numbers suggests that it is much more likely that company Y will be able to expand by 50% than company X.

No matter how you look at it, this is not an example of the law of large numbers. Whoever wrote this entry does not understand the definition they provided.

Added: I missed the mistake in Investopedia's definition of the law: It's not an assumption, it's a theorem.

More Added: For the sake of comparison, here is Felix Salmon's example of a correct use of the law of large numbers in a business setting:

If you run a bunch of casinos with hundreds of thousands of punters coming and and betting hundreds of millions of dollars, then you can predict with high accuracy the amount of money you're going to make at the end of the quarter.

They wouldn't do that, would they?

Let's start off the blog's new incarnation with a fresh installment of the Non Sequitur of the Month.

The largest oppisition party in Poland, called Law and Justice, has recently put forward a vague proposal of a new tax which they call "banking tax". No details are available, but judging from media rumors, it's essentially supposed to be some sort of tax on financial transactions. Predictably, the proposal has been criticized on the grounds that, contrary to Law and Justice's rhetoric, this new tax will be a burden not only on rich financial executives, but also on ordinary customers of banking institutions, since those institutions will most likely respond by raising prices of their services.

Now for the non sequitur part. In a TV interview, Law and Justice's press spokesman responded to this criticism by saying that the whole argument is a "myth," and that the critics must not really trust free markets as much as they say they do because if we have free markets in the banking sector, then banks have to compete for customers and will therefore not try to shift the burden of the new tax on them.

This is exactly like saying that the belief that an increase in world prices of crude oil will drive up prices of gas at the pump is a "myth". After all, car fuel is sold on a free market by multiple distributors who have to compete for customers.

The beauty of this argument is that it's such a pure example of a non sequitur: the conclusion does not follow from the premise. If the Polish banking sector is indeed a competitive free market, then prices of banking services are always equal to their marginal costs. A new tax on a service increases its marginal cost. The end.

Non sequitur of the month: Stunning economics

This installment is inspired by a piece of breathtaking simple-mindedness exhibited by a writer for the Polish leftist site Krytyka Polityczna (it means "political critiques;" think of it as of Polish Huffington Post, with all applicable differences of scale of course). Some guy named Kapela has a review of the movie "The Social Network" in which he writes:

Where does Facebook's worth of 25 billion dollars come from? (...) it means that each account is worth about $50. How come I can't sell it then? I have over 1000 friends, which is more than average. So maybe my account is worth more than $50. No one wants to buy it though.

To clear things up first, the actually relevant reason you can't sell your account is that you don't legally own it; Facebook does. But let's imagine you would be legally able to sell your account. You still wouldn't actually be able to, and the reason is that your account isn't worth $50. It's worth next to nothing. Most of Facebook's revenue comes from selling ads. The possibility of reaching hundreds of millions of people is worth a lot to advertisers, so they are willing to pay a lot for it. The possibility of reaching "over a thousand" of individuals through one account, on the other hand, isn't worth a dime to advertisers, which means a single account would bring zero revenue, which means that it's worth about zero dollars. Now this isn't necessarily true of all accounts; I'm sure that if, say, Barack Obama were to sell his account on ebay, he could get a lot of money for it. But in those cases the account's worth is due in large part to things other than possible future revenue. Assuming that since 500 million accounts are worth $25 billion, 1 account is worth $50, is just dumb.

Elections bring out the stupid in people

So, elections are upon us. Ever heard the phrase "If you don't vote, you have no right to complain?" Always uttered with much self-righteousness and absolutely no reflection, it's one of the most blatant non sequiturs in circulation at the moment. There's absolutely no logic to it. The probability that your particular vote will be pivotal in terms of the outcome of state or national elections is so small that it's essentially zero. Given the fact that multiple millions of other voters turn out at the polls, the influence that your individual vote has on the outcome is zero, which is about the same as the influence you have on the outcome if you don't even bother to vote in the first place. Ergo, either everyone has the right to complain or no one does.

At any rate, I have recently stumbled upon an original rephrasing of this peculiar argument due to a Marc Hedlund writing for O'Reilly.com. Hedlund starts with a simple paraphrase of the original:

Once, many years ago, I was waiting in line at the Post Office on election day. One postal worker asked another if she had voted, and the second responded, "Hell yes, I voted. If you don't vote, you can't bitch, and I am not giving up my right to bitch!"

So far so bad. Therefore, Hedlund decides to make it worse by inventing his own analogy of the problem:

I was thinking about that the other day when trying to decide whether to buy a new iPhone 4 or wait to see what happens with Verizon at some point in the future (...) If you're buying an iPhone 4 tomorrow, you already know AT&T is almost universally considered the weakest aspect of the phone's experience. You're signing up for that (...) So here's the deal: if you buy that phone right now, you're giving up your right to bitch about AT&T for the next two years. No, I mean it! Complaints will be returned to sender unread.

(...)

I'm undecided about what to do, myself. I'll probably cave. But if I do, I won't bitch, and you shouldn't either. The single strongest message you could send to Apple and AT&T would be to vote with your wallet against AT&T's crappy service. If you don't vote, then you're getting what you paid for.

OK, so this is actually a slight variation on the original, because what Hedlund seems to be saying is that you have the right to complain about Obama if and only if you voted for McCain. If you voted for Obama, you can't complain because you got what you've signed up for, and if you didn't vote at all, you can't complain because blah blah blah whatever (I'm not sure what the argument is there).

However, none of this changes the fact that Hedlund's is one of the worst analogies in the history of bad analogies. If I decide to buy an iPhone instead of a Verizon-based smartphone, i will have an iPhone. If I decide to buy a Verizon phone, I will have that. See where this is going? What happens to me phone-wise depends on my decision and my decision only. By contrast, if I decide to vote McCain, I may get McCain for president, but then again I may not. I may get Obama. What happens to me president-wise depends not just on what I do, but also on what about 215 million other people do. 215 million people on whom I have no influence whatsoever. In other words, while I decide what type of phone I will use (or if I will use any type of phone at all, for that matter), I do not decide what type of president I will use. That is decided for me. Sometimes I will like the outcome, and sometimes I will not, but either way the fact is that I did not choose the outcome. Tell me again, then, what exactly does the act of voting have to do with the right to complain?

Now because people have full control over what phone they use, they tend to choose between phones based on which one they think will be best for them. In other words, if they buy an iPhone and not a Verizon phone, they do it because they like iPhone's features so much that they are willing to get over its inferior signal reception; if they buy a Verizon phone, they do it because they value signal reception so much that they are not willing to sacrifice it for additional features. Hedlund suggests a novel reason for making a choice like that: if you don't like Apple's crappy phone service, buy Verizon in order to induce Apple to change their strategy. Actually, perhaps this is a good analogy of voting after all: thinking that your individual decision as to where to allocate $300 can change the behavior of a corporation whose annual revenue is some $43 billion is just as delusional as thinking that your individual vote matters for an outcome of an election with over 200 million participating voters. If you're forgoing iPhone 4 because you want to change Apple's strategy with respect to phone service, you have a serious problem with respect to your grip on reality.

Non sequitur of the month: One of the most bizarre attempts at argument I've seen

It's due to Magda Figurska, a Polish blogger I've written about once before. She commented on a blog post whose author quoted a statistic which says that in the second quarter of 2010, annualized GDP growth in Poland was 3.4%. Figurska wrote: "What you're writing is just like communist propaganda. You write about GDP growth of 3.5% in the second quarter but forget to mention that, in the same quarter, inflation was 3%." So far so bad. Figurska is unaware that the 3.4% annualized GDP growth statistic is already adjusted for inflation (in fact, published GDP statistics always are). That's stunning ignorance, but it isn't a non sequitur just yet. Inevitably, another commenter pointed her mistake out to her, to which she replied: "Really? Then look here. Inflation in Germany 0.00%." (The link she gave was simply a table with a bunch of economic statistics for the second quarter of 2010 for a bunch of countries.)

In other words, the fact that Germany experienced no inflation in April, May and June of 2010 somehow proves that GDP growth statistics are not adjusted for inflation. Other commenters in that thread asked Figurska to elaborate on this rather peculiar inference but she ignored them. So what we're left with is a truly stunning non sequitur.

One more about Newt

A few more words about Gingrich's statement I wrote about in my previous post. Was he equating Muslims with Nazis? No, he wasn't. His was making an analogy, but the analogy was not between Muslims and Nazis. He said that a mosque is to the World Trade Center site what a swastika would be to the Holocaust museum. From this, it doesn't follow that a mosque is like a swastika. Of course it's possible that deep down he does believe the latter to be true. But it's not what he said or implied, so he cannot be held responsible for having this belief.

In short, everyone is wrong. Gingrich and those who agree with him are wrong about the law. Those who criticize him for saying Muslims are like Nazis are wrong about logic.

Good decision, bad result. Bad decision, good result.

The aftermath of the Warsaw Uprising was absolutely horrifying. Because the Soviets deliberately decided to not help the Polish resistance (Home Army) in their fight with the Nazis--which, militarily, they were in a position to do--the Home Army has found itself grossly outmatched and alone in its plight. The Uprising went on for 63 days and, on average, each one of these days saw more Polish casualties then the total number of victims of the 9/11 attacks. The Home Army did not have enough resources to protect Warsaw's civilians from Nazi genocide; overall, it is estimated that the Nazis murdered about 200,000 Polish civilians. Most of those murders occurred in mass executions in which people were rounded up, building by building, block by block, and shot by the SS troops. Indiscriminately of anything. Old people were shot, as were young people, men, women, 10-year-olds, infants, pregnant women, disabled people; in hospitals, the sick and wounded were being burned alive on their beds. For example, during the 48 hours between August 4 and August 5 ("Black Saturday") of 1944, the SS murdered 35,000 helpless civilians in indiscriminate street executions (see The Wola Massacre). This was all in accordance with a personal order from Hitler. After the Home Army capitulated, its troops and almost all remaining city residents were sent to concentration camps in Germany. In the Summer of 1939, Warsaw's population was roughly 1,300,000. In the Fall of 1944, after the Uprising was over, 90% of the city's buildings lay in ruins; amongst those ruins there were about 1,000 survivors. That's what Warsaw, before the war a densely populated, modern, affluent, vibrant and culturally sophisticated city, was reduced to in the aftermath of the Uprising. Not only did the Uprising end in apocalyptic destruction, it also did not help Polish people to bargain any type of independence from the Soviets whose plans were to turn Poland into its satellite state. Which they did. Despite fighting the Nazis and bearing unimaginable costs of this fight, Poland was not granted by the Soviets any more independence than Hungary or Bulgaria--countries which not only never fought the Nazis, but were actually allied with them.

In my opinion, the Uprising's end was the worst possible scenario realized. I can't imagine how things could have gone any worse. And of course to this day, the discussion as to whether or not the decision to fight was a mistake, is extremely emotional. For now, I'd like to side step those emotions and concentrate on something admittedly very dry: how are we to decide if the Uprising was a mistake or not. Six years ago, which was the sixtieth anniversary of the Uprising, one of the largest Polish newspapers Rzeczpospolita ("The Republic") ran a series of articles which debated the question of whether the Uprising was a mistake or not. The one quote I remember is from a historian Dr. Dariusz Stol who said

Of course, the decision to launch the Uprising was a mistake; any decision that leads to a catastrophe is by definition a mistake.

This statement will be my non sequitur of the month. It is an example of hindsight bias, that is, a propensity to overestimate the ex-ante probability of occurrence of those events that actually occurred. Dr. Stol's version of hindsight bias is quite extreme; he seems to think that the outcome of the Warsaw Uprising that was actually observed was inevitable (had an ex-ante probability of occurrence equal to 1).

Nothing in life is certain or impossible. Life is a random variable. There is a certain deterministic component to it, sure; but there's a random one as well. And this is why decisions cannot be judged by their one-time consequences, because decisions can only be right or wrong on average. What follows from this is that it's possible to make a right decision that leads to disaster, and conversely, a bad decision that leads to good outcomes. For example, in poker, making large bets to chase an inside straight on the river is a very bad decision. Even if luck is on your side, you do get your straight on the river and win a large pot, you have still made a bad decision. And conversely, betting "all in" when you have a set of aces and you think your opponent only has an inside straight draw is a good decision; and even if he gets that Jack on the river and beats you, you made the right decision, and he has made a wrong one.

I do, actually, believe that the Home Army commanders made a mistake when they started the Warsaw Uprising. I believe they underestimated the probability that Soviets will not help at all. But there remains also the possibility that they were not mistaken at all: they may have estimated the relevant probabilities correctly, then simply gambled, and lost. At any rate, if they did make a mistake, it certainly was not a mistake "by definition." The actual result of the Uprising was not inevitable. The commanders didn't think the Uprising would end the way it did. The Home Army headquarters did not have a death wish: if they knew the Uprising would end the way it actually did, they would not have started it.

The naturalistic fallacy, subtle style

Via PZ Myers: An adjunct professor at the University of Illinois Kenneth Howell has written an email to his students in which he defended the Catholic position on homosexuality. Here's an excerpt:

But the more significant problem has to do with the fact that the consent criterion is not related in any way to the NATURE of the act itself. This is where Natural Moral Law (NML) objects. NML says that Morality must be a response to REALITY. In other words, sexual acts are only appropriate for people who are complementary, not the same. How do we know this? By looking at REALITY. Men and women are complementary in their anatomy, physiology, and psychology. Men and women are not interchangeable. So, a moral sexual act has to be between persons that are fitted for that act. Consent is important but there is more than consent needed.

This reasoning is an instance of the naturalistic fallacy, albeit in a somewhat subtler form. In its naive version, the naturalistic fallacy states that what is natural is moral and vice versa. This is grotesquely absurd: rape and murder are natural, for example, so they would have to be adjudicated as moral. Howell doesn't go as far as to say that morality and naturalness are equivalent; he adds consent as another criterion for judging an act to be moral. More precisely he (seems to be) saying that consent is a necessary, but not sufficient, condition for an act to be qualified as moral. (IOW, all moral acts are consensual but not all consensual acts are moral.) To fulfill the morality definition, an act must be consensual and natural (as he says, "be a response to REALITY").

But this definition is absurd too. I'm sure any of my readers could come up with an indefinite number of examples of consensual acts that pretty much everyone on the planet agrees are not immoral but which nonetheless are highly unnatural. To give just one example: If he's consistent in his beliefs, Howell must think that flying in an airplane is immoral. It doesn't matter that people do it voluntarily; flying is not appropriate for people. How do we know this? By looking at REALITY. There is nothing in human anatomy or physiology that would suggest humans are meant to fly. So, a moral act of flying has to be left to species that are fitted for that act.

Non sequitur of the month: pure and simple

This one is also communicated personally. In a conversation I've had with a friend I argued it's silly to avoid travelling to Middle East if the reason for it is fear of being a victim of a terrorist attack because the probability of that happening to you is negligibly low. His answer was: "Yeah, but those statistics mean a lot less if it's you who's the victim, don't they?"

Some non sequiturs are really other fallacies in disguise. Not this one; this is a non sequitur in its purest. There isn't much you can say in ways of explaining what is wrong with it except it just does not follow.

Non sequitur of the month: personal communication

I'll describe this month's entry in abstract terms first, then provide two examples. Suppose you have two sets A and B. They're both non-empty and they overlap; other than that, no additional assumptions about them are made. Our non sequitur of the month is the following implication:

If the number of elements that belong to the intersection of A and B constitutes x percent of the number of all elements of A, then it also constitutes x percent of the number of all elements of B.

This is true if and only if A and B are exactly equal in size (or, as a special case, if they are one and the same set). But no such assumption was made, so it is definitely not true; in fact, it is a stunning non sequitur.

I've encountered this baffling mistake in two real-life contexts; describing those may help highlight how incredibly wrong this reasoning is. First context: There are studies trying to estimate the percentage of homosexual men among pedophiles. Any study that comes up with such an estimate (say 10 percent) will invariably be quoted somewhere in the media as saying that 10 percent of homosexual men are pedophiles--as if it were the same thing. Second context is a guy I knew named Bob (not his actual name). Bob is a very successful individual. He holds a Ph.D. in one of the social sciences, and among many things he does for a living is teaching a course in quantitative research methods at an Ivy League university. One of Bob's research interests is the connection between homelessness and foster care. There's research out there estimating that roughly 20 percent of youth that "age out" of foster care experience a significantly long spell of homelessness. Bob is convinced that this means that 20 percent of homeless people have aged out of foster care. I am not making this up; both myself and several other people have asked him to articulate his conviction as clearly as possible, and it is indeed how I've just described it. All attempts to show him the absurdity of said conviction have failed miserably.

Think about what this logic implies. Suppose 5 percent of smokers get lung cancer; this means you should believe that 5 percent of people with lung cancer are smokers. If 95 percent of men have a high school diploma, then 95 percent of people with a high school diploma are men. If half of the cats are black then half of all black things are black cats. Etc., etc. I think the sheer stupidity of this particular non sequitur is clear enough by now.

There are errors in reasoning that are subtle: it's easy to make them if one is not careful enough, and even if avoided, it's easy to understand where the mistake originates. None of the non sequiturs of the month fall into that category. Try as I might, I cannot come up with a plausible line of reasoning that could lead to them.

Non sequitur of the month: Very amusing, completely unimportant

This time it's a line from a TV commercial. It's so funny (in a non sequitur kind of way) that I just couldn't help but quote it. Some company is advertising its product by giving away free samples. One of the customers participating in the commercial says

They're giving it away for free? It must be good!

As Paul Krugman is fond of saying, not only is it not true, it's the exact opposite of truth.

Non sequitur of the month: "either-or" means "fifty-fifty"

Very frequently I experience an urge to write a post with no other purpose than to just mock some piece of extraordinary stupidity that I've encountered in the media. I try to fight that urge, as writing posts that do nothing but attack an easy target is kind of cheap. However, sometimes the temptation is just too strong. For one thing, shooting fish in the barrel can occasionally be fun. For another, psychologists tell me that it's unhealthy to suppress anger, and non sequiturs make me extremely angry. I take them very personally.

Therefore, in order to channel that anger, I decided to start a periodical feature called "Non Sequitur of the Month." In it, I'll try to present truly spectacular errors in reasoning. Fireworks of stupidity, if you will. And since pretty much any fallacy can in principle be stated as a non sequitur, I don't anticipate having trouble keeping things going.

First up is one Anna Cieślak, a journalist working for the second-largest Polish daily newspaper, Rzeczpospolita (title means "The Republic"). She has a blog, and on that blog, there's a post on unemployment. The very first two sentences of that post, in translation, read

From the micro point of view, i.e. from the point of view of an individual, it makes no practical difference whether the unemployment rate will climb to 12.5 or to 13.5 percent. You'll either get fired or you won't, and therefore your risk of becoming unemployed is 50 percent anyway.

Yes, you read that right. Apparently, the author believes that whenever the outcome variable is binary, it necessarily means that each alternative occurs with probability one-half. I should immediately start playing lottery; since I can either win the jackpot or not, it must mean that I have a fifty percent chance of winning.

This is truly a remarkable, monumental piece of stupidity. I believe the first entry in the Non Sequitur of the Month category may well turn into a non sequitur of the year.