Monday, June 29, 2015

Glossip v. Gross: Getting the Logic Right?

If we agreed that “no woman shall ever launch a nuclear weapon” does it then follow that it is okay for a man to do so? It seems that Scalia answers in the affirmative.

Scalia (pdf):

Mind you, not once in the history of the American Republic has this Court ever suggested the death penalty is categorically impermissible. The reason is obvious: It is impossible to hold unconstitutional that which the Constitution explicitly contemplates. The Fifth Amendment provides that “[n]o person shall be held to answer for a capital… crime, unless on a presentment or indictment of a Grand Jury,” and that no person shall be “deprived of life… without due process of law.”
He argues that because the Constitution says the government cannot execute unless if meets certain conditions, that the Constitution also permits it to execute so long as those conditions are met. I am willing to concede that the Founders thought that the death penalty was constitutional. And it would be one thing if the Constitution enumerated the power (”The government shall have the power to deprive a person of life withstanding due process of law.“

However, Scalia argues that the Constitution permits it to execute even if the government can find no way around some other constitutional barrier. There must always be a constitutional way for it to kill. From the Opinion of the Court (emphasis added):

Our decisions in this area have been animated in part by the recognition that because it is settled that capital punishment is constitutional, “[i]t necessarily follows that there must be a [constitutional] means of carrying it out.” Id., at 47. And because some risk of pain is inherent in any method of execution, we have held that the Constitution does not require the avoidance of all risk of pain. Ibid. After all, while most humans wish to die a painless death, many do not have that good fortune. Holding that the Eighth Amendment demands the elimination of essentially all risk of pain would effectively outlaw the death penalty altogether.
From Breyer’s dissent:
The relevant legal standard is the standard set forth in the Eighth Amendment. The Constitution there forbids the “inflict[ion]” of “cruel and unusual punishments.” Amdt. 8. The Court has recognized that a “claim that punishment is excessive is judged not by the standards that prevailed in 1685 when Lord Jeffreys presided over the ‘Bloody Assizes’ or when the Bill of Rights was adopted, but rather by those that currently prevail.” Atkins v. Virginia, 536 U. S. 304, 311 (2002). Indeed, the Constitu­tion prohibits various gruesome punishments that were common in Blackstone’s day. See 4 W. Blackstone, Com­mentaries on the Laws of England 369–370 (1769) (listing mutilation and dismembering, among other punishments).
Notice the conflicting arguments:
  • Scalia: It is constitutional, therefore there must be a constitutional means of execution.
  • Breyer: If there is no constitutional means, then it is unconstitutional.
Need I say that I find Breyer’s the more compelling argument? Regardless, if Scalia had not assumed his answer, he might have argued “If it is constitutional, then there must be a constitutional means.” But that would not differ in logic from Breyer’s argument.

I am not prepared to argue that the government is barred from exerting any inferred power which may come into conflict with any part of the Constitution. There are always questions of power and rights, but I find Scalia’s argument very strange.

What is the Conflict Between Uninformed Priors and Gambling?

Via Noah Smith at Twitter, Deborah G. Mayo asks “What do hard-nosed Bayesians like Gelman really mean by posterior probability?” In reading the piece, I grew concerned that Mayo left out something important to Andrew Gelman.

The following example pretty well describes the issue at hand.

Suppose I ask you show me a coin from your pocket and then ask you to flip it once. If it comes up heads, are you now willing to stake \$3 for a chance to win \$4 if among the next 1,000 flips at least 500 are heads? Or do you still think the coin is fair and you expect this bet on average would lose you \$1?
This illustrates the difference between an uninformed prior and a strong prior. If you began with a very weak prior– that is, prior to flipping the coin you believed the bias of the coin was equally likely to be always coming up heads as always coming up tails as anywhere in between– then you might take the bet. If on the other hand your experience is such that you do not frequently wind up with heavily biased coins in your own pocket then you might not think much of that first coin flip.

I think Gelman’s point is that however useful you may find it to employ an uninformative prior to communicate science, that does not mean your personal prior is necessarily uninformed. Therefore you might not make personal decisions based on a posterior derived from an uniformed prior– or, presumably, any prior much different than your own.

I am of two minds about the implications for reporting. My instinct is that biases should be made clear and so perhaps personal priors should be used in place of uniformed. On the other hand, a strong prior may greatly reduce the power of the study. If I believe that 999 of 1,000 coins yield tails 99 times out of 100, then a single flip of heads will do little to convince me that the coin is not biased toward tails. Is it really useful for me to report that I believe the coin is almost certainly biased toward tails? Is that science or opinion? It seems therefore that the likelihood– rather than the posterior– is the important scientific result of the study. In this case, the likelihood is identical to the posterior derived from the uniform prior, so there is no choice between these two.

To take an example from my own research, consider the audit of the April 14, 2013 Venezuelan election. There, a very extensive audit– 53 percent of more than 39,000 voting machines– turned up zero discrepancies between the numbers on the machines and paper ballots counted by hand. What conclusions may be drawn from this result? If you have uninformed priors, this seems overwhelmingly to suggest that the election was free of any meaningful audit-detectable fraud. If you believe strongly that the election results were fraudulent in a manner detectable by the audit, then it seems more sensible that the audit itself was a fraud. That is, your conclusion respecting the election result depends on your priors regarding possible fraud in the audit as well as the election. Scientifically, the important conclusion is that the audit results were not consistent with detectable fraud.1

Coming back to the original posts, then, it seems to me that Mayo missed Gelman’s point that priors do matter. I think Gelman is suggesting that relatively uninformed priors are reasonable for the basis of scientific reporting; in going beyond a study the reader must construct their own posteriors. To whatever extent possible, apply your Bayesian inference to your own priors rather than allowing someone else to insert their own.

1 To the point that a full audit of all voting machines would change nobody’s conclusions respecting the election results.