POSNER, Circuit Judge.
In 1992 the plaintiff, who was then 48 years old, was convicted in a Wisconsin state court of having sexually assaulted a boy repeatedly for five years beginning [931]*931when the boy was eight years old. (The plaintiff was and is a resident of that state and his crimes occurred there.) Oddly, he was given only a year in jail and probation for these assaults, but before the period of probation expired he was convicted of having in 1988 sexually assaulted a nine-year-old girl. Sentenced to 10 years in prison for that crime, he was paroled after 6 years. But his parole was revoked a year later after he admitted. that he had had sexual fantasies about two girls, one four years old and the other five, and that he had “groomed” them for sexual activities and would have molested them had he had an opportunity to do so.
Scheduled to be released from prison in 2005, instead he was civilly committed to the Sand Ridge Secure Treatment Center in 2004 as a “sexually violent person,” Wis. Stat. ch. 980, after a civil trial in which he was found to be “dangerous because he ... suffers from a mental disorder that makes it likely that [he] will engage in one or more acts of sexual violence.” Wis. Stat. §§ 980.01(7), 980.06; see id. §§ 980.01(2), (6). He was released in 2010 on the basis of the opinion of a psychologist that he was no longer more likely than not to commit further sexual assaults. But in 2006 Wisconsin had enacted a law requiring that persons released from civil commitment for sexual offenses wear a GPS monitoring device 24 hours a day for the rest of their lives. Wis. Stat. § 301.48. The statute applied to any sex offender released from civil commitment on or after the first day of 2008 and thus applied (and continues to apply) to the plaintiff. And therefore ever since his release from civil commitment he has been forced to wear an ankle bracelet that contains a GPS monitoring device.
His suit, which is against officials of the Wisconsin Department of Corrections who administer the monitoring statute, claims that the statute violates both the Fourth Amendment to the Constitution -and Article I, § 10, cl. 1 of the Constitution, the latter being the prohibition of states’ enacting ex post facto laws — laws that either punish people for conduct made criminal only after they engaged in it or increase the punishment above the maximum authorized for their crime when they committed it. (In the district court he also argued that he’d been denied equal protection of the laws, but he’s abandoned that argument on appeal.) The district judge held the Wisconsin monitoring statute unconstitutional on both grounds, precipitating this appeal by the defendants (in effect by the state). Although the judge wrote a long opinion, it omits what seem to us the crucial considerations in favor of the constitutionality of Wisconsin’s requiring the plaintiff to wear the ankle bracelet for the rest of his life.
Anyone who drives a car is familiar with GPS technology, which enables the driver to determine his geographical location, usually within a few meters. The GPS ankle bracelet (more commonly referred to as an ankle monitor or anklet monitor; we’ll use the latter term), shown below, likewise determines the geographical locar tion of the person wearing it, within an error range of no more than 30 meters. The most common use of such monitors is to keep track of persons on probation or parole; the device that Wisconsin uses is advertised specifically for those purposes. But such devices are also used by some parents to keep track of their kids or elderly relatives and by some hikers and mountain climbers to make sure they know where they are at all times or to track their speed.
The type of anklet worn by the plaintiff is waterproof to a depth of fifteen feet, so one can bathe or shower while wearing it. It must however be plugged into a wall [932]*932outlet for an hour each day (while being worn) in order to recharge it. There are no restrictions on where the person wearing the anklet can travel, as long as he has access to an electrical outlet. Should he move away from Wisconsin, he ceases having to wear it. And while he’s supposed to pay a monthly fee to compensate for the cost of the anklet, the plaintiff in this case does not pay it and the Department of Corrections appears not to have tried to compel him to do so,
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When the ankleted person is wearing trousers the anklet is visible only if he sits down and his trousers hike up several inches and as a result no longer cover it. The plaintiff complains that when this happens in the presence of other people and they spot the anklet, his privacy is invaded, in violation of the Fourth Amendment, because the viewers assume that he is a criminal and decide to shun him. Of course the Fourth Amendment does not mention privacy or create any right of privacy. It requires that searches be reasonable but does not require a warrant or other formality designed to balance investigative need against a desire for privacy; the only reference to warrants is a prohibition of general warrants. And although the Supreme Court has read into the amendment a qualified protection against invasions of privacy, its recent decision in Grady v. North Carolina, — U.S. -, 135 S.Ct. 1368, 1371, 191 L.Ed.2d 459 (2015) (per curiam), indicates that electronic monitoring of sex offenders is permitted if reasonable, cf. Samson v. California, 547 U.S. 843, 848-50, 126 S.Ct. 2193, 165 L.Ed.2d 250 (2006); Vemonia School District 47J v. Acton, 515 U.S. 646, 652-53, 115 S.Ct. 2386, 132 L.Ed.2d 564 (1995); Skinner v. Railway Labor Executives’ Ass’n, 489 U.S. 602, 618-24, 109 S.Ct. 1402, 103 L.Ed.2d 639 (1989)—and that standard is satisfied in this case.
Having to wear a GPS anklet monitor is less restrictive, and less invasive of privacy, than being in jail or prison, or for that matter civilly committed, which realistically is a form of imprisonment. The plaintiff argues that because he is not on bail, parole, probation, or supervised release, and so is free of the usual restrictions on the freedom of a person accused or convicted of a crime, there is no lawful basis for requiring him to wear the anklet monitor. But this misses two points. The first is the nature of the crimes he committed— sexual molestation of prepubescent children. In other words the plaintiff is a pedophile, which, as the psychologist who evaluated him explained, “predisposes [the plaintiff] to commit sexually violent [933]*933acts.... [I]t is well understood in my profession that pedophilia in adults cannot be changed, and I concluded that Mr. Belleau had not shown that he could suppress or manage his deviant desire.” The compulsive nature of such criminal activity is recognized in Rules 414 and 415 of the Federal Rules of Evidence, which in contrast to the rules governing cases involving other crimes allow evidence of the defendant’s other crimes, or acts, of sexual molestation of children to be introduced in evidence in a criminal or civil case in which the defendant is accused of such molestation.
The plaintiff in our case is about to turn 73, however, and he argues that he has “aged out” of pedophilic acts. There is evidence that the arrest rate of pedophiles declines with age, and from this it can be inferred that pedophilic acts probably decline with age as well, though there are no. reliable statistics on the acts, as distinct from the arrests for engaging in the acts. There is no reason to think that the acts decline to zero. Most men continue to be sexually active into their 70s, and many remain so in their 80s and even 90s. Stacy Tessler Lindau et al., “A Study of Sexuality and Health among Older Adults in the United States,” 357 New England J. Medicine 762-74 (Aug. 23, 2007). And even if not physically capable of the common forms of male sexual activity, older men can still molest and grope young children.
The psychologist who recommended that the plaintiff be released from civil commitment opined that the risk of the plaintiffs being charged or convicted of further sex crimes against young children had been 16 percent when he was released from civil commitment and could be expected to be about 8 percent at the time of the district judge’s summary judgment order this past September. It is important to understand however that such estimates, based on personal characteristics, such as age, number of past convictions, and type of victim, pertain only to the odds that the released offender will subsequently be arrested for or convicted of — in short, detected — committing further sex crimes. Gregory De-Clue & Denis L. Zavodny, “Forensic Use of the Static-99R: Part 4. Risk Communication,” 1 Journal of Threat Assessment & Management 145, 149 (2014). In the words of the psychologist, “actuarial scales ... underestimate the risk an offender will commit an offense over [his] lifetime.”
There is serious underreporting of sex crimes, especially sex crimes against children. A nationwide study based on interviews with children and their caretakers found that 70 percent of child sexual assaults reported in the interviews had not been reported to police. David Finkelhor, Heather Hammer, & Andrea J. Sedlak, “Sexually Assaulted Children: National Estimates & Characteristics,” Juvenile Justice Bulletin 8 (August 2008). The true level of underreporting must be even higher, because the study did not account for sexual assaults that go unreported in the interviews. Another study finds that 86 percent of sex crimes against adolescents go unreported to police or any other authority, such as a child protective service. U.S. Dept, of Justice, Office of Justice Programs, “Youth Victimization: Prevalence and Implications” 6 (April 2003); see also Candace Kruttschnitt, William D. Kalsbeek & Carol C. House (eds.), National Research Council, Estimating the Incidence of Rape and Sexual Assault 36-38 (2014).
And even if we credit the 8 and 16 percent figures the plaintiff can’t be thought just a harmless old guy, Readers of this opinion who are parents of young children should ask themselves whether they should worry that there are people in their community who have “only” a 16 [934]*934percent or an 8 percent probability of molesting young children — bearing in mind the lifelong psychological sears that such molestation frequently inflicts. See, e.g., Christina Rainville, “Using Undiagnosed Post-Traumatic Stress Disorder to Prove Your Case: A Child’s Story,” 31 Child Law Practice 97 (2012); Beth E. Molnar, Stephen L. Buka & Ronald C. Kessler, “Child Sexual Abuse and Subsequent Psychopathology: Results from the National Comorbidity Survey,” 91 American J. Public Health 753 (2001). The Supreme Court in Smith v. Doe, 538 U.S. 84, 103, 123 S.Ct. 1140, 155 L.Ed.2d 164 (2003), remarked on “the high rate of recidivism among convicted sex offenders and their dangerousness as a class. The risk of recidivism posed by sex offenders is ‘frightening and high.’ McKune v. Lile, 536 U.S. 24, 34, 122 S.Ct. 2017, 153 L.Ed.2d 47 (2002); see also id. at 33, 122 S.Ct. 2017 (‘When convicted sex offenders reenter society, they are much more likely than any other type of offender to be rearrested for a new rape or sexual assault’ (citing U.S. Dept, of Justice, Bureau of Justice Statistics, Sex Offenses and Offenders 27 (1997); U.S. Dept, of Justice, Bureau of Justice Statistics, Recidivism of Prisoners Released in 1983, p. 6 (1997))).”
Of child molesters released from prison in 1994, 39 percent were rearrested (though not necessarily for child molestation) within three years. U.S. Dept, of Justice, Bureau of Justice Statistics, Recidivism of Sex Offenders Released from Prison in 1994, P- 17, tab. 10 (Nov.2003). Although non-sex offenders had a higher rearrest rate (68 percent) than sex offenders and only 3 percent of child molesters were rearrested for a child-molestation offense, id. at 14, 17, these numbers don’t take account of the very high rate of un-derreporting of sex offenses. If only 20 percent of child molestations result in an arrest, the 3 percent recidivism figure implies that as many as 15 percent of child molesters released from prison molest again. That’s a high rate when one considers the heavy punishment they face if caught recidivating, and thus is further evidence of the compulsive nature of their criminal activity.
In short, the plaintiff cannot be certified as harmless merely because he no longer is under any of the more familiar kinds of post-imprisonment restriction. As for his civil commitment having been terminated on the basis of a psychologist’s determination that he was not more likely than not to molest children any longer, we doubt that the community would or should be reassured by a psychologist’s guess that a pedophile has “only” (say) a 49 percent chance of reoffending, or even the 16 percent chance estimated in this case — especially given all the accompanying negatives in the psychologist’s report. His affidavit states that the plaintiff is a pedophile and that “pedophilia in adults cannot be changed, and ... [the plaintiff] had not shown that he could suppress or manage his deviant arousal,” “had not reduced his sexual deviance and had not shown that he could suppress or manage his deviant arousal,” “had a mental disorder that predisposed him to commit sexually violent acts,” and “was not eligible for supervised release because he had not made significant progress in treatment.” There is the further problem that the 16 percent figure is just a guess, and the even more serious problem that the figure implies that of every six pedophiles with characteristics similar to those of the plaintiff in this case one will resume molesting children after his release from prison. Assuming that the anklet would (for reasons we’ll explain) deter that person, requiring that it be worn is a nontrivial protection for potential victims of child molestation.
The focus must moreover be on the incremental effect of the challenged statute [935]*935on the plaintiffs privacy, and that effect is slight given the decision by Wisconsin— which he does not challenge — to make sex offenders’ criminal records and home addresses public. These records are downloaded by private websites such as Family Watchdog that enable anyone with access to the Internet to determine whether a sex offender — more precisely anyone who has ever been convicted of a sexual offense serious enough to be made public by the state — lives near him. One of the members of this appellate panel, out of curiosity stimulated by another sex offender privacy. case, visited Family Watchdog and learned that there were several (one hopes reformed — but it is only a hope) sex offenders living on his street.
So the plaintiffs privacy has already been severely curtailed as a result of his criminal activities, and he makes no challenge to that loss of privacy. The additional loss from the fact that occasionally his trouser leg hitches up and reveals an anklet monitor that may cause someone who spots it to guess that this is a person who has committed a sex crime must be slight.
For it’s not as if the Department of Corrections were following the plaintiff around, peeking through his bedroom window, trailing him as he walks to the drug store or the local Starbucks, videotaping his every move, and through such snooping learning (as the amicus curiae brief of the Electronic Frontier Foundation would have it) “whether he is a weekly church goer, a heavy drinker, a regular at the gym, an unfaithful husband,” etc. (quoting United States v. Maynard, 615 F.3d 544, 562 (D.C.Cir.2010)). The fruits of such surveillance techniques would be infringements of privacy that the Supreme Court deems serious. See, e.g., Kyllo v. United States, 533 U.S. 27, 33-36, 121 S.Ct. 2038, 150 L.Ed.2d 94 (2001); see also Katz v. United States, 389 U.S. 347, 361; 88 S.Ct. 507, 19 L.Ed.2d 576 (1967) (Harlan, J., concurring); United States v. Jones, — U.S. -, 132 S.Ct. 945, 954-56, 181 L.Ed.2d 911 (2012) (Sotomayor, J., concurring); id. at 963-64 (Alito, J., concurring). But nothing of that kind is involved in this case, quite apart from the fact that persons who have demonstrated a compulsion to commit very serious crimes and have been civilly determined to have a more likely than not chance of reoffending must expect to have a diminished right of privacy as a result of the risk of their recidivat-ing — and as Justice Harlan explained in his influential concurrence in the Katz case, the only expectation of privacy that the law is required to honor is an “expectation ... that society is prepared to recognize as ‘reasonable.’ ” 389 U.S. at 361, 88 S.Ct. 507.
Rather, every night the Department of Corrections makes a map of every anklet wearer’s whereabouts that day so that should he be present at a place where a sex crime has been committed, or be hanging around school playgrounds or otherwise showing an abnormal interest in children not his own, the police will be alerted to the need to conduct an investigation. But the main “objective of the searches [the mapping, in this case] was [not] to generate evidence for law enforcement purposes,” as in Ferguson v. City of Charleston, 532 U.S. 67, 83, 121 S.Ct. 1281, 149 L.Ed.2d 205 (2001) (emphasis in original), but instead to deter future offenses by making the plaintiff aware that he is being monitored and is likely therefore to be apprehended should a sex crime be reported at a time, and a location, at which he is present.
The plaintiffs argument that his monitoring violates the Fourth Amendment is further weakened when we consider the concession by his lawyer at oral argument that the Wisconsin legislature could, with[936]*936out violating the Fourth Amendment, make lifetime wearing of the anklet monitor a mandatory condition of supervised release for anyone convicted of sexual molestation of a child. That would be a likely, and seemingly an unassailable, response of the legislature to a decision by this court upholding the district court’s invalidation of the GPS-monitoring statute — which is to say that for pedophiles to prevail in cases such as this would give them only a hollow victory.
It’s untrue that “the GPS device burdens liberty ... by its continuous surveillance of the offender’s activities,” Commonwealth v. Cory, 454 Mass. 559, 911 N.E.2d 187, 196-97 (2009); it just identifies locations; it doesn’t reveal what the wearer of the device is doing at any of the locations. And its “burden” must in any event be balanced against the gain to society from requiring that the anklet monitor be worn. It is because of the need for such balancing that persons convicted of crimes, especially very serious crimes such as sexual offenses against minors, and especially very serious crimes that have high rates of recidivism such as sex crimes, have a diminished reasonable constitutionally protected expectation of privacy.
So let’s recapitulate the gain to society from GPS monitoring of convicted sexual molesters. Every night as we said a unit of the Department of Corrections downloads the information collected that day by the anklet monitor and creates a map showing all the locations at which the wearer was present during the day and what time he was present at each location. Should a sexual offense be reported at a location and time at which the map shows the person wearing the anklet to have been present, he becomes a suspect and a proper target of investigation. But by the same token if he was not at the scene of the crime when the crime was committed, the anklet gives him an ironclad alibi. Missing this point, the amicus curiae brief of the Electronic Frontier Foundation in support of the plaintiff criticizes anklet monitoring for its accuracy!
A study of similar GPS monitoring of parolees in California found that they were half as likely as traditional parolees to be arrested for or convicted of a new sex offense. Stephen V. Gies, et al., “Monitoring High-Risk Sex Offenders with GPS Technology: An Evaluation of the California Supervision Program,” Final Report, pp. 3-11, 3-13 (March 2012). There is no reason to think that GPS monitoring of convicted child molesters in Wisconsin is any less efficacious.
Given how slight is the incremental loss of privacy from having to wear the anklet monitor, and how valuable to society (including sex offenders who have gone straight) the information collected by the monitor is, we can’t agree with the district judge that the Wisconsin law violates the Fourth Amendment, The plaintiff argues that monitoring a person’s movements requires a search warrant. That’s absurd. The test is reasonableness, not satisfying a magistrate. Consider a neighborhood in which illegal drug dealing is common. There will be an enhanced police presence in the neighborhood and, probably more important, several former or present drug dealers whom the police have enlisted as undercover agents. The result will be surveillance of the drug scene. No one (unless 'it’s the plaintiffs lawyer in this case) thinks that such surveillance requires' a warrant.
Or suppose police place hidden cameras in traffic lights to detect drivers who run red lights. That is investigative surveillance similar to what the Wisconsin Department of Corrections is doing with regard to potential sex offenders, yet no warrant is required for traffic surveillance. It would be odd to think that the Depart[937]*937ment of Corrections could not use GPS monitoring to determine the plaintiffs location at all times, but could have one of its agents follow him whenever he left his house.
It would 'be particularly odd to think that all searches require a warrant just because most of them invade privacy to a greater or lesser extent. The terms of supervised release, probation, and parole often authorize searches by probation officers without the officers’ having to obtain warrants, and the Supreme Court has held that such warrantless searches do not violate the Fourth Amendment as long as they are reasonable. Samson v. California, supra; United States v. Knights, 534 U.S. 112, 118-120, 122 S.Ct. 587, 151 L.Ed.2d 497 (2001). The “search” conducted in this case via the anklet monitor is less intrusive than a conventional search. Such monitoring of sex offenders is permissible if it satisfies the reasonableness test applied in parolee and special-needs cases. Grady v. North Carolina, supra, 135 S.Ct. at 1371. Wisconsin’s ankle monitoring of Belleau is .reasonable.
We conclude that there was no violation of the Fourth Amendment, and so we turn to whether the GPS-monitoring statute is an ex post facto law, as it took effect after the plaintiff had committed the crimes for which he had been convicted. A statute is an ex post facto law only if it imposes punishment. Smith v. Doe, supra, 538 U.S. at 92-96, 123 S.Ct. 1140. The monitoring law is not punishment; it is prevention. See, e.g., id. at 97-106, 123 S.Ct. 1140; Mueller v. Raemisch, 740 F.3d 1128, 1133-35 (7th Cir.2014); Doe v. Bredesen, 507 F.3d 998 (6th Cir.2007); cf. Connecticut Dept, of Public Safety v. Doe, 538 U.S. 1, 123 S.Ct. 1160, 155 L.Ed.2d 98 (2003); City of Indianapolis v. Edmond, 531 U.S. 32, 44-46, 121 S.Ct. 447, 148 L.Ed.2d 333 (2000); Michigan Dept, of State Police v. Sitz, 496 U.S. 444, 110 S.Ct. 2481, 110 L.Ed.2d 412 (1990); Skinner v. Railway Labor Executives’ Ass’n, supra, 489 U.S. at 620-21, 630, 109 S.Ct. 1402. The plaintiff does not quarrel with his civil commitment; even though it took away his freedom and was in most respects indistinguishable from confining him in prison, it was not ex post facto punishment because the aim was. not to enhance the sentences for his crimes but to prevent him from continuing to molest children. In Kansas v. Hendricks, 521 U.S. 346, 368-69, 117 S.Ct. 2072, 138 L.Ed.2d 501 (1997), the Supreme Court held that civil commitment of sex offenders who have completed their prison sentences but are believed to have a psychiatric compulsion to repeat such offenses is not punishment as understood in the Constitution; it is prevention. The aim of the anklet monitor statute is the same, and the difference between having to wear the monitor and being civilly committed is that the former measure is less likely to be perceived as punishment than is being imprisoned in an asylum for the criminally insane. So if civil commitment is not punishment, as the Supreme Court has ruled, then a fortiori neither is having to wear an anklet monitor. It is not “excessive with respect to [the nonpunitive] purpose,” Smith v. Doe, supra, 538 U.S. at 97, 123 S.Ct. 1140, for Wisconsin to conclude that all formerly committed sex offenders pose too great a risk to the public to be released without monitoring.
Having to wear the monitor is a bother, an inconvenience, an annoyance, but no more is punishment than being stopped by a police officer on the highway and asked to show your driver’s license is punishment, or being placed on a sex offender registry, held by the Supreme Court in Smith v. Doe, supra, and by our court in Mueller v. Raemisch, supra, 740 F.3d at 1133, not to be punishment. But while citing Smith v. Doe the district judge in this case did not properly apply that deci[938]*938sion, but instead embraced the hyperbolic statement in Riley v. New Jersey State Parole Bd., 219 N.J. 270, 98 A.3d 544, 559 (2014), that “the tracking device attached to Riley’s ankle identifies Riley as a sex offender no less clearly than if he wore a scarlet letter.” No, the aim of requiring a person who has psychiatric compulsion to abuse children sexually to wear a GPS monitor is not to shame him, but to discourage him from yielding to his sexual compulsion, by increasing the likelihood that if he does he’ll be arrested because the Department of Corrections will have incontestable evidence that he was at the place where and at the time when a sexual offense was reported to have occurred.
To return to our traffic analogy briefly: no one thinks that a posted speed limit is a form of punishment. It is a punishment trigger if the police catch you violating the speed limit, but police are not required to obtain a warrant before stopping a speeding car. The anklet monitor law is the same: it tells the plaintiff — if you commit another sex offense, you’ll be caught and punished, because we know exactly where you are at every minute of every day. Similar statutes in other states have reduced sex-crime recidivism. And though no one doubts the propriety of parole supervision of sex criminals though it diminishes parolees’ privacy, a study by the National Institute of Justice finds that GPS monitoring of sex criminals has a greater effect in reducing recidivism than traditional parole supervision does. Gies et ah, supra, at vii, 3-11, 3-13.
Reversed