William M. Briggs

Statistician to the Stars!

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Jumping The Infinity Shark: An Answer To Senn; Part Last

Lazy eights everywhere.

Read Part V

From his page 55 (as before slightly edited for HTML/LaTex):

Consider the case of a binary event where the two outcomes are success, S, or failure F and we suppose that we have an unknown probability of success \Pr(S) = \theta. Suppose that we believe every possible value of \theta is equally likely, so that in that case, in advance of seeing the data, we have a probability density function for \theta of the form f(\theta) = 1.

And \theta lives on 0 to 1. “Suppose we consider now the probability that two independent trials will produce two successes. Given the value of \theta this probability is \theta^2. Averaged over all possible values of \theta” this is 1/3 (the integral of \theta^2d\theta).

A simple argument of symmetry shows that the probability of two failures must likewise be 1/3 from which it follows that the probability of one success and one failure in any order must be 1/3 also and so that the probability of success followed by failure is 1/6 and of failure followed by success is also 1/6.

This is a contradiction or paradox and a glaring one which causes subjective Bayesians to cower (rightly). (I skip over the difficulties covered before with the idea of “independent trials”.) Where does the fault lie? Here:

Suppose that we believe every possible value of \theta is equally likely…

What could that possibly mean? Nothing. Sure, it’s easy to write down a mathematical answer, but this does not make it a true or useful answer. First: how many numbers are there between 0 and 1? Uncountably many. It is impossible for any being short of God to assign a probability to each of these. Second: even if somebody could, because there are uncountably many answers, it is impossible that any should be the right one. Recall the probability of seeing any actual observation with any continuous (i.e. infinity-beholden) distribution is always 0, a daily absurdity to which we always shut our eyes.

We have jumped the infinity shark. Jaynes warned us about this (in his Chapter 15; though he didn’t always obey his own injunction). I think his caution goes unheeded because the calculus is so easy to demonstrate and to work with. What’s easier than integrating a constant?

As shown in the original series, we must begin with a real-world finite conception of each problem and only after we’ve sorted out what is what can we take a limit, and only then for the sake of ease and approximation. We must not fall prey to the temptation of reifing infinity.

(If there is sufficient interest, I’ll show the solution for Senn’s example another day: it’s a simple extension of the problem in the original series.)

Jaynes himself should have followed his own advice in the derivation of a (two-dimensional) normal distribution. He began with a premise (something like this; I don’t have the book to hand) when measuring a star’s position errors are possible in any direction. But he took “any direction” to mean a continuum of directions. This isn’t possible.

Suppose all we have to measure a star’s position (on a plane) is a compass which points only in the cardinal directions. Then our measured error can only be a finite number of possibilities. There would be nothing Gaussian about the probability distribution we use to quantify our uncertainty in this error. Right?

Next suppose we double the precision of our compass, so that it points eight directions. Still nothing Gaussian. Finally suppose we set the precision to whatever is the precision of today’s finest instrument. This would still be finite and non-Gaussian. We have nothing, and will never have anything, which can measure to infinite precision in finite time. This goes for star’s positions, salaries, ages, weight, and anything else you can think of. We’re always limited in our ability to see.

Acknowledging this “solves”—actually does away with—the long-standing problem of putting “flat priors” on (unobservable) parameters of distributions like the normal. These are called “improper” priors because they aren’t real probabilities, they’re only mathematical objects to which we assign an improper meaning. Since they aren’t real probabilities you’d guess people would abandon them. You’d guess wrong.

The other problem with infinite probabilities is measurement units: probabilities can change just by a change in unit, say from feet to centimeters, an absurdity if probability has a constant meaning. This problem also disappears when we remain this side of infinity.

Anyway, time to stop. Logical probability Bayes always lands on its feet. Plenty of mistakes enter with subjective Bayes, it’s true, or even in LPB when people (wrongly) insist on quantifying the unquantifiable. There are many misunderstandings when toying with infinity.



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Bayes Always Works: An Answer To Senn; Part V

Wikipedia chart trying to say something about probability. Nice colors, but it’s screwy.

Read Part IV.

We’re almost done. Only one more after this.

There are examples without number of the proper use of Bayes’s Theorem: the probability you have cancer given a positive test and prior information is a perennial favorite. You can look these up yourself.

But be cautious about the bizarre lingo, like “random variable”, “sample space”, “partition”, and other unnecessarily complicated words. “Random variable” means “proposition whose truth we haven’t ascertained”, “sample space” means “that which can happen” and so on. But too often these technical meanings are accompanied by mysticism. It is here the deadly sin of reification finds its purchase. Step lightly in your travels.

Let’s stick with the dice example, sick of it as we are (Heaven will be the place where I never hear about the probability of another dice throw). If we throw a die—assuming we do not know the full physics of the situation etc.—the probability a ‘6’ shows is 1/6 given the evidence about six-sided objects, etc. If we throw the same die again, what is the probability a second ‘6’ shows? We usually say it’s the same.

But why? Short answer is that we do so when we cannot (or will not) imagine any causal path between the first and second throws.

Let’s use Bayes’s theorem. Write E_D for the standard premises about dice (“Six-sided objects, etc.), T_1 means “A ‘6’ on the first throw”, and T_6 means “A ‘6’ on the second throw”. Thus we might be tempted to write:

\Pr(T_2 | T_1 E_D) = \frac{\Pr(T_2 | E_D) \Pr(T_1 | T_2 E_D )}{\Pr(T_1 | E_D)}.

In this formula (which is written correctly), we know \Pr(T_1 | E_D) = 1/6 and say \Pr(T_2 | E_D) = 1/6. Thus is must be (if this formula holds) \Pr(T_2 | T_1 E_D) = \Pr(T_1 | T_2 E_D ). This says given what we know about six-sided objects and assuming we saw a ‘6’ on the first throw, the probability of a ‘6’ on the second is the same as the probability of a ‘6’ on the first toss assuming there was a ‘6’ on the second toss. Can these be anything but 1/6, given \Pr(T_1 | E_D) = \Pr(T_1 | E_D) = 1/6? Well, no, they cannot.

But there’s something bold in the way we wrote this formula. It assumes what we wanted to predict, and as such it’s circular. It’s strident to say \Pr(T_2 | E_D) = 1/6. This assumes, without proof, that knowledge of the first toss does not change our knowledge of the second. Is that wrong? Could the first toss change our knowledge of the second. Yes, of course.

There is some wear and stress on the die from first to second throw. This is indisputable. Casinos routinely replace “old” dice to forestall or prevent any kind of deviation (of the observed relative frequencies from the probabilities deduced with respect to E_D). Now if we suspect wear, we are in the kind of situation where we suspect a die may be “loaded.” We solved that earlier. Bayes’s Theorem is still invoked in these cases, but with additional premises.

Bayes as we just wrote it exposes the gambler’s fallacy: that because we saw many or few ‘6’s does not imply the chance of the next toss being a ‘6’ is different than the first. This is deduced because we left out, or ignored, how previous tosses could have influenced the current one. Again: we leave this information out of our premises. That is, we have (as we just wrote) the result of the previous toss in the list of premises, but E_D does not provide any information on how old tosses affect new ones.

This is crucial to understand. It is we who change E_D to the evidence of E_D plus that which indicates a die may be loaded or worn. It is always us who decides which premises to keep and which to reject.

Think: in every problem, there are always an infinite number or premises we reject.

If it’s difficult to think of what premises to use in a dice example, how perplexing is it in “real” problems, i.e. experiments on the human body or squirrel mating habits? It is unrealistic to ask somebody to quantify their uncertainty in matters which they barely understand. Yet it’s done and people rightly suspect the results (this is what makes Senn suspicious of Bayes). The solution would be to eschew quantification and rely more on description until such time we have sufficient understanding of the proper premises that quantification is possible. Yet science papers without numbers aren’t thought of as proper science papers.

Conclusion: bad uses of probability do not invalid the true meaning of probability.

Next—and last—time: the Trials of Infinity.



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Fingers Lakes Wine Tasting Notes

Medals for everybody!

Medals for everybody!

A few years ago the boss of Guinness toured American micro-breweries and congratulated them for their enterprise, but he also gave them a spot of advice, which, paraphrased, was that they should concentrate on making just one great beer rather than on many indifferent ones.

Hosmer Winery, the first of our three stops along Cayuga Lake, had available for tasting at least two dozen varieties, and they produce a couple more. Knapp Winery & Vineyard Restaurant listed 38 wines and spirits. And Lucas Vineyards listed 24.

These statistics are worth mentioning because none of these wineries are major concerns: all exist on minimal acreage, so it’s a wonder how this many wines can be produced. After the tastings, somebody explained this by starting the undoubtedly scurrilous rumor that the native-grown grapes are supplemented with water and high fructose corn syrup.

And then none of the wines they sell are cheap. The least expensive bottles were $8.99 at Lucas, but the average is around fifteen bucks, topping out around thirty. Obviously, these places subsist on the tourist trade. Tastings are three to five bucks, so they’re breaking even there. But each shop sells tchotchkes or they have small restaurants. And everybody buys a bottle or two, just for the fun of it.

In short, and with exceptions, you’re not going to these wineries for the wine. Instead, the trip is ventured for the sake of the trip, for the beautiful vistas on a gorgeous day. And to see the shining gold and silver glint in the sunlight. These reflections are provided by the multitudinous medals lining the walls. Since these wineries are by Ithaca—which Utne Magazine once called the “Most Enlightened City in America”—every wine is a winner. Each goes home with a prize and a hug.

Following is a selection of my tasting notes.

Hosmer

A small barn with a vineyard not too much bigger. Specializes in the sweet stuff, especially Raspberry Bounce, a Faygo Redpop simulacrum.

2012 Dry Riesling. Smells like cheddar cheese from the supermarket. Sour. Except for the smell, indistinguishable from the 2011 Riesling, the “Double Gold winner.” $15.

2009 Lemberger. Dusty, sweet scent. Drank, but taste disappeared instantly. Immediately forgetful. $18.

2010 Cabernet Franc. Cheap barbershop cologne. Awful. Oh My God. Awful. $18.

Estate Red. Thin. Not sweet. Reasonable plonk. $10.50. (I bought two bottles; shared them out on bus.)

Knapp

The only place we visited with a distillery. Tasting room nicely decorated with barrels. We had their barbecue of overcooked chicken. They like it sweet too, advertising Jammin’ Strawberry which will “flood your palate and bombard your senses.” I believed them and didn’t try it.

Cabernet Sauvignon ’11. Almost no smell? Sour, thin; dries the mouth. “King of reds.” $18.95.

Sangiovese ’10. What is this? Aha! Nail polish remover. Tastes of day old apple cider made with peels. $16.95.

Meritage ’11. Like flat, not-too-sweet root beer. $22.95.

Pasta Red Reserve. Smells like road construction. Too sweet. $10.95.

Brandy. Fumes good replacement for nose hair trimmer. Stings the tongue. Couldn’t swallow. Aged what? Two days? $24.95.

Serenity. Passable. Tasted like a bin red wine. $12.95. (Bought bottle, shared out over lunch.)

Lucas

For no apparent reason, a nautical-themed winery (it’s nowhere near any water). Sorority hangout? The picture of medals is from here. The wines were pre-selected for us.

Miss Chevious. My Grandma Briggs would have liked this: but she never paid more than two dollars a bottle. Sour as vinegar. $8.99. (Apparently if you buy some, you won a sticker “I got Naughtie at Lucas”. Several bridesmaids parties had these. “Gold Medal Winner!”)

Blues 2010. Cheap. God. Muck. Undrinkable. $8.99. (Nobody in our party could finish.)

Semi-Dry Riesling 2010. Compared with neighbors and, yes, Off! Smells just like the bug spray. Didn’t dare taste. $13.99. (“Gold Medal Winner!”)

Butterfly. Smells like one of those junior artists paint set; kind which have ten paints in little joined plastic pots. Tastes exactly like Play Doh. $8.99. (“Gold Medal Winner!”)

Tug Boat Red. Smells and tastes like a red bank sucker, the kind tellers used to hand out to children. $8.99. (“Gold Medal Winner!”)

Cabernet Franc Limited Reserve 2009. Puts me in mind what a diet alcohol would taste like. (This wasn’t on the scheduled flight. I asked pourer if we could try something that wasn’t sweet. I asked for boldest, best red. He suggested this. “Gold Medal Winner!”)

I didn’t buy anything from Lucas, but took a nap in their grass out front while waiting for our bus.


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Highs And Lows Of Summer Skirts: Guest Post by the Blonde Bombshell

Somebody please feed this poor woman

Somebody please feed this poor woman

One thing that is admirable about the well-attired man is that if he were to time travel, he would be at home in nearly any era. He may have to make some adjustments to better fit in (perhaps cast off the jacket and leave only the vest; or manipulate his tie into more of a cravat), but he will have the tools at his disposal that he needs to make a good impression on short notice. Sure, he might draw the odd glance from Henry VIII, but, on the whole, many of his items of clothing would be recognized as functional.

The same cannot be said of a woman, because “well-attired” has many nuances, especially in the summer months. Imagine a woman wearing a “hi-lo” skirt (which is the sartorial equivalent of the mullet, and perhaps can be best described as “party in the front and all business in the back”) and flip-flops hitching a ride in a time machine to 1860. Our poor time traveler will stick out like the proverbial sore thumb, and her mission does not portend to end well.

One of the misfortunes of the hi-lo skirt, or any garment that reveals a woman’s knees, is that a woman’s knees are revealed. Women’s knees, as a whole, are not beautiful. I am sorry to be a bearer of bad news, especially as the temperatures are soaring and as the winter’s tights and woolens are being cast off, but this is an incontrovertible fact.

Part of the problem with the knee itself is the anatomy of a woman’s leg. The basic shape is an inverted triangle, with the point of the triangle buried somewhere in the region of the mid-calf. The knee, in real life, does not at all resemble that of a knee of a mannequin propped up in a store window. I have never been in a designer’s atelier, but I have seen dressmaker’s dummies, which are torso shapes affixed to some sort of pole—with no legs. Any fashion that can be dreamed up is going to look much better on a leg-less dummy than on a flesh-and-blood human being that has to make her way in the world hobbling around on a pair of inverted triangles.

The poets support this view. They are silent on the shining beauty of Cleopatra’s knees and they entirely mute when it comes to the lower limb joints of the fair Helen. Fortunately for them both, they had the sense to cover them up in mixed company, or at least in the presence of poets.

It took millennia for hemlines to rise to the level they are now. My mother, a daughter of the 1950s, was a firm believer that a proper hemline for the female of the species was “two inches below the knee”. And my mother was literal in her interpretation of “knee” in that it was the horizontal center of that particular joint. “Two inches below” fell, in my view, right below the knee, and to be “two inches below” would require an additional two inches.

In the space between my mother’s and mind accounting of the “two inches” there was still enough of the knee’s characteristics on display to make the legs look their unflattering worst.

As a result, I was perhaps the only teenager who cried to my mother to lower my hems. I felt, and still feel, that the most attractive hem for a female is mid-calf, just where the flesh swells.

Many women declare that they admire the style of Grace Kelly, Audrey Hepburn, and Lauren Bacall, but then they go to their closets and come out looking like a second-string actress heading for rehab.

The problem isn’t necessarily with the women themselves, but more with what’s on offer. The shop windows are full of skirts that can be characterized as “eight inches above the knee.” By not offering a variety of hemlines, manufacturers and retailers are doing a grave disservice to women who want to save an unsuspecting public from having to absorb the shock of having to look at their puckered, wrinkled, discolored, but otherwise very useful knees.*

*Please don’t mention that they should wear trousers. Trousers could prove problematic for time travel.



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