William M. Briggs

Statistician to the Stars!

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Subjective Versus Objective Bayes (Versus Frequentism): Part I

Pierre-Simon, marquis de Laplace, Chance Master


We first have to define what subjectivity and objectivity are and from these see what happens. For those unused to reading long stretches of prose, here is the conclusion, which will be proved in due time:

Like in The Highlander, “There can be only one”…correct interpretation of probability (uncertainty), but many approximations. Objective Bayes is the correct interpretation. Subjective Bayes fails in theory but often works in practice because subjective Bayesians act like objectivists. Frequentism fails except when it overlaps objective Bayesian methods (and when it does it works “in the long run” too). All others fail or are approximations too. Despite the labels, all three groups operate at least partly subjectively, partly objectively.

To assert without argument is to be subjective. There are two divisions to subjectivity, the internal and external. Internal subjective propositions are those which you believe (either not at all, somewhere in-between, or fully), whether via argument or observation or both. External subjective propositions are those which you seek to convince another to believe or disbelieve or to give weight to. It is subjective to say “I think I might be catching a cold1,” “I am hot,” or “I dislike rice pudding”. If I am feeling or thinking these things, and I believe them and am not trying to fool my listener, then these propositions are true, given my internal state.

Externally, however, these kinds of propositions may be doubted. If you hear your neighbor say “My favorite color is red” and if you add to that the premises “My neighbor is sane and honest and sane and honest people faithfully report their beliefs” then it follows your neighbor’s favorite color is red. But other premises are possible which cause you to doubt your neighbor or to disbelieve him. Thus internal subjective statements of one person can be argued against externally by another. Think of a cop grilling a suspect. For example if I gave prior evidence of relishing rice pudding, you can put this to me. Or it may be freezing out which makes you question why I report being hot. Thus depending, or conditional, on the premises which are or aren’t accepted by the parties, the proposition in question can be true for one and false for another, or somewhere in between.

“Garfield was a better president than Tyler” is an internally subjective statement which has as much evidentiary support as the statement “Tyler was a better president than Garfield”, which is to say none except the implicit force of the speakers. That force is a premise, but difficult to clarify. To hear one and believe it and not the other is therefore to be subjective.

This next argument is also subjective, but only in part, and illustrates the sort of internal argument which happens when one hears a raw assertion like “Garfield’s better.” “Garfield was a better president than Tyler because Garfield’s vice president went on to be president himself and better presidents see their vice presidents go on to be presidents.” This is an (inverted) valid syllogism, and therefore the conclusion (“Garfield’s better”) is true given the premises. Not everybody will agree with the conclusion because not everybody will agree with the premises. The premises are merely asserted, and are therefore subjective. The argument as a whole is then also subjective, but interior to the argument there is no subjectivism; interior is it objective.

In order for an argument to avoid being subjective, and therefore be completely objective, it has to begin with premises which are true (and which has unambiguous terms) and ends with a conclusion which validly follows from the premises. Mathematical theorems are like this, though they are not usually presented in that fashion. Instead a theorem will begin with premises which are accepted but not proven in place as true. The premises are accepted as true because they have earlier been derived via other arguments, themselves with premises which are accepted but not proven (in place) as true. This chain eventually ends at axioms and rules of deduction which everybody believes are true. This is how we can all agree that propositions like “2 + 7 = 9″ are true. The premises which led to this are suppressed, but always there.

Though with many propositions (like “2 + 7 = 9″), the premises are different for different folks, and hence so are the arguments. Quite a lot of people cannot prove these propositions true, but believe them anyway. But this is because they use a valid argument from authority, trusting in the experts who told them about propositions like this. This shows that there can be more than one path to a true conclusion.

The shorthand of only presenting a few premises when deducing new mathematical theorems is handy, but its use can be dangerous in other kinds of argument. Arguments can appear purely objective, like mathematical theorems are, but which are really not because the chain of premises do not lead back to unassailable axioms but instead to propositions which are dubious or do not share universal approval.

This is why propositions like “Garfield was a better president than Tyler” are often contentious. It is too difficult, especially seated at a bar stool or in a blog combox, to expose and agree to the exact and relevant chain of premises which lead unambiguously to proving the proposition certainly true or certainly false. It is even too difficult for sober scholars who devote their entire lives to studying the subject. What is the precise, unambiguous list of qualities and circumstances which defines presidential quality? It is also difficult or impossible to give a complete list of premises which lead me to say “I feel hot” or “I like this Mozart sonata better than that one.” Sometimes all I can do is just state it and believe it. Thus the hardest and simplest propositions are often irremediably subjective.

It isn’t always hopeless. Simple propositions are often proved via short chains of premise-arguments. It isn’t a long road to “One should do good and eschew evil” or “I am mortal.” And here lies the problem. Too often when seeking to understand a subject we try to peel off the more difficult problems first, like “What values do we give this prior?” when we should be starting with simple ones like we’ll see next time.

Part II.


1Note: the proposition “I think I might be catching a cold” differs from “I will catch a cold by tomorrow.” One is a report of health, the other is a forecast. The forecast is a “somewhere in-between” kind of belief, i.e. uncertain, i.e. probabilistic.

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Poor Statistics Undermine The Reliability Of Neuroscience

I'm all aglow

I’m all aglow

Note carefully the picture which accompanies this post. The right-most glow is centered on the upper-middle-fifth amygdalic cingulatum region of your author’s brain. Statistics show that this region is “associated” with feelings of joy; more specifically, the shivers of delight one experiences when saying, “I told you so!”

(The other smaller glow to the left is “associated” with pleasant thoughts of Myers Dark rum, now ridiculously expensive.)

The synaptic juices started flowing and the glow glowed after I read “Many Neuroscience Studies May Be Based on Bad Statistics” in Wired, which opens:

The fields of psychology and cognitive neuroscience have had some rough sledding in recent years. The bumps have come from high-profile fraudsters, concerns about findings that can’t be replicated, and criticism from within the scientific ranks about shoddy statistics. A new study adds to these woes, suggesting that a wide range of neuroscience studies lack the statistical power to back up their findings.

The study is “Power failure: why small sample size undermines the reliability of neuroscience” in Nature Reviews Neuroscience by Katherine Button, John Ioannidis, and several others. Best thing about that paper was a short guide to terms researchers ought to know. My favorite:

Winner’s curse
The winner’s curse refers to the phenomenon whereby the ‘lucky’ scientist who makes a discovery is cursed by finding an inflated estimate of that effect. The winner’s curse occurs when thresholds, such as statistical significance, are used to determine the presence of an effect and is most severe when thresholds are stringent and studies are too small and thus have low power.

Button and team did a meta-meta analysis of fMRI studies and the like and discovered what will be no secret to regular readers: the statistics of these works ain’t too hot. Specifically, many (most?) have very low power. “The consequences of this include overestimates of effect size and low reproducibility of results.”

They looked at 48 meta-analyses, which comprised “730 individual primary studies.” The median power was 18%. If you don’t have a feel for that, the “normal” power for medical studies is 80%+. That’s the level grant granters want, anyway. Button’s finding means half the studies are worse than anemically powered.

The Scientist quotes Hal Pashler, a psychologist at the University of California, San Diego, as saying, “This paper should help by revealing exactly how bad things have gotten.” Can’t go too much by that, because it’s standard journalistic practice to fetch a quote from somebody who didn’t write the paper (and often didn’t read it). But in this case Pashler is right.

Or maybe I’m just happy to agree with him. Here’s why I do.

Point one, I did an extensive (maybe too extensive) critique of Sam Harris’s paper “The neural correlates of religious and nonreligious belief.” One of the worst papers, in a series of bad papers, that I’ve ever read. Shoddy experimental design, editorialism masked as science, data mysteriously disappearing, biases galore, et cetera.

If you’re only going to read two things, read these entries of the review “Can fMRI Predict Who Believes In God?” Part I and Part Last.

Points two and higher: click to one of these two reviews: Yet Another Study ‘Proves’ Liberal, Conservative Brain Differences, Brain Atrophy Responsible For Religious Belief?

I’ve done many more, but these capture the gist. Strange that these “studies” have a sort of theme to them, no?

Wired had the sense to ask why so many bad studies? Reason one: studies are too expensive. But since scientists must publish lest they perish, reason two: “[T]he pressure on scientists to publish often, preferably in high-profile journals, to advance their careers and win funding from the government.”

Since that pressure will not be lifted even after Button’s identification of systematic flaws, it is rational to expect a continuation of systematic flaws. Gives me a kind of job security, though.

IDing poor science doesn’t pay as well as generating it, however,. Actually it pays not at all. That’s why I think warm thoughts about rum: to keep my brain lit up.


Thanks to Mike Flynn for pointing us to this fine news.

General Ethics: Peter Kreeft’s Summa Philosophica Part VII

Evil and Good, together again.

Evil and Good, together again.

Read Part VI.

Remember, we’re doing summaries of summaries here; only bare sketches are possible. Buy his book for more detail.

Question VII is fun: General Ethics. That morality really exists and such forth. Specific rules follow next time.

Article 1: Whether moral laws are objectively real?

Oh my yes. But isn’t it the case that “moral laws are not agreed to by everyone, but argued about, in all times and places”? And isn’t it true that moral laws are values and not facts? Or maybe morals are nothing but subjective feelings?

On the contrary, all men, in all times, places, and cultures, argue about whether certain acts…are morally right or wrong. But we argue only about objective truths, not subjective feelings…

Kreeft likes to say no one “argues about private, subjective feelings. No one responds to ‘I feel well’ with ‘No, you feel sick.'” Disagreement about objective matters does not prove subjectivity. That would make every scientific disagreement subjective. If you can find anybody who disagrees with any mathematical judgment, and if mere disagreement made topics subjective, then all of mathematics is subjective.

It’s true that “[a]ll judgments exist only in a subjective consciousness, including judgments about material things like size, judgments about mathematical truths, and judgments about moral laws. But the objects of these judgments, unlike feelings, are objective.”

We’re not empiricists, because if we were we’d have to toss out all of mathematics, logic, and all other metaphysical judgments like the existence of other minds. Positivism is dead.

Since there are objectively real moral laws, it is our task to discover them.

Article 2: Whether there are any universal, exceptionless norms?

Yes, afraid so.

Whenever an exception is made to a moral law, there is always a more general moral law that justifies the exception…The commandment forbids murder, not killing. The attempted murderer forfeits his right to life by threatening other lives.

Also, “[m]otives and situations cannot make an act that is wrong in itself to be right. They can only make an act that is otherwise right to be wrong. As T.S. Eliot said…about motives, ‘The last temptation is the greatest treason:/ to do the right thing for the wrong reason.'”

The clich´ says the exception proves the rule, but what it means is “that an exception presupposes a rule. Some rules have exceptions and some do not. If all rules have exceptions, what about that rule?”

Article 3: Whether ‘ought’ is a kind of ‘is’?

Yes. Yet Hume says an ‘is’ does not entail an ‘ought’ and Hume cannot be praise too much.

Now it is “common-sensical to argue that what you ought to be and do depends on what you are and on what is. For instance, ‘be a man’ is meaningless if addressed to an animal.” And it is only sensible to tell a man what to do what can be done. There cannot be a moral commandment to ride unicorns or to wipe the sauce dropped from flying spaghetti monsters.

“What is” is ambiguous. It can mean an empirical fact as distinct from a value…or it can mean, more broadly, any objective truth at all, including the truth that values are objective, and including truths about what these objective values are…

When we say justice “is” good, we are not merely using a logical copula as in “a unicorn is a mythical beast,” but we are asserting something about reality, about what really is.

Even better:

Moral arguments have two premises, not one. One of them is about values and one about facts. Thus there is no non sequitur. E.g. “Killing an innocent person is morally wrong; abortion is killing an innocent person; therefore abortion is morally wrong” derives the specific ethical conclusion from a more general ethical premise combined with a metaphysical premise about the real nature of abortion and its object.

Article 4: Whether there is a natural law?

Yes. And not in the biological sense of “We saw this group of men do this and that under such and such conditions.” While biology is important, men are not pandas or centipedes. From the anthropological observation that many men murder, we do not derive the moral law that murder is just. There’s more to us than our bodily workings.

Now anyone “can discover the natural law simply by an honest attention to his conscience.”

Since it is part of man’s essential nature to have free choice of will, rather than to be determined unfreely, the natural law for man is a law that prescribes rather than describes, or which describes what man ought to do rather than what man necessarily does.

Stating there is a natural law does not, of course, show us what the natural laws are. We do these next time.

Article 5: Whether evil is real?

Yes: don’t you read books? A popular non-argument is to say, “If we treat people as if they were good, they will respond by being good.” But this assumes people can respond oppositely, which is to say evilly, therefore there is evil.

Article 6: Whether evil is ignorance?

No. If you think so, try telling the cop who pulled you over you didn’t know the speed limit was only 35 MPH.

A brother fallacy of the preceding one is “Raising Awareness”, a radioactive byproduct of the Enlightenment which says that if only we expose people to “the truth” (which often just means “our belief”) they will automatically act properly or in a way which we desire. If this were true, every one of you reading this post would become saints (and so would your author). As much as I’d love that, smart money says it ain’t gonna happen.

Article 7: Whether mankind is insane to choose evil over good?

Yes, amen. Kreeft includes a third Enlightenment-driven counter-argument which claims that to broadcast we’re all bat-guano crazy is “harmful to our self-esteem, which is a necessary precondition to motivate us to improvement. Therefore it is counterproductive and should not be made.” There is no modern (non) crime worse than to hurt somebody’s feelings.

“When one is repeatedly given the opportunity for joy or misery and repeatedly chooses misery, it is not too severe to call this disorder insanity”. Now insanity has degrees; grips on reality or more and less firm. Those wearing sunglasses in the dark can see better than those with their heads wrapped in tinfoil covered in duct tape. So that just because we’re all nuts, doesn’t mean we can’t recognize people nuttier than us.

Article 8: Whether there is more good than evil in man?


Evil cannot be greater than good because evil is a corruption of good and can exist only in a subject that is good by nature. If it corrupted all the good, it would cease to exist, since it would have no subject to exist in, like a parasite who killed its host.

As St Thomas Aquinas famously said, “Good can exist without evil, whereas evil cannot exist without good.”

Article 9: Whether virtue always brings happiness?

It does. Eventually. Though in the moments of sin, it feels so good. There is however always (as in always) a price.

[A]nyone can verify in his own experience, by experiment, so to speak, the fact that practicing virtue, especially charity, always produces deep happiness, while sinning against the virtues, especially charity, always produces deep discontent and inquietude.

Boethus: “All things seek again their proper courses and rejoice when they return to them.” And there’s no problem deriving pleasure from being virtuous.

It is not self-serving to desire one’s own happiness, or the “good of delight” (the bonum delictabile) together with the good of moral virtue (the bonum honestum), because they are consummated together. Proper self-love is not wrong but right, and proper self-love and proper altruism should coincide: thus “love your neighbor as yourself.”

Article 10: Whether sanctity is the key to ontology?

Yes—and hadn’t you thought about sanctity and ontology? Consider “Heidegger’s new and radical idea…that the mode of be-ing…[which] is usually called personhood or subjecthood, i.e. being an I rather than an It, is the key to being itself…and thus to ontology.”

The two realms of the personal subject and the ontological object, or person and being, are joined in God, who is the standard for both, and for their relationship. This when He revealed His own true, eternal name as “I AM,” He revealed the unity of personhood (“I”) and being (“AM”).

Also worthwhile is Kreeft’s speeach “Is Anything Really Right or Wrong?” (youtube video).

Read Part VIII.

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