William M. Briggs, Statistician

UPDATE: Reader Stephen Dawson has kindly shown where I made a very stupid error. This error caused me to label the y-axis incorrectly in the third picture below. It also causes the fifth picture to change dramatically. I will leave my original analysis untouched, except to indicate in bold where it is wrong. See the post from 23 November 2008 for an update on these two important figures, where I will give the proper interpretation. Thanks again to Dawson.

It’s obvious that, as time has gone on, the Federal Government has spent more and more and more money. Since a reasonable proxy of government control over the lives of its citizens is the outlay of funds from its treasury, a sane observer might wonder about this increasing trend.

A raw plot of the Federal outlay by year will not do as a measure, however. At least two adjustments have to be made.

A government ruling over 1000 people will obviously have to spend more than one ruling over 10 people, so we have to adjust by population size, which has also been increasing. We can be reasonably sure we are measuring population to, say, the nearest million, which is close enough. The budget is also reasonably well measured.

Then there is inflation, the phenomenon whereby a loaf of bread costs $1.00 ten years ago becomes $1.89 this year. But inflation is difficult to measure because of many reasons. For one, that loaf of bread probably isn’t the same as the loaf now: it has different ingredients, uses changed baking technology, improved packaging—who knows what has changed in that ten years. The population, too, which has increased over this period also tends to drive prices higher because it makes certain commodities scarcer. Plus, nobody knows which are the ideal items to track to measure cost increase: bread? cars? Eliot Spitzer’s hobbies? We’ll use inflation adjusted dollars in some of the plots, but we have to remember that these pictures are a lot more uncertain.

The first picture is the Outlay per Capita: that is, the dollars spent per citizen since 1901 (data from the US Budget Office and the US Census).

I have also colored the years red for Republican presidents, and blue for Democrat presidents. The years from 2009-2012 are obviously projections, so should not be taken too seriously. Not too much can be noticed, except for the obvious exponential increase in government control, plus the two blips for World Wars I and II.

Since the rate of increase is exponential, we can see things clearer by showing the picture on a logarithmic scale:

The two war-time era increases now pop out, with WW I showing the biggest increase. The after-war decreases are also more obvious. And we can see the small blip for the Korean war and a smaller build-up for Vietnam (all these increases are in the blue areas). The steady increase after Vietnam is also clear: where you can see a higher rate of increase in George W. Bush’s years because of the Iraq/Afghan wars, but certainly not a giant surge. Of course, I do not parse how much of any spending is due to military and civilian funding.

The big, but maybe not so obvious. point is that 2008 spending is about $10,000 per person. That means the government is spending $10-grand per head. That also means, in some loose sense, that if you pay more than this in taxes—if your personal bill is more than $10k—then you are paying more than your equal share. This implies, then, that if you are paying less than $10k you are not paying your equal share. You are requiring those that are better off to support the bulk of the government.

Now, if you are a Lefty, then you probably like this idea. “Let the rich pay their ‘fair’ share!” But to say this ignores Briggs’s Doctrine of Unintended Consequences. To see what I mean, let’s look at the same picture adjusted for inflation. The inflation adjustment index is from Oregon State University.

This is adjusted to 2008 dollars. Suppose I were to declare that every citizen had to pay $10 to the treasury. If you, for example, were Dad and the only worker in a family of four, your bill would be $40. The last time this happened was in the 1940s (remember: this is 2008 dollars, not 1940 dollars, so $40 was affordable).

This analysis is broadly correct, but the y-axis was off. You can see that the cost in 1940 was about $700 per head, or $2800 per family of 4 (all in estimated 2008 dollars). Still affordable, but to as many families as $40.

Everybody can afford this (with the trivial exception of a handful of people). Everybody would contribute an identical amount and would, morally at least, be entitled to an equal say in government. “But, wait! The rich will still have more money, and with money comes influence!” Yes, true. It is a tautology to say the rich will have more money, and it is obvious that with more money comes more influence. But this is not a good argument, my Lefty friend. Because look at 2008, where the bill is $10k per head. Only a small percentage of the population (about 5%) can afford this. Those 5% of course have more money. They further are aware of where that money is going. They will therefore have plenty of motivation to control the outflow, which means controlling the laws, rules, and regulations—controlling the government—which say where the money is to go. This small minority will use their money to align the government to their views.

Now, the rich certainly would have done this to some extent had everybody had to pay the same share, but they will have orders of magnitude more motivation to do it when they are paying nearly all the bill. And—here’s the kicker, so pay attention—they will still have plenty of money left over to have the same influence over other non-governmental matters, influence they already had before this tax structure started asking more of them.

About the only thing this confiscatory tax policy will do is to take enough money from the just-rich, to make them no longer rich. Thus, more control will flow into the hands of fewer and fewer people. This is inevitable. And it’s happening at an exponential pace. The noble idea of having those with more pay for those with less guarantees that those with more will have even more, and those with less will have even less, plus they will suffer a corresponding loss of influence and control over government.

Disproportionately taxing the rich to grow government, and doing so at an increasing, exponential pace, thus guarantees the creation of a oligarchic ruling class. Supporting these tax laws, then, will have the exact opposite effect of your intent.

I use the term “Lefty” not to indicate “Democrat”, as will be clear in the next two pictures:

These are the year-to-year change in outlay per captia. The first is unadjusted, the second is adjusted to inflation.

The unadjusted shows the blips due to the wars, plus the accompanying decreases in the budgets after the wars ended. Most of the wars, WW I, WW II, Korea, and Vietnam happened under Democrat administrations. But there was only moderate growth until Nixon was president in 1969, then the increases began with real vigor, and it has rarely abated since (only one year in Reagan’s presidency did the budget not increase significantly).

The scarier picture is this one, adjusted by inflation:

This shows the contest between R and D more clearly. Nixon (R) had a modest rate of increase, but Carter (D) really showed how it was done with a stellar increase. Reagan (R) did his best, but could never match Carter. Clinton (D) was also just an average player. Bush (R) beat them all. No taxpayer left behind. Again, Obama’s (D) tenure is just a wild guess by the budget office; however he has often boasted of increasing taxes on “the rich”, so we can guess that his rate will be Carter-like.

My comment below about my not being an economist is right on: I am not and made a fundamental error here. The new figure shows the changes more clearly—they bounce around 0 a lot more than I originally thought. Be sure to see the 23 November 2008 post for more on this Figure.

I am not historian or economist enough to say why the rapid increase in government control really got going with Nixon, but we have some hints in his social spending policies. The funny thing is the opposite of common wisdom appears to be true. Most, but not all, of the increases in spending for the military have come from Democrats (the wars just mentioned); and most, but not all, of the increase in spending on social causes have come from Republicans. Each side, as we all know, is continuously accusing the other of the opposite! It might be a case of projected guilt all politicians feel (at some level; I cannot really guess why this is so).

Even if you don’t agree with me on anything, it must be clear that this rate of increase cannot continue indefinitely. It cannot even continue for very much longer. Roughly, every 20 years brings an order of magnitude increase in government control. So in 40 years, in the trend continues, the bill will be about $1 million per head, an impossibly high number. Power would be coalesced into the hands of a very, very few.

I don’t know about you, but I plan a two-pronged strategy: (1) to never vote for anybody, D or R, who I think will raise taxes, and (2) to be one of those who can afford the tax, because I’d rather have the control than not.

Michael Crichton, as you will have heard by now, is dead. Unfortunately.

The Wall Street Journal today reprinted an excerpt of a speech Crichton gave called “Aliens Cause Global Warming.” Regular readers of this blog will know Crichton’s opinion on the certainty of man-made catastrophic climate change. Just a reminder (from his speech):

No longer are [climate] models judged by how well they reproduce data from the real world — increasingly, models provide the data. As if they were themselves a reality. And indeed they are, when we are projecting forward. There can be no observational data about the year 2100. There are only model runs.

This fascination with computer models is something I understand very well. Richard Feynman called it a disease. I fear he is right. Because only if you spend a lot of time looking at a computer screen can you arrive at the complex point where the global warming debate now stands.

Nobody believes a weather prediction twelve hours ahead. Now we’re asked to believe a prediction that goes out 100 years into the future? And make financial investments based on that prediction? Has everybody lost their minds?

To explain why he was flummoxed, Crichton first made a point about SETI, the Search for Extraterrestrial Intelligence. A lot of people in that field make reference to the Drake Equation, originated by SETI big cheese Frank Drake. That equation is

• N = R ^* x f_p x n_e x f_l x f_i x f X L

.

We want to solve for N, which is the number of civilizations in our galaxy with which intelligent communication is possible. N depends on the rate of star formation R ^*, the fraction f_p of those stars that have planets, and all those other things you can look up.

Crichton says:

This serious-looking equation gave SETI a serious footing as a legitimate intellectual inquiry. The problem, of course, is that none of the terms can be known, and most cannot even be estimated. The only way to work the equation is to fill in with guesses. And guesses — just so we’re clear — are merely expressions of prejudice. Nor can there be “informed guesses.” If you need to state how many planets with life choose to communicate, there is simply no way to make an informed guess. It’s simply prejudice.

The Drake equation can have any value from “billions and billions” to zero. An expression that can mean anything means nothing. Speaking precisely, the Drake equation is literally meaningless, and has nothing to do with science. I take the hard view that science involves the creation of testable hypotheses. The Drake equation cannot be tested and therefore SETI is not science. SETI is unquestionably a religion.

The fact that the Drake equation was not greeted with screams of outrage — similar to the screams of outrage that greet each Creationist new claim, for example — meant that now there was a crack in the door, a loosening of the definition of what constituted legitimate scientific procedure. And soon enough, pernicious garbage began to squeeze through the cracks.

I agree with him that none of these terms can be known exactly, or even sufficiently precisely to calculate a quantitative answer for N. I also agree that the pursuit of N can take on religious qualities.

But I can’t agree that SETI itself is worthless, nor can I agree that interest in it loosens the definition of “legitimate scientific procedure.” SETI is not just the Drake equation.

Now, I will not attempt to defend even one procedure that SETI workers use, nor will I comment on any statement made by any of its proponents. I cannot say, for example, that searching nearby stars for signals in the hydrogen line makes any sense. But I will say SETI is not the same as religion

I am interested in saying something about the probability of this proposition:

    S = “Intelligent/sentient life besides that on planet Earth exists”

Because we must calculate the probability of S is conditional on some evidence, I offer this blog. Yes, because this blog—because you and I—exist, it means that the universe is set up to allow at least one species of sentient life. Therefore, it is rational to believe that the probability of S given this evidence is greater than 0. I have no idea how much larger than 0 it is. If you are a fan of the reasoning behind the Fermi Paradox, you might say that the probability, while non-zero, is trivially small.

The Fermi Paradox basically says that, since the universe is about 10-13 billion years old, and the one sentient-life example we know of only took about 4-5 billion years to evolve, and since there are plenty of stars and galaxies, there should be sentient life all over the place. That is, SETI should be easy, and since it isn’t, since we haven’t made contact yet, this implies that we are the first or only sentient species. There are obvious subtleties to each stage of that argument that I glossed over, but that’s the gist.

The Fermi Paradox is also conditional on information not articulated. One obvious item is the proposition that all sufficiently advanced civilizations would want to make contact with us. Not just with other species, but with us. That’s a mighty big supposiion. Another hidden assumption is that we ourselves are sufficiently advanced enough to detect messages aimed at us, or have the ability to intercept messages meant for other beings. Pretty big guess, especially with the knowledge that the more efficient a message gets, the more it looks to an outside like noise (basic information theory; deep ties with probability and statistics there), and so civilizations more advanced than us might have communications which are impenetrable to us.

That argument cuts both ways, of course. If the messages are too complex, any search for them is fruitless. And, well, you get the idea. It’s complicated, so much so that it is not an open and shut judgment that SETI is valueless.

Though we have to be careful. Wishcasting is always a danger here, as everywhere. A lot of people—me included—want S to be true and this naturally clouds our judgment.

It is a cliche, but it is true, that academics must publish or perish. Papers, and more papers, and more papers still, are what makes for a professorial life.

It’s often—it’s very often—not the quality of the papers that counts, it’s the count of the papers that qualifies one for promotion, tenure, and other glories. In many, or even most, places an informal target number exists, saying have this many or it’s out the door you go.

So it should not be surprising that people eek out every ounce of information from a study and try to write it up in as many ways as possible. If you run, say, a clinical trial, and you can only get one paper out of it, then, if you’ll pardon me saying so, there is something wrong with you.

What usually happens, in for example clinical trials, is that a paper is written describing the trial methodology, even if the study design is no different than dozens of hundreds of other studies. Another paper is written with the main results. Then as many as can are done on subsets of the data, or on the data with various “scales” that are added on to pad out a trial. A “scale” is a questionnaire about a subject like quality of life. Any good clinical trial should be able to generate a minimum of eight papers, and a dozen or more is not unheard of.

The only problem with this, is that the civilians are starting to catch on, pace this article in The Guardian.

The author, Ben Goldacre, caught a drug company publishing a trial result twice. There was one main paper, and another tying the main results to a “depression scale”. Goldacre was aghast and said that some people will look to the journals and say, about this new drug, “There are two studies showing its efficacy.”

In other words, they will be more certain than is actually warranted. Goldacre also frets about meta-analysis, and how that strange technique will be fooled, too. Meta-analysis is a tool that gathers studies together to show an effect is real even though the effect was not found in most or any of the individual studies (we’ll talk about this subject another day).

But that drug company was doing nothing unusual. The people who work for it need papers too.

Paper churning—for that is the informal name of the phenomenon—is not limited to medicine. We all have stories of the professor down the hall who has been publishing the same paper for years, here and there adding a small twist to make it seem different. There are so many journals, and new ones appearing regularly, that there is always a market for his work.

This has the result that, in my own field of statistics, there are about 100—yes, 100—monthly or quarterly journals. Each has roughly 10-30 articles. In medicine, there are about 2000-3000 regular journals. There is no way to keep up. It is impossible to read more than a tiny fraction of papers. Most, and it pains me to say this, are not worth reading anyway.

Every now and then, academics will gather and beat their breasts and say “No more! From now on we shall also value teaching and service and not just quantities of papers published!” But their resolve lasts only until the end of the meeting. The next day, they go back to tallying.

There doesn’t appear to be a solution. You can’t limit numbers of papers that people are allowed to publish. You could, I guess, insist that all publications go to open source journals, where the authors are required to pay for “page charges”, that is, pay for publication. Professors aren’t rich and don’t have unlimited research funding, so this would slow the rate of papers. But what about graduate students or independents who have no funding? Forcing them to pay is silly.

The old fashioned filter, allowing the creme to rise to the top—in the form of books, usually—is probably all that will work. Unless we somehow can return to the roots of what a university is meant to be. We’ll leave that for another time.

I’ll be placing the order for the books tomorrow (Monday) morning. I’ll order a few more in case anybody else orders a signed copy. I imagine I’ll have the books by the end of the week, at which time I’ll mail them out. Look for updates here. Naturally, everybody who ordered will get an email.

Thanks for your support everybody!

It is finally done!

You may order directly from the publisher here¹. The book will also be available on Amazon, Barnes and Noble, etc. in about a month. I’ll update this post with the links when the book is in the distribution channels. Order as many copies as humanely possible.

Signed copies
I have had several requests for signed copies. I’ll be happy to do this for people. If you want a signed book, please email me at matt@wmbriggs.com. Please use “SIGNED COPY” as a subject line, and include your address in the body of the email. I’ll buy a few books from the publisher and then re-ship them out to people who want a copy. The charge will be the same as the publisher’s plus the same as they charge for Media Mail shipping and handling ($5.90), plus $1.15 cents (to cover tax). This makes the cost an even US$32.00. Payment will be arranged through PayPal (apparently, you don’t have to have an account to pay this way). I’ll send those who email me a PayPal “Request for Payment”; after that is received, I’ll ship the book (anywhere in the world).

Because I first have to order copies, sign them and then mail them out, it will of course take longer for you to get your book. I will wait a couple of days to see how many people email so I have a rough idea of how many books I should order.

I have two permanent places for news of the book:

1. My books tab (see upper right of screen): general news and information
2. Code page: free R code examples, erratum, links to papers, data, etc.

.

Why is this book different?

Statistics has traditionally been taught decade after decade in a fashion that is long outdated. This book presents a brand new way of understanding probability and statistics at the introductory level. The approach taken does not require mindless memorization. There is very little math, and what there is requires nothing beyond multiplication and division. This book takes busy work out of standard statistics and puts insight back in.

Preface excerpt:

The regular readers of my blog, where parts of this book previewed piece by piece, provided razor sharp editing and keen questioning and kept me from making major blunders. So thanks to (screen names) Mike D, JH, Harvey, Joy, Noahpoah, Harry G, Bernie, Lucia, Luis Dias, Noblesse Oblige, Charlie (Colorado), Dan Hughes, Mr C Physics, Jinnah Mohammed, Ari, Steve Hempell, Wade Michaels, Raphael, TCO, Sylvain, Schnoerkelmanon, and many others (sorry if I left you out!). Any mistakes left are obviously their fault.

What’s next?
I use the book in my own classes, of course, and a few other professors have been either using a draft or have expressed interest in the book for their classes. If, by some miracle, the book becomes popular, I’ll start working on a “Answers to Selected Exercises” or, given that I get substantial comments from actual class use, a Second Edition. But that is all in the far, far future.

If you are a professor of a statistics class and want to chat about the book, send me an email at matt@wmbriggs.com and we can set up a time to talk. I have had great success with this approach for beginning students and can let you know how I run the class. Some guidelines are also given in the Preface.

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¹The cover art looks terrible on the publisher’s page. They have scaled it down from an enormous PDF to a small JPEG and it is pixelated. It looks great when printed, however.

This is a bit of a preview of a paper my friend Russ Zaretzki are working on.

Take a gander at his pic:

This is the probability that John McCain wins the election given only the historical evidence of Republican/Democrat elections, and the fact that there will be just 1, 2, …, up to 38 more Republican/Democrat elections. Let me explain.

Since Democrat James Buchanan ran against Republican John C. Fremont in 1857, United States presidential elections have been dominated by these two parties. From that first contest, Democrats have won 16 elections and Republicans 22. This year we have another election in which the two parties are again featured. Now, this means that the number of elections of this type has so far been finite, and history strongly suggests that this series of elections itself will be finite; that is, some day it will not be Democrats versus Republicans, or even might even be that there will be no elections¹.

How many more elections there will be is, of course, an open question. But let us suppose that the one before us is the last election between the two parties. Then, conditional only on the past elections, the probability that the victor will be a Republican is 0.577. The standard Bayesian (continuous-value approximation) estimate gives 0.575. The classical guess is 0.579.

Our new method of guessing is based on knowing that the number of elections has been and will continue to be finite, that is, that it will not be without number, going on forever. It is important to recognize that traditional methods make this assumption. That is, that the number of “trials” (elections) will be infinite.

Ok, ok. These don’t seem like very big differences—and for this problem, they are not. But let’s suppose that instead of this being the final election, we’ll have two more. Then the probability McCain wins is just over 0.575. If we think there will be 9 more elections, then the probability McCain wins this one is only 0.570. Once the number of future elections becomes “large”, our guess matches the standard Bayesian one. That’s what the dashed, black horizontal line is. The red dot-dashed line is the classical estimate.

Eh, not a very big difference either, but it could be enough of one if you were, say, making a bet. And in some other problems, the differences are enormous; but this problem is a lot more fun.

The probability is over 50%. It obviously does not account for anything except previous elections. But it’s enough to raise a smile.

Incidentally, the math for all this is very heavily related to Laplace’s probability of succession. Google that. We introduce a twist that makes solving it sensible for certain problems. The surprise is that the probability depends on knowing the future number of trials (that’s the big difference).

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¹ Ever notice that at the Democrat rallies you hear “Obama! Obama! Obama!”, while at the Republican ones you hear “USA! USA! USA!”?

My book is coming!

It’s almost there, so let me tell you how modern math publishing works these days.

The author of course writes the work, and we all do it in a typesetting language called Latex (some just use Tex). Google it. It’s not different in spirit from web pages, which are content surrounded by “markup code” that tells the words where to go.

We can extend the analogy. Web pages are written in a markup code which is further subject to cascading style sheet rules. The style sheet rules say how big headlines are, what background images to use, and so on. In Latex, these are called class files (or “.cls” files).

Point of all this is that we write the words and math and the publisher provides us with a class file that does all the typesetting for them. Builds the Table of Contents, numbers all the pages and formulas, lays the footnotes properly, and so on, all automatically. Latex is sweet and orders of magnitude better than other word processing programs, such as MS Word.

But, unless you are a really famous author (not me), you are even given the privilege of writing your own Index! So, in math/physics/etc. books written with Latex, there is nothing for the publisher to do. They don’t even—again, unless you are famous—provide any direct copy editing. They let the authors do that, too.

Since I’m doing everything, I decided, a la Tufte¹, to bring out the book myself. Most of the copies I sell will be to the students who are forced—er, elect—to take my class. This way I can keep the price way down.

When I was a visiting professor at CMU, the textbook cost, if you bought the “Solutions Pack” and “Calculator Guide” (or whatever it was called), was well north of $100. 100 bucks! That’s nuts. Mine will be $24.95.

The rest is done automatically, including uploading the text and sending it to printers, everything is actually pretty quick. The real time is in getting the book out to the distribution channels. So while my book will be available first from the publisher’s site, it will take from 1 to 2 months to show up on Amazon.com etc.

What do you do if you can’t wait? You can check out this book. My attempt at inserting skepticism into a strange field.

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¹ Tufte does statistical graphics. If you haven’t seen his work, you should. His books, which are famous, are also non-traditional since there are, unfortunately, few statistical graphics courses at colleges. Still, he’s done OK with the books.

Here, by the guy who does Wayward Robot, is the front cover for the book (which should—with hope—be out of my hands by the end of this week).

(Pretend, if you have, that you haven’t read my first weak attempt. I’m still working on this, but this gives you the rough idea, and I didn’t want to leave a loose end. I’m hoping the damn book is done in a week. There might be some Latex markup I forgot to remove. I should note that I am more than half writing this for other (classical) professor types who will understand where to go and what some implied arguments mean. I never spend much time on this topic in class; students are ready to believe anything I tell them anyway. )

For frequentists, probability is defined to be the frequency with which an event happens in the limit of “experiments” where that event can happen; that is, given that you run a number of “experiments” that approach infinity, then the ratio of those experiments in which the event happens to the total number of experiments is defined to be the probability that the event will happen. This obviously cannot tell you what the probability is for your well-defined, possibly unique, event happening now, but can only give you probabilities in the limit, after an infinite amount of time has elapsed for all those experiments to take place. Frequentists obviously never speak about propositions of unique events, because in that theory there can be no unique events. Because of the reliance on limiting sequences, frequentists can never know, with certainty, the value of any probability.

There is a confusion here that can be readily fixed. Some very simple math shows that if the probability of A is some number p, and it’s physically possible to give A many chances to occur, the relative frequency with which A does occur will approach the number p as the number of chances grows to infinity. This fact—that the relative frequency sometimes approaches p—is what lead people to the backward conclusion that probability is relative frequency.

Logical probabilists say that sometimes we can deduce probability, and both logical probabilists and frequentists agree that we can use the relative frequency (of data) to help guess something about that probability if it cannot be deduced¹. We have already seen that in some problems we can deduce what the probability is (the dice throwing argument above is a good example). In cases like this, we do not need to use any data, so to speak, to help us learn what the probability is. Other times, of course, we cannot deduce the probability and so use data (and other evidence) to help us. But this does not make the (limiting sequence of that) data the probability.

To say that probability is relative frequency means something like this. We have, say, observed some number of die rolls which we will use to inform us about the probability of future rolls. According to the relative frequency philosophy, those die rolls we have seen are embedded in an infinite sequence of die rolls. Now, we have only seen a finite number of them so far, so this means that most of the rolls are set to occur in the future. When and under what conditions will they take place? How will those as-yet-to-happen rolls influence the actual probability? Remember: these events have not yet happened, but the totality of them defines the probability. This is a very odd belief to say the least.

If you still love relative frequency, it’s still worse than it seems, even for the seemingly simple example of the die toss. What exactly defines the toss, what explicit reference do we use so that, if we believe in relative frequency, we can define the limiting sequence?². Tossing just this die? Any die? And how shall it be tossed? What will be the temperature, dew point, wind speed, gravitational field, how much spin, how high, how far, for what surface hardness, what position of the sun and orientation of the Earth’s magnetic field, and on and on to an infinite list of exact circumstances, none of them having any particular claim to being the right reference set over any other.

You might be getting the idea that every event is unique, not just in die tossing, but for everything that happens— every physical thing that happens does so under very specific, unique circumstances. Thus, nothing can have a limiting relative frequency; there are no reference classes. Logical probability, on the other hand, is not a matter of physics but of information. We can make logical probability statements because we supply the exact conditioning evidence (the premises); once those are in place, the probability follows. We do not have to include every possible condition (though we can, of course, be as explicit as we wish). The goal of logical probability is to provide conditional information.

The confusion between probability and relative frequency was helped because people first got interested in frequentist probability by asking questions about gambling and biology. The man who initiated much of modern statistics, Ronald Aylmer Fisher³, was also a biologist who asked questions like “Which breed of peas produces larger crops?” Both gambling and biological trials are situations where the relative frequencies of the events, like dice rolls or ratios of crop yields, can very quickly approach the actual probabilities. For example, drawing a heart out of a standard poker deck has logical probability 1 in 4, and simple experiments show that the relative frequency of experiments quickly approaches this. Try it at home and see.

Since people were focused on gambling and biology, they did not realize that some arguments that have a logical probability do not equal their relative frequency (of being true). To see this, let’s examine one argument in closer detail. This one is from Sto1983, Sto1973 (we’ll explore this argument again in Chapter 15):

Bob is a winged horse
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Bob is a horse

The conclusion given the premise has logical probability 1, but has no relative frequency because there are no experiments in which we can collect winged horses named Bob (and then count how many are named Bob). This example, which might appear contrived, is anything but. There are many, many other arguments like this; they are called couterfactual arguments, meaning they start with a premise that we know to be false. Counterfactual arguments are everywhere. At the time I am writing, a current political example is “If Barack Obama did not get the Democrat nomination for president, then Hillary Clinton would have.” A sad one, “If the Detroit Lions would have made the playoffs last year, then they would have lost their first playoff game.” Many others start with “If only I had…” We often make decisions based on these arguments, and so we often have need of probability for them. This topic is discussed in more detail in Chapter 15.

There are also many arguments in which the premise is not false and there does or can not exist any relative frequency of its conclusion being true; however, a discussion of these brings us further than we want to go in this book.⁴

Haj1997 gives examples of fifteen—count `em—fifteen more reasons why frequentism fails and he references an article of fifteen more, most of which are beyond what we can look at in this book. As he says in that paper, “To philosophers or philosophically inclined scientists, the demise of frequentism is familiar”. But word of its demise has not yet spread to the statistical community, which tenaciously holds on to the old beliefs. Even statisticians who follow the modern way carry around frequentist baggage, simply because, to become a statistician you are required to first learn the relative frequency way before you can move on.

These detailed explanations of frequentist peculiarities are to prepare you for some of the odd methods and the even odder interpretations of these methods that have arisen out of frequentist probability theory over the past ~ 100 years. We will meet these methods later in this book, and you will certainly meet them when reading results produced by other people. You will be well equipped, once you finish reading this book, to understand common claims made with classical statistics, and you will be able to understand its limitations.

(One of the homework problems associated with this section)
{\sc extra} A current theme in statistics is that we should design our procedures in the modern way but such that they have good relative frequency properties. That is, we should pick a procedure for the problem in front of us that is not necessarily optimal for that problem, but that when this procedure is applied to similar problems the relative frequency of solutions across the problems will be optimal. Show why this argument is wrong.

———————————————————————
¹The guess is usually about a parameter and not the probability; we’ll learn more about this later.

²The book by \citet{Coo2002} examines this particular problem in detail.

³While an incredibly bright man, Fisher showed that all of us are imperfect when he repeatedly touted a ridiculously dull idea. Eugenics. He figured that you could breed the idiocy out of people by selectively culling the less desirable. Since Fisher also has strong claim on the title Father of Modern Genetics, many other intellectuals—all with advanced degrees and high education—at the time agreed with him about eugenics.

⁴For more information see Chapter 10 of \citet{Sto1983}.

(This is a modified excerpt from my forthcoming—he said hopefully—book, on the subject of why probability cannot be relative frequency. This is to be paired with the essay on why probability cannot be subjective. I particularly want to know if I have made this excruciatingly difficult subject understandable, and what parts don’t make sense to you.)

For frequentists, probability is defined to be the frequency with which an event happens in the limit of “experiments” where that event can happen; that is, given that you run a number of “experiments” that approach infinity, then the ratio of those experiments in which the event happens to the total number of experiments is defined to be the probability that the event will happen. This obviously cannot tell you what the probability is for your well-defined, possibly unique, event happening now, but can only give you probabilities in the limit, after an infinite amount of time has elapsed for all those experiments to take place. Frequentists obviously never speak about propositions of unique events, because in that theory there can be no unique events.

There is a confusion here that can be readily fixed. Some very simple math shows that if the probability of A is some number p, and you give A many chances to occur, the relative frequency with which A does occur will approach the number p as the number of chances grows to infinity. This fact, that the relative frequency approaches p, is what lead people to the backward conclusion that probability is relative frequency.

The confusion was helped because people first got interested in frequentist probability by asking questions about gambling and biology. The man who initiated much of modern statistics, Ronald Aylmer Fisher, was also a biologist who asked questions like “Which breed of peas produces larger crops?” Both gambling and biological trials are situations where the relative frequencies of the events, like dice rolls or ratios of crop yields, very quickly approach the actual probabilities. For example, drawing a heart out of a standard poker deck has logical probability 1 in 4, and simple experiments show that the relative frequency of experiments quickly approaches this. Try it at home and see.

Since people were focused on gambling and biology, they did not realize that all arguments that have a logical probability do not all match a relative frequency. To see this, let’s examine some arguments in closer detail. This one is from Stove (1983; we’ll explore this argument again in Chapter 16).

Bob is a winged horse
———————–
Bob is a horse

(Screen note: this is to be read “Bob is a winged horse, therefore Bob is a horse: stuff above the line is the evidence, stuff below is the conclusion.)

The conclusion given the premise has logical probability 1, but has no relative frequency because there are no experiments in which we can collect winged horses named Bob (and then count how many are named Bob). This example might appear contrived, but there are others in which the premise is not false and there does or can not exist any relative frequency of its conclusion being true; however, a discussion of these brings us further than we want to go in this book.

A prime difficulty of frequentism is that we have to imagine the experiments that pertain to an argument if we are to calculate its relative frequency. In any argument, there is a class of events that are to be called “successes” and a general class of events that are to be called “chances.” Think of the die roll: success are sixes and chances are the number of rolls. While this might make sense in gambling, it fails spectacularly for arguments in general. Here is another example, again adapted from Stove.

(A)
Miss Piggy loved Kermit
—————————–
Kermit loved Miss Piggy

What are the class of successes and chances? The success cannot be the unique event “Kermit loved Miss Piggy” because there can be no unique events in frequentism: all events must be part of a class. Likewise, the chances cannot be the unique evidence “Miss Piggy loved Kermit.” We must expand this argument to define just what the success and chances are so that we can calculate the relative frequencies. It turns out that this is not easy to do. This argument has three different choices! The first

(B)
Miss Piggy loved X
—————————–
X loved Miss Piggy

or,

(C)
Y loved Kermit
—————————–
Kermit loved Y

and finally,

(D)
Y loved X
—————————–
X loved Y

Evidence (from repeated viewings of The Muppet Show) suggests that the logically probability and frequency of (A) is 0. Any definition of successes and chances based on this argument (so that we can actually compute a relative frequency) should match the logical probability and relative frequency of (A). Now, because of Miss Piggy’s devotion, the relative frequency of (B) seems to match that of (A) where we have filled in the variable X for Kermit, a perfectly acceptable way to define the reference classes. But we are just as free to substitute Y for Miss Piggy. However, the relative frequency of (C) is about 0.5 and does not, obviously, match that of (A) or (B). Finally, under the rules of relative frequency, we can substitute variables for both our protagonists and see that the frequency of (D) is nothing like the frequency of any of the other arguments. Which is the correct substitution to define the reference class? There is no answer.

It’s worse than it seems, too, even for the seemingly simple example of the die toss. What exactly is the chance class? Tossing this die? Any die? And how shall it be tossed? What will be the temperature, dew point, wind speed, gravitational field, how much spin, how high, how far, for what surface hardness, and on and on to an infinite progression of possibilities, none of them having any particular claim to being the right class over any other. The book by Cook (2002) examines this particular problem in detail. And Hajek (1996) gives examples of fifteen—count `em—fifteen more reasons why frequentism fails, most of which are beyond what we can look at in this book.

These detailed explanations of frequentist peculiarities are to prepare you for some of the odd methods and the even odder interpretations of these methods that have arisen out of frequentist probability theory over the past ~100 years. We will meet these methods later in this book, and you will certainly meet them when reading results produced by other people. You will be well equipped, once you finish reading this book, to understand common claims made with classical statistics, and you will be able to understand its limitations.

——————————————-
¹While an incredibly bright man, Fisher showed that all of us are imperfect when he repeatedly touted a ridiculously dull idea. Eugenics. He figured that you could breed the idiocy out of people by selectively culling the less desirable. Since Fisher also has strong claim on the title Father of Modern Genetics, many other intellectuals—all with advanced degrees and high education—at the time agreed with him about eugenics.

²Stolen might be a more generous word, since I copy this example nearly word for word.