Christian-Science Monitor article

"Case of soup-kitchen thief fuels critics of three-strikes laws" (August 19th, 2010) discusses California's three-strikes law, a law that requires an individual be sentenced to life imprisonment after being convicted of three felonies. It leaves relatively undiscussed the statistical reasons for such a law.

Los Angeles Superior Court Judge Peter Espinoza said Taylor’s sentence was one of many third-conviction cases that brought “disproportionate” sentences and “resulted in, if not unintended, then at least unanticipated, consequences.”

[...]

But stories like Taylor’s are useful in illustrating the problems behind the law's implementation, says Ms. Levinson. The challenge for critics will be trying to prove that society does not benefit from decades-long or even life sentences for nonviolent – or at least not serious – crimes.

Corrections suggests that it may make sense to permanently incarcerate individuals who have been convicted of only minor crimes previously, because the likelihood that they have committed

*unobserved* and

*serious* crimes is high. It's worth noting that, truncating the distribution of claims at the 90th percentile, burglars appear to commit about 38.1 burglaries per year (

Visher 1986, reported by NSW Bureau of Crime Statistics and Research). The Senate Congressional record from 2004 indicates that rapists commit between 8 and 10 rapes on average, before being caught (

Congressional Record, page 22999). Car Thieves in England appear to steal 45 cars before they're 18, and 94 when they're older (

Car Theft: The Offender's Perspective, Light, Nee and Ingham, 1993, page 11). Between 545 and 707 metric tons of cocaine was shipped toward the United States in 2007. Two-hundred and fifty-nine metric tons were caught by U.S. authorities, either internally, at arrival zones, or in transit. Given the discrepancy, Corrections conjectures that distributors of cocaine go through hundreds, or thousands of transactions without being caught.

Perhaps some of these statistics are individually not dependable, but their consistency and magnitude serve as more general suggestive evidence. Given someone has been convicted of one crime, the likelihood that they have committed an order of magnitude more without having been caught appears rather large. The idea that crimes someone is convicted for are simply a

*signal* for the crimes they have committed is the impetus for the point Corrections makes here.

Let us imagine, before we have caught and convicted an individual, that we have some probability distribution that we believe they are sampled from, some likelihood that they are a career criminal, irredeemable recidivists. This is our "prior probability," a distribution between zero and one, that someone will be a recidivist, displayed graphically below (

the image is rather small, click to enlarge).

Someone's criminal act or acts provide a data distribution that the person will be a recidivist. One example of this might be their being

*caught* and

*convicted* of breaking into a building. Our hypothetical probability distribution we might place that someone has committed a major crime given they have been caught committing a serious crime like breaking an entering is displayed graphically below

We have distribution of the data on the probability of another criminal act, displayed graphically below (

click to enlarge).

We can combine our data on our parameters of interest with our prior beliefs about the parameters, in this case the likelihood that someone will commit another criminal act (

click to enlarge)

We can summarize bayesian update by displaying our prior, data, and posterior in one graph, displayed graphically below (

click to enlarge). For those interested, in this case we took data to be drawn from a binomial distribution, and assumed its conjugate prior distribution, a beta distribution (therefore making the prior distribution is a beta distribution--of course, our assumptions are without loss of generality, merely for mathematical convenience).

To summarize the point Corrections is making: given that someone has committed and for two felonies previously and has been convicted of a third felony, the likelihood that they have committed dozens, or hundreds of felonies and crimes is high. Convicting and imprisoning these career criminals for life, for their three observed, and dozens of unobserved felonies and crimes, may simply be the optimal response of a justice system that is sentencing people for their likely crimes (the likely crime they were convicted, and the likely crimes they were not caught for).

For those concerned that we are convicting individuals for crimes they have only

likely committed, we remind them that this is precisely what the judicial system does currently. The question is how to decide the

threshold for punishment.

Corrections finally notes that it is possible punishments already take this into account--that is, the penalty for marijuana distribution, in locations (excepting those with marijuana legalization or decriminalization) are deliberately punishing crimes for which one has not yet been caught. Our discussion took place under the assumption that punishments reflect only the crime for which one was convicted.