Skip to content

Author: Briggs

October 3, 2008 | 38 Comments

Why probability isn’t relative frequency

(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.

October 1, 2008 | 12 Comments

Random global warming nuttiness

Many people have been sending me various tidbits about rampant global warming insanity, but it’s taking me a long time to get round to posting them. This darn book of mine….it never wants to be done!

So here are just some quick links for you to mull on.

The Who’s-Nuttier-Than-A-Psychologist? Department

Walking outside rather than inside—even for just 15 minutes—makes you feel happier, more energetic and more protective of the environment. So says psychologists at a recent APA meeting. How do you design a “study” to test environmental protectiveness intensity? With an instrument of course. “Instruments” are the learned name for “questionnaires.”

It’s not that I dispute the “finding” that walking outside is good for you. It’s just that it’s yet another in an endless line of unnecessary useless “studies” done either for the sole purpose of generating papers, or because the “researchers” just aren’t that bright. See the link for more studies of similar intensity and validity.

Hot Blooded

You just cannot stop “researchers.” Some of them are claiming that hot weather—that caused by evil global warming—will lead to more blood contamination. Their logic? Blood spoils in the heat (for various reasons), global warming will warm the globe, therefore more blood will spoil.

A Nobel is on the way to Prof Dunstan, a specialist in emerging infectious diseases at Curtin University in Perth, for thinking up this one. Google the New Zealand Journal of Public Health to read more of his exploits. I haven’t the heart.

No, It’s the Fit People!

In a study that flatly contradicts the findings of Dr Harrister, PhD, “researchers” at the London School of Hygiene & Tropical Medicine in a letter to the medical journal Lancet. Apparently, these prescient folks figured that fat people eat more food than skinny ones, and that food costs money to transport, and that transportation releases more green house gases, which…well, you know the rest.

But our scientific study clearly showed the opposite. Extremely fit people release more CO2 into the air than do lazy slobs.

Nobody ever asked us for an interview.

Please Release Me

“Researcher” James Walsh says Divorce — yes, the dissolution of marriage — is what causes global warming.

This one is so incredibly stupid that I cannot think of something silly to say about it.

Dr X Invokes The Lord

Dr X, whom we met before advocating lawlessness to combat global warming, said that if we do not “act” now we will “destroy the creation.” That’s awfully close to religious language—he was in Kansas when he uttered those words—and one thing lefties cannot abide is any hint of Christian religion. New Age yes, Muslim yes, vague yoga-nistics sure, Mother Earth and soaring hawks as spiritual messengers absolutely, but Christianity? Not a chance.

Thin ice here, Dr X. Best stick to anti-USA type rhetoric. This God stuff can bite you in the ass.

Flesh Eating Bipeds

“Researchers” at the Food Climate Research Network, based at the University of Surrey want to limit your meat intake to four portions a week. This, they say, will surely cool the planet.

Eating less meat will also make you cooler. Nothing hipper than a vegetarian.

Do not laugh, you absurd ill-informed fools! Did you not know that cows are “four-legged weapons of mass destruction”?

That you did not shows that you lack the kind of enlightened education provided to ivory-towered researchers the world over.

September 30, 2008 | 31 Comments

Let them fail

The same experts, in Congress and out, who did not foresee and who promulgated the current banking/credit crisis are the same ones assuring us their plan for salvation is just the thing.

Is it rational to believe that these creatures have finally figured out what is best for us? Or is better to say: Stop! Just let things fall out where they may. Let the people who caused this pay the price for their own mistakes.

Analysis so far suggests that the entire mess was brought on by Congressional prompting, in the form of laws which would penalize banks for not making risky loans, and by unscrupulous financiers who figured how to game the system. Also to blame are the people who bought absurdly constructed loans, the kind which they knew they would not be able to eventually afford. It is ridiculous to claim that these people were duped by forces more powerful than themselves. Nobody coerced anybody into buying a house.

And who twisted arms of bank executives to pay their failing CEOs millions? What rewards were given to financial engineers who packaged and sold the most creative sub-prime mortgage-backed securities?

There are many guilty parties here, not the least of which are our own elected representatives who reflexively believe that throwing money—as quickly as possible—at any problem is always the solution.

Thus, the belly-aching speech about partisanship by the appalling Nancy Pelosi after the failed vote was particularly galling. Here are two statistics of interest about yesterday’s “bailout” vote:

40% of House Democrats voted no.

33% of House Republicans voted yes

Which is to say, a fairly uniform rejection by both sides of the aisle.

Thank God for that. I in no way want to give any of my money to either over-confident corporate executives or to the people who will lose their homes. I am utterly unconvinced that I should feel any sense of responsibility for any of this mess.

In this current “bailout”, I feel the same way as when asked to contribute money to people who built their houses on a coastal area well known to be in a location of frequent hurricanes. How is their stupidity my problem?

Whatever the solution is, it is not more government.

Incidentally, it’s been little reported so far, but the Fed has already started printing more money to “infuse cash into the system.” One figure I read is $600 million. Next stop: inflation. Some bailout!

September 29, 2008 | 19 Comments

Next prohibition: salt

Here is a question I added to my chapter on logic today.

New York City “Health Czar” Thomas Frieden (D), who successfully banned smoking and trans fat in restaurants and who now wants to add salt to the list, said in an issue of Circulation: Cardiovascular Quality and Outcomes that “cardiovascular disease is the leading cause of death in the United States.” Describe why no government or no person, no matter the purity of their hearts, can ever eliminate the leading cause of death.

I’ll answer that in a moment. First, Frieden is engaged in yet another attempt by the government to increase control over your life. Their reasoning goes “You are not smart enough to avoid foods which we claim—without error—are bad for you. Therefore, we shall regulate or ban such foods and save you from making decisions for yourself. There are some choices you should not be allowed to make.”

The New York Sun reports on this in today’s paper (better click on that link fast, because today could be the last day of that paper).

“We’ve done some health education on salt, but the fact is that it’s in food and it’s almost impossible for someone to get it out,” Dr. Frieden said. “Really, this is something that requires an industry-wide response and preferably a national response.”…”Processed and restaurant foods account for 77% of salt consumption, so it is nearly impossible for consumers to greatly reduce their own salt intake,” they wrote. Similarly, regarding sugar, they wrote: “Reversing the increasing intake of sugar is central to limiting calories, but governments have not done enough to address this threat.”

Get that? It’s nearly impossible for “consumers” (they mean people) to regulate their own salt intake. “Consumers” are being duped and controlled by powers greater than themselves, they are being forced to eat more salt than they want. But, lo! There is salvation in building a larger government! If that isn’t a fair interpretation of the authors’ views, then I’ll (again) eat my hat.

The impetus for Frieden’s latest passion is noticing that salt (sodium) is correlated—but not perfectly predictive of, it should be emphasized—with cardiovascular disease, namely high blood pressure (HBP). This correlation makes physical sense, at least. However, because sodium is only correlated with HBP, it means that for some people average salt intake is harmless or even helpful (Samuel Mann, a physician at Cornell, even states this).

What is strange is that, even by Frieden’s own estimate (from the Circulation paper), the rate of hypertension in NYC is four percentage points lower than the rest of the nation! NYC is about 26%, the rest of you are at about 30% If these estimates are accurate, it means New York City residents are doing better than non residents. This would argue that we should mandate non-city companies should emulate the practices of restaurants and food processors that serve the city. It in no way follows that we should burden city businesses with more regulation.

Sanity check:

[E]xecutive vice president of the New York State Restaurant Association, Charles Hunt…said any efforts to limit salt consumption should take place at home, as only about 25% of meals are consumed outside the home.

“I’m concerned in that they have a tendency to try to blame all these health problems on restaurants…This nanny state that has been hinted about, or even partially created, where the government agencies start telling people what they should and shouldn’t eat, when they start telling restaurants they need to take on that role, we think its beyond the purview of government,” Mr. Hunt said.

Amen, Mr Hunt. It just goes to show you why creators and users of statistics have such a bad reputation. Even when the results are dead against you, it is still possible to claim what you want to claim. It’s even worse here, because it isn’t even clear what the results are. By that I mean, the statements made by Frieden and other physicians are much more certain than they should be given the results of his paper. Readers of this blog will not find that unusual.

What follows is a brief but technical description of the Circulation paper (and homework answer). Interested readers can click on. Continue reading “Next prohibition: salt”