WILLIAM BLACKSTONE'S LOSS FUNCTION.

In statistical inference, the loss function is, loosely, a way of quantifying the consequences of a deviation of your estimate from reality.  I was looking for something precise and intuitive with the help of a search engine, and down the rabbit hole that quickly goes as apparently the people writing artificial intelligence algorithms make all sorts of use of loss functions, and whether they put more weight on positive or negative deviations of algorithm from reality might be an art form.  Ah, the joys of being pensioned off.  Here's my fortysomething self.

In statistical inference, there are two serious mistakes a researcher can make. The first type, which practitioners call Type I error, involves the rejection of a true null hypothesis. In many circumstances, people will describe such a case as a "false positive," because the null hypothesis is commonly absence of something (a disease, a price effect) and its rejection involves a mistaken diagnosis or a "statistically significant" coefficient (particularly troublesome if said coefficient also has the "wrong sign.")

The second type, per corollary a Type II error involves the failure to reject a false null hypothesis. Here, a sick person tests as healthy, or a coefficient that ought to differ from zero prints out as "statistically insignificant."

Sometimes, a researcher will speak of the power of a hypothesis test against Type I (or Type II) error, something that might be of great value if you don't want sick people tested as healthy, or as innocent people tested as guilty.

In a hypothesis test, the alternative hypothesis is the statement that the null hypothesis is false. If the alternative hypothesis is actually true, the power is the probability that one will correctly reject the null hypothesis. The most meaningful application of statistical power is to decide before initiation of a clinical study whether it is worth doing, given the needed effort, cost, and in the case of clinical experiments, patient involvement. A hypothesis test with little power will likely yield large p values and large confidence intervals. Thus when the power of a proposed study is low, even when there are real differences between treatments under investigation, the most likely result of the study will be that there is not enough evidence to reject the H₀and meaningful clinical differences will remain in question. In that situation a reasonable question to ask would be, was the study worth the needed time and effort to get so little additional information.

The usual question asked involving statistical power is: what sample size will result in a reasonable power (however defined) for the primary hypothesis being investigated. In many cases however, a more realistic question would be: what will the statistical power be for the important hypothesis tests, given the most likely sample size that can be obtained during the duration of the proposed study?

In statistical inference, "significance" and "confidence intervals" are artifacts of sample size, and generally loss functions and calculations of power are symmetric with respect to the divergence of estimator from reality.  Thus, for example, "least squares" regression, minimizing the sum of the squared residuals (a fancy way of saying estimator differing from reality) more harshly penalizes larger errors compared with smaller.

If you're testing for a disease, would you be more troubled with a test that incorrectly flagged people as infected or with a test that missed many infections?  If so, you might be more interested in the power of a test against either Type I or Type II errors alone.  So too with William Blackstone's "Better that ten guilty persons escape, than that one innocent suffer," analyzed with a footnote, "In this case, Type 1 errors would mean convicting innocent people of crimes they didn't commit or subjecting them to added scrutiny despite their innocence, and Type 2 errors would mean failing to convict guilty people of crimes they did commit and allowing them to go free unpunished," and Mr Blackstone might have capital punishment top of mind in seeking more power against a Type I error as there's no undoing a death sentence once it's carried out.  National Review's Kathryn Lopez elaborates.  "I don’t believe there is a circumstance in the United States in this day and age that warrants the death penalty."  She has the Christa Pike case in mind.  Town Hall's Demetrius Minor offered, a year ago, a more general argument based on a loss function against Type I error.

If there is one function that our government simply cannot afford to get wrong it is in wielding the ultimate government power: arresting, charging, trying, convicting, imprisoning, and ultimately executing an American citizen. We've learned a lot about the death penalty in the last 40 years. Tragically, innocent people are sentenced to die far more often than any of us should be comfortable with. Since 1976, 200 people have been exonerated from death row nationally.

It might be, as Charles Lane argues in The [Not Detroit] Free Press, that participants in those exoneration projects have tight priors of their own, and thus weight their loss functions heavily against any hint of a false positive.

For instance, some of the “exonerated” had had their convictions overturned not because their innocence had been proven, but because of a technicality. Others had had death sentences set aside while they were still in prison, well before their exoneration.

On the other hand, in the same forum, Noah Bernstein interviews former Oregon penitentiary superintendent Frank Thompson, who developed doubts large and small during the course of implementing the protocols to conduct executions, which is to say, doing his job.

It’s very difficult to go from the specific to the general in this execution process. All of them are unique circumstances. Training supplies or medical issues can go awry. You lose the real important question: Is the death penalty just?

Justice calls for a consistent application of outcomes, and the death penalty cannot be administered consistently across the board, or else we wouldn’t be having this conversation. Its history is replete with mistakes, with innocent people being killed. This incident raises an entire truckload of questions about justice not being applied.

There is no system contrived by humans that is as consistent in its application as the death penalty would need to be, and it needs to be above reproach. Maybe not necessarily perfect. But I don’t think human beings can put together that kind of a system when there are so many victims based on discrimination, based on mistakes. Given the flaws of human nature, I don’t think it’s possible to raise the death penalty to those kinds of standards it needs to meet to be just.

All of these difficulties in training, dear reader, arose out of attempts by government to, as National Review's Andrew T. Walker notes, make execution look like an outpatient procedure.

I write as a professor of Christian ethics and as a “retentionist” — meaning that I believe the death penalty is just and called for by the demands of political justice. Capital punishment has a storied place in Christian theology. See, for instance, Genesis 9:6: “Whoever sheds the blood of man, by man shall his blood be shed, for God made man in his own image.” Also consider Romans 13:4: “For he [government] is God’s servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God’s wrath on the wrongdoer.” Western law has long relied upon the natural law foundations of Christianity to justify capital punishment.

Yes, but carrying out God's wrath has morphed from the stake to the gurney.

Yet it is the visceral nature of what capital punishment is that runs afoul of how the modern state practices it — by medicalizing it and sanitizing it. We use gurneys, IVs, and the technical vocabulary of modern medicine to portray it as a “procedure.” We invite medical doctors to certify death or find the necessary veins to administer the deadly elixir. We make a spectacle of the ordeal — highlighting last meals and final statements, accompanied by press conferences.

Sanitizing capital punishment erases the deterrence it is supposed to evoke. By medicalizing capital punishment, we try to protect the public conscience from what is really taking place. Capital punishment ought to be a public act of justice. But great effort undertaken to obscure what is happening shortchanges the moral point of capital punishment in the first place. If a society authorizes capital punishment in law, refusing to reckon with it should make us question why we practice it at all.

Thus he echoes the sentiments of many Militant Normals: the noose, or the firing squad, in public.

We're doing the Lord's work, only let us not be dramatic about it?  Reason's Billy Binion dissents.

Lethal injection was first authorized by Oklahoma in 1977 and first used in Texas in 1982. Its introduction was part of a tortured history of trying, in part, to increase the humanity of killing people. Before a fatal cocktail of drugs was the electric chair, the gas chamber, the gallows, and the firing squad.

Reservations specifically with the electric chair are easy to understand. The contraption made its debut in the late 1800s after New York Gov. David Hill, who said hanging had "come down to us from the dark ages," commissioned a report "to investigate…the most humane and practical method of carrying into effect the sentence of death." The conclusion it came to: death by electricity.

The benefit of hindsight would have been helpful. It takes several minutes to shock the heart into submission. In reality, the electric chair has become synonymous with torture, both for the condemned and for those condemned to watch.

Graduating to lethal injection removed much of the latter—it is, of course, less of an assault on the eyes. It looks like a medical procedure, something that could be routine were it not for the setting and the last words. But in once again seeking to make killings humane, the government settled on a method that can still be intensely painful for the individual, as Pike's experience suggests. It can cost taxpayers hundreds of thousands of dollars for one dose. And it holds the unenviable distinction of having the highest botched-execution rate.

That Thomas Edison was electrocuting animals, including an elephant called Topsy, to illustrate the hazards of Westinghouse's alternating current, and that one Doctor Joseph Guillotin sought a "humane and practical method" more precise than the broadsword or the axe.  Mr Binion isn't sure treating an execution as an outpatient procedure is sensible.

This debate, for more than a century, has focused on how to do that humanely. It doesn't really exist. Killing people is, by definition, at odds with the meaning of the word—something that is true whether you're an abolitionist or a person who understandably views the ultimate penalty as a moral imperative for heinous acts against others. Pike received the humane option.

I'll give Nicholas Clairmont, also in The Free Press, the closing argument.  "If we can’t stomach a killing that looks like a killing, we shouldn’t do it at all."  He's making a case for firing squads, not guillotines, but in that "we can't stomach" there might be echoes of Blackstone's loss function: what certainty must a jury, let alone the population from which the jury is summoned, have that the person convicted of a capital crime is well and truly deserving of that punishment?