May 17th, 2013 (Permalink)
The Puzzle of the Library Books
Adam, Beth, and their daughter, Cathy, like to visit their local public library to check out books. They always check out either mysteries or puzzle books, but never more than one book apiece. Either Adam or Beth will select a mystery novel. If Adam picks a mystery then Cathy will choose a puzzle book. If Cathy selects a puzzle book then Beth will check out a mystery. Adam and Beth won't both select the same kind of book. Who checked out a mystery on one trip to the library but a puzzle book a different time?
May 15th, 2013 (Permalink)
These are two current headlines about two different studies that came out at about the same time. Have you ever wondered why you see so many conflicting news reports, such as these, about scientific studies on health and diet? Philosopher Gary Gutting has a worthwhile article explaining this puzzling phenomenon (see the Source, below). Hint: The problem isn't so much with the studies as it is with the reporting. Check it out.
Source: Gary Gutting, "What Do Scientific Studies Show?", The Stone, 4/25/2013
May 13th, 2013 (Permalink)
Massimo Pigliucci has an interesting article in the most recent issue of Skeptical Inquirer magazine on the notion of the burden of proof (BoP). The onus probandi is probably most familiar from the law where―at least in the U.S.―the onus in a criminal trial is on the prosecution. Pigliucci doesn't mention it, but there is a presumption corresponding to the onus; in criminal law, it is that the defendent is presumed to be innocent. Of course, such a presumption can be overcome, otherwise no one would ever be convicted, but to do so the prosecution must present sufficient evidence of guilt to shift the onus to the defense. Exactly how much evidence and of what kind is necessary to overcome the presumption of innocence is, of course, a legal question that I won't go into since I'm not a lawyer. But, at the very least, placing the onus and presumption where they are means that the prosecution must present some evidence, while the defense needn't present any if the prosecution fails to shoulder its burden.
Outside of the law, is there a presumption corresponding to the BoP? Yes, because this relation is a logical one: to say that there's a burden on those that argue P is to say that there's a presumption that P is false, and to say that there's a presumption that P is true is to say that there's a burden on those that argue not-P. Pigliucci is right that where the BoP lies is not a simple matter of who makes a positive claim; rather, it's determined by the degree of plausibility of the claim. Carl Sagan's famous slogan that "extraordinary claims require extraordinary evidence" is a specific application of the BoP: an "extraordinary claim" is one that is highly implausible because it goes against what we already know, so that the BoP is heavily weighted against it, that is, it requires "extraordinary evidence" to overcome.
So, I like the idea of applying Bayes' Theorem to determine where the load lies and how heavy it is, but I think that Pigliucci makes a mistake in the following sentence: "…[I]f we set our priors to 0 (total skepticism) or 1 (faith), then no amount of evidence will ever move us from our position…". He's certainly right about a prior probability of zero, since that would cause the posterior probability of the hypothesis on the evidence to also be zero. However, setting the prior probability of a hypothesis to one means that the prior drops out of the equation, so that―using Pigliucci's notation―P[T|O] = P[O|T]/P[O]. If P[O|T] = 0, then the posterior probability of the hypothesis would also equal zero. P[O|T] would be equal to zero in case the theory, T, implied that the observation, O, would be false. So, even if we started out with "faith" in the hypothesis, if the hypothesis implies something that turns out to be false then the probability of the hypothesis will drop to zero.
For this reason, I think that it's wrong to call setting the prior probability of a hypothesis to one "faith" in the hypothesis. Rather, faith in a hypothesis would probably manifest itself in refusing to update the prior probability in the face of counter-evidence. In other words, "faith" is a disposition to irrationally cling to a belief in the face of counter-evidence: a person who has "faith" in a particular theory will choose to reject the evidence against it rather than to reject the theory.
In contrast, setting a prior probability to zero does involve a kind of closed-mindedness that should be avoided, except in cases where the hypothesis is self-contradictory: in that case, the prior should indeed be set to zero, since there's no possibility that the hypothesis can be true. However, any consistent hypothesis has a non-zero, though perhaps very small, probability of being true. In general, outside of logical truths and falsehoods, no priors should be set to the extreme values of zero or one.
This means that the BoP and its corresponding presumption are not all-or-nothing, as they may seem to be, but matters of degree. For instance, if someone claimed that there was a deer in my backyard, I would take that person's word for it, as this is not an implausible claim. In contrast, if someone else claimed that an elephant was in the backyard, the BoP would be on that claim, as I presume that elephants don't frequent my backyard―and so far this presumption has proven true. However, if the claim was that there was a unicorn in my backyard, the burden would be heavier, that is, it would take weightier evidence to convince me that a unicorn was in my backyard than that an elephant was there, because elephants exist but there's no such thing as a unicorn. So, some BoPs are heavier than others.
Turning now to the relation between the BoP and logical fallacies, Pigliucci is right about the fallacy of ad ignorantiam, as well as the fact that it's not always a fallacy to appeal to ignorance―a point that I emphasize in the entry for that fallacy. For instance, if someone claimed that there was an elephant in my backyard but offered no evidence, I would not commit a fallacy of appeal to ignorance by rejecting the claim for lack of evidence.
I suppose that circularity may be thought to fail to shift the BoP because the premisses are as implausible, or even more implausible, than the conclusion. However, explaining what's wrong with begging the question in terms of the BoP seems to miss what's distinctive about circular reasoning, namely, its circularity.
I wish that Pigliucci had chosen to explain the relationship between the BoP and the ad hominem fallacy, rather than leaving it as an exercise for the reader. I suppose that an ad hominem fails to shift the BoP, but so would any fallacious argument. Other than that, I'm stumped.
Via: Mark David Barnhill, "New and Noteworthy: Continuity Problem Edition", To Believe or Do, 5/1/2013
May 11th, 2013 (Permalink)
New Book: Naked Statistics
I haven't read all of Charles Wheelan's new book Naked Statistics yet, so I can't give a full review of it or an unqualified recommendation. However, from what I have read, it's very clearly written and the explanations are easy to understand. Wheelan is also the author of a previous book called Naked Economics, which I also haven't read, but I suppose that explains the odd title.
In the "Acknowledgments" (p.xvii), Wheelan writes that he was inspired by How to Lie with Statistics, but Naked Statistics is a much longer and apparently much more thorough discussion of the basics of statistics than Huff's skinny book. Wheelan concentrates on explaining the "whys" of statistics, exiling the math to appendices at the ends of chapters. However, I would recommend not skipping the mathematical appendices, even if you think you're bad at math. The math isn't especially hard and, in the appendices that I've read, Wheelan does an excellent job of explaining it.
The book discusses a number of familiar topics from these pages: "Assuming events are independent when they are not" (pp. 100-101), the gambler's fallacy (pp. 102-103), the prosecutor's fallacy (pp. 104-105), regression to the mean (pp. 105-107); correlation ≠ causation (pp. 215-216); there are explanations of several important biases, including publication bias (pp. 120-122); and there's an entire chapter on polls (ch. 10).
Source: Charles Wheelan, Naked Statistics: Stripping the Dread from the Data (2013)
May 5th, 2013 (Permalink)
Charts & Graphs: Three-Dimensional Pie
Pie charts are one of the most common and useful types of statistical graph. Often, the "pie" is simply a circle that has been divided into slice-shaped wedges, but sometimes a third dimension is introduced by slanting the pie away from the viewer, so that its edge can be seen.
Why add a third dimension to usually two-dimensional graphs? Typically, the added dimension adds no additional information, rather it seems to be done simply to make a prettier picture. Of course, there's nothing wrong with making a chart look better, but such changes should not distort the information contained in it.
For an example, take a look at the pie chart above and to the right, which is angled as if it were sitting on a table in front of you. From the thickness of its edge, it's obviously a deep-dish pie. Its slices represent the percentages of total greenhouse gas emissions from different types of activity. Notice, however, that the "Food" and "Local Passenger Transport" slices of the pie are an identical 12% of the whole, but the "Food" slice looks bigger because perspective puts it closer to you and you can see its edge, which adds area. Similarly, the smallest slice, "Inter-city Passenger Transport", looks larger than it should for the same reasons as the "Food" slice next to it―just imagine how much less noticeable it would be to the rear of the pie where you couldn't see its edge. Finally, the "Building Energy Use" slice is almost a third of the pie, but the angle at which it is shown makes it look like no more than a quarter.
Now, you might defend this chart on the grounds that the percentages represented by each slice are printed right by them, so how misleading can it be? However, the whole point of such a chart is to make it possible for the viewer to visually compare the sizes of the different slices of the pie, so it shouldn't be necessary to read the percentages so as not to be fooled. If you have to provide the numbers to avoid misleading the viewer, then it would be better to just give the numbers and leave out the chart.
The example chart may not have been intended to be misleading, but may have resulted from a graphics program that makes it easy to construct a chart in three dimensions. So, if you're a pie maker, and can't resist adding a third dimension to it, try to keep the angle of the pie away from the viewer as shallow as possible. We should be looking almost directly down at the pie, which should appear to be close to a circle. Also, the edge of the pie should not be too thick, as the thicker the edge the larger the pieces at the front of the pie will appear to be.
If you're a consumer of pie, be on the lookout for the three-dimensional kind. Keep in mind the visual distortions that may result from looking at the pie from a shallow angle, and pay close attention to the actual percentages if they are provided.
Previous Entries in this Series:
April 26th, 2013 (Permalink)
There ain't no such thing as a free lunch in logic!
Ted Grant emails the following question:
As far as I can tell, logical arguments can be used to prove logical things exist but they can't be used to prove physical things exist. For example, it might be possible to prove that negative numbers exist or that a solution to an equation exists, but such arguments have no direct connection with the physical world. You cannot prove an apple exists using logic alone; there has to be some physical evidence that the apple in question exists. So it seems to me that some of the arguments for the existence of God must be fallacious because they are logical arguments and God is supposed to be a real physical being, not just some logical construct. So assuming you agree with this line of reasoning, what kind of fallacy am I discussing?
Let's get a couple of things straight before we try to answer your question:
How is it possible for the premisses of a deductive argument to necessitate its conclusion? This happens because the information contained in the conclusion is already semantically included in the premisses, so that if the information in the premisses is true then that in the conclusion must also be true. This point is easiest to see through examples of deductive reasoning, though it's by no means always obvious―if it were, then deductive reasoning would be unnecessary, since we would simply "see" all of the information contained in the premisses. A deductive argument is useful for making explicit in its conclusion information that is only implicit in its premisses.
So, in a deductive argument, you pay in the premisses for what you get in the conclusion; or, to put it another way, there's no free lunch in deduction. There's a trade-off between the strength of the logical connection between premisses and conclusion and the extent to which the conclusion goes beyond the premisses. In deduction, the conclusion is connected to the premisses in the strongest possible way―namely, with necessity―but it achieves this connection by not going beyond the premisses at all. In induction, the conclusion goes beyond what's contained in the premisses, but at the cost of a weaker connection with them: probability only.
At this point, we can see what's wrong with deductive arguments for the existence of God, even those that are valid: anyone who doubts the conclusion of such an argument should doubt the premisses as much or more. In other words, such arguments are circular―they beg the question. A valid deductive argument cannot force you to accept its conclusion; rather, it forces you to choose: either accept the conclusion or reject at least one of the premisses. If the conclusion is sufficiently implausible, that means that there's something equally implausible in the premisses. Find it!
This is why attempting to prove the existence of God in the same way that a theorem in logic or mathematics is proved is a fool's errand: even if such an argument were valid, it should fail to convince anyone skeptical about its conclusion. However, most of the traditional deductive arguments for the existence of God are not even valid, for various reasons that I won't go into here―if you're interested, check out John Allen Paulos' book Irreligion for a short, non-technical introduction.
Fallacy: Begging the Question
April 20th, 2013 (Permalink)
An Unprecedented Contextomy
An ad in yesterday's New York Times for the new documentary War on Whistleblowers includes the blurb: "UNPRECEDENTED!"-Tom Devine, THE GUARDIAN.
I've never seen this single word used in a movie blurb before, so I guess that makes this an unprecedented blurb. However, I wonder exactly what it's supposed to mean in this contextomized context; for instance, is it even necessarily a compliment to call a movie "unprecedented"? In what way is it unprecedented"? Could it be unprecedentedly bad?
I've cautioned before to be wary of the one-word blurb, since it's so easy for a single word to be taken misleadingly out of context, and that's certainly the case here. Here's the passage from the article in which the word has its sole occurrence:
In a film out this week, War on Whistleblowers, the New York Times' David Carr says: "The Obama administration came to power promising the most transparent administration in history…and began prosecuting [whistleblowers] every which way." … As this important documentary exposes, so far the tactic has been prosecuting or harassing national security whistleblowers. … The Whistleblower Protection Enhancement Act (WPEA) provides unprecedented employment protections. … [Highlighting added, of course.]
So, the word doesn't refer to the film at all, but to the WPEA. In fact, the article from which it comes is not a review of the movie, but a critique of the Obama administration's policies on whistleblowing, and only mentions the documentary at the beginning. However, it's puzzling that the ad didn't use the one word from the article that did characterize the movie: "IMPORTANT!"
Sometimes the ways of the ad writer are mysterious.
April 17th, 2013 (Permalink)
The $604 Question
Rob Rhinehart: "How I Stopped Eating Food"
The above headline is from a San Francisco Chronicle article about a man who supposedly "stopped eating food". Of course, it's possible to stop eating―for awhile. However, the headline is one of those misleading tabloid headlines aimed at tricking you into reading the article by suggesting something surprising but false. No, the man in question, Rob Rhinehart, is not a "breatharian". He did not stop consuming food, though he did stop "eating" it―he's drinking it, instead. Rhinehart created a beverage in his kitchen that supposedly supplies all of the protein, fat, carbohydrates, vitamins and minerals that the human body needs. Oddly enough, he calls this stuff "soylent", though this soylent isn't people.
Assuming it doesn't kill you, one of the supposed advantages of this diet, according to its creator, is economic:
The average American spends $604 per month on food. But given that Soylent only costs Rhinehart about $50 per month, he's very excited at the prospect of feeding people in developing countries.
Before we nominate Rhinehart for the Nobel prize, let's examine these claims a little closer. These two sentences should raise a lot of questions in your mind. For instance, do you spend $604 a month on food? If you're like me, you probably don't know exactly how much you do spend, but $20 per day seems rather high. I suppose that you might spend that much if you eat out every single day, or cook gourmet meals at home, but how many Americans do that and how "average" would they be?
Also, who's an "average American"? The answer is that no one is an "average" American (see the Resource, below): what the author of the article is trying to say is that Americans on average spend $604 for food in a month. That number is very precise, though, which may impress people into thinking it must be very scientific. But if it's an average, then it must be based on a sample survey of Americans―after all, who could afford to find out from every American how much they spend on food? So, what's the margin of error, and isn't it likely to be greater than four dollars? If so, why wouldn't the author write that Americans spend approximately $600 a month for food? That wouldn't sound so impressive, but it would actually be more scientific.
Moreover, in what sense is our "average American" average? Is it the mean, median, or the mode? As I've explained in the Resource, each of these statistical measures can be called an "average", and when used to measure matters involving money, such as income or spending, these three "averages" may be quite different.
So, let's turn to the source given by the article to see if we can answer some of these questions. At the point where the article makes the claim in question, it links to a report from the Gallup polling company (see Source 2, below). The most remarkable thing about this survey is that the question upon which the $604 average is based is the following: "On the average, about how much does your family spend on food each week? [Emphasis added.]" The mean is $151, and four weeks per month explains where $604 came from. However, that's $604 per family, not per "average American". Given that there must be at least two people per family, the mean for the individual American will be no more than $302 per month, and since the "average family" probably has more than two members, that mean will be lower still.
The Gallup report also answers our other questions: The margin of sampling error for the spending average is ±$7, so that $604 definitely gives an illusion of accuracy. $604 is the mean, whereas the median is only $500 per month. The median is often a more accurate measure of typical spending since the mean is pulled higher by big spenders, such as those who eat out a lot or choose expensive foods. Also, given that the question asked about families, large families are likely to spend more on food than small ones. In fact, we see this pattern in Gallup's results: the distribution has two peaks, one at $100-$124 and the other at $200-$299. So, given that the median is a better average than the mean, the "average American" that we're concerned about would actually spend no more than $250 a month on food, and probably less. That's a lot less than $604, but it's still five times as much as what Rhinehart claims to spend on his soylent.
This raises another question about the article: given that even a quick glance at the source of the $604 claim would reveal that it refers to an "average family", and not an "average American", how did this get published in The San Francisco Chronicle? Even if the newspaper can't afford fact-checkers, aren't journalists themselves expected to do at least minimal fact-checking of their stories? And if something like this managed to slip by the reporter, aren't editors supposed to check the basic facts before printing the story?
One explanation, though not a satisfying one, is that the article was taken from a "blog" post written by Rhinehart himself (see Source 3, below), apparently without even minimal checking. Here's what the post currently says about the expense of soylent compared to food:
Monthly I was spending about $220 on groceries, and another $250 eating out for lunch and the occasional dinner. The average american spends $604/month on food, about half of which is groceries. … Consuming only Soylent costs me about $50/month…. At scale the cost would be even lower.
Somehow this edit didn't make it into the Chronicle story, even as a correction. At least they're consistent! Interestingly, Rhinehart himself claims to spend less on food per month than that incorrect average. However, the corrected cost of soylent is actually only $100 less per month than the corrected average, rather than the original $550 difference. How much do you want to bet that if this stuff is ever manufactured and sold it will cost more to live on than to eat regular food? It will probably be marketed as healthier or more convenient than solid food, rather than as cheaper.
So, this example is a warning not to believe everything you read in the media, even respectable newspapers. Nowadays you have to be your own fact-checker.
Resource: "Average" Ambiguity, 11/4/2002
April 4th, 2013 (Permalink)
Charts & Graphs: The Gee-Whiz Bar Graph
Bar graphs are useful for visually comparing quantities when the heights of the bars are proportional to the quantities they represent. However, if a bar chart is truncated―that is, the bottom part of the chart, including the zero baseline, is removed―the heights of the bars will no longer be in proportion to the quantities they represent, with the effect of exaggerating the differences between those quantities. We've seen examples of this type of chart before (see the Resources, below) and it seems to be one of the most common types of misleading graph, if not the most. As an example, see the graph to the right (from Source 1, below).
At a glance, it appears that gas prices in both Florida cities, the state as a whole, as well the country, have "soared" about 50%. This impression results from comparing the heights of the bars with the naked eye, and the impression is reinforced by the chart's caption. However, a closer look shows that the baseline of the graph is not zero: in fact, it's hard to tell exactly where the baseline is because the chart fades away at the bottom, but the lowest value that can be read on the scale is $3. The actual growth of gas prices in all four areas is around 10% so that the chart visually exaggerates the rise by about fivefold. This is a graphical manifestation of the news media's desire to make a big deal out of every rise in gas prices, which we've seen previously in their failure to adjust prices for the effects of inflation.
So, the moral of this installment is similar to that for the first: when looking at a bar graph, be sure to check that the baseline starts at zero. If not, then you can't compare the sizes of the data by the sizes of the bars, which is the whole point of such a chart. A truncated bar chart is almost bound to be a misleading one.
Previous Entry in this Series: Charts and Graphs: The Gee-Whiz Line Graph, 3/21/2013
April 3rd, 2013 (Permalink)
New Book: Math on Trial
The new book Math on Trial: How Numbers Get Used and Abused in the Courtroom, by mathematician Leila Schneps and her daughter, looks very interesting. I haven't read it yet, but judging from the chapter titles and the Amazon "Look Inside!" feature, it deals primarily with probability theory and statistics, which makes sense given its subject matter. The first chapter concerns the tragic case of Sally Clark, and a later one the Lucia de Berk case, both of which I've alluded to previously (see the Resources, below). There's also a chapter on the Amanda Knox case that's been back in the news recently (see Source 1, below). Statistical fallacies that play a role in these cases include the so-called "prosecutor's" fallacy―which needs a more accurate name, since not only prosecutors commit it―and the mistake of multiplying non-independent probabilities―which is more a definition than a name. This book may also answer the question "where's the harm?", since it shows that fallacious reasoning can put innocent people in prison or leave the guilty free to commit more crimes.
April 1st, 2013 (Permalink)
An April Fool's Puzzle
Will the puzzle you solve before you solve the puzzle you solve after you solve this one be harder than the puzzle you solve after you solved the puzzle you solved before you solved this one?