
    UNITED STATES of America, Appellee, v. Avery LUNDQUIST, Defendant-Appellant.
    Docket No. 11-5379-cr.
    United States Court of Appeals, Second Circuit.
    Argued: Feb. 27, 2013.
    Decided: Sept. 9, 2013.
    
      Richard A. Friedman (Lanny A. Breuer, Assistant Attorney General, Gregory D. Andres, Acting Deputy Assistant Attorney General, Richard S. Hartunian, United States Attorney for the Northern District of New York, Tamara Thomson, Assistant United States Attorney, on the brief), United States Department of Justice, Washington, D.C., for Appellee.
    James P. Egan (Lisa A. Peebles, Federal Public Defender, Melissa A. Tuohey, Assistant Federal Public Defender, on the brief), Federal Public Defenders, Syracuse, NY, for Defendanb-Appellant.
    Before: CHIN and LOHIER, Circuit Judges, and GARDEPHE, District Judge.
    
    
      
       The Honorable Paul G. Gardephe, United States District Judge for the Southern District of New York, sitting by designation.
    
   CHIN, Circuit Judge:

In this case, defendant-appellant Avery Lundquist was convicted of receiving and possessing child pornography. Among the images in his possession was one of “Amy,” the pseudonym for a young woman who was sexually abused by her uncle when she was four years old. The uncle photographed his abuse of Amy, and disseminated those images on the Internet.

Amy is now in her twenties, and the pornographic images her uncle took of her continue to be traded on the Internet. Some 113 individuals — including Lund-quist — have been convicted of possessing images of her. The questions presented are whether Lundquist may be ordered to make restitution to Amy and, if so, in what amount.

The district court (Suddaby, J.) concluded that Lundquist proximately caused $29,754.19 of Amy’s losses, but decided he should be held jointly and severally liable, along with all others convicted of possessing Amy’s images, for her total losses of $3,381,159. We conclude that there was sufficient evidence to support a finding of proximate cause and that the district court reasonably estimated the share of Amy’s losses to be attributed to Lundquist as her total loss divided by the number of persons convicted of possessing her images at the time of the restitution request. The district court abused its discretion, however, by including in its calculations losses that Lundquist could not have proximately caused and by holding Lundquist jointly and severally liable for harm caused by defendants who were not before the court. Accordingly, we affirm in part, vacate in part, and remand for recalculation of the amount of restitution.

STATEMENT OF THE CASE

A. The Facts

1. Lundquist’s Possession of Child Pornography

On March 5, 2010, law enforcement found Lundquist in possession of child pornography and arrested him. He later admitted that he had downloaded the pornography from the Internet on or around February 14, 2010. On December 22, 2010, he pled guilty to a two-count indictment charging him with receiving and possessing child pornography, in violation of 18 U.S.C. § 2252A(a)(2)(A) and (a)(5)(B). The district court sentenced him principally to 210 months in prison. Among the images of child pornography in his possession was an image of Amy.

2. Amy’s Images

As noted above, Amy was sexually abused by her uncle when she was four years old. He photographed his abuse of her and disseminated those images on the Internet. In the late 1990s, law enforcement was able to track these images back to her uncle and prosecute him for these crimes. He was convicted of related charges in both federal and state court.

Amy began undergoing psychological treatment for symptoms resulting from her uncle’s abuse in 1998, at age nine. At the time, Amy responded so well to treatment that her therapist declared that she was “back to normal” within a year and her treatment was discontinued. Joyanna Silberg, Report of Psychological Consultation 2 (Nov. 21, 2008) [hereinafter 2008 Report], In 2008, however, expert psychologist Joyanna Silberg, Ph.D., concluded that the earlier prognosis was overly “optimistic” because Amy’s symptoms had re-emerged during her adolescence and her “history [had] conformed] to the expected trajectory of victims like herself who experience early sexual abuse.” Id at 2-3, 8.

Among other factors causing the reemergence of her symptoms, in 2005, when she was seventeen, Amy received her first notice that another person had been found in possession of the images that her uncle had created and disseminated on the Internet. Since 2005, Amy has received hundreds of similar notices. During a psychological evaluation conducted in 2008, Amy told Dr. Silberg that she felt as if “each discovery of another defendant that has traded her image re-traumatizes her again.” Id. at 3. In her 2008 Report, Dr. Silberg explained why Amy felt this way:

[R]eeovery from post-traumatic stress requires foremost a sense of safety that the trauma is over and that the past will not be replayed in the present[.] Yet, a victim of child pornography whose pictures remain present on the internet can never really have that sense of safety, or separation of the past and present. The past, in fact continues to be repeated in the present over and over again.... Specifically, Amy’s awareness of these pictures, [and] knowledge of new defendants being arrested become ongoing triggers to her.

Id. at 9 (citation omitted).

After consulting with an attorney, Amy began to seek restitution from persons convicted of possessing her images, pursuant to the mandatory restitution provision in the Violence Against Women Act of 1994. Amy submitted her first request for restitution in 2008 and has since submitted requests in more than 100 cases. See United States v. Lundquist, 847 F.Supp.2d 364, 375 (N.D.N.Y.2011).

3. Amy Learns About Lundquist

After law enforcement agents arrested Lundquist on March 5, 2010, they submitted the pornography found in his possession to the Child Victim Identification Program of the National Center for Missing & Exploited Children (“NCMEC”) for comparison with known child victims from other criminal investigations. The NCMEC identified Amy in one of the images in Lundquist’s collection. The government then notified Amy of the pending proceedings against Lundquist so she could assert her rights.

Between the dates of Lundquist’s arrest and his guilty plea, Amy visited Dr. Sil-berg twice for psychological re-evaluations. On August 17, 2010, Amy met with Dr. Silberg to determine whether she was still suffering from the symptoms documented in the 2008 Report. After this consultation, Dr. Silberg concluded in a report dated October 21, 2010;

It is clear that many of the symptoms that Amy evidenced in the initial evaluation [in 2008] remain, and some have worsened....
She continues to have post-traumatic symptoms, such as being triggered by the basement door at her uncle’s house....
Amy continues to struggle with making academic and vocational progress, is paralyzed by shame and struggles with feelings of victimization, and had begun to recapitulate this re-victimization. Despite feelings of guilt and shame she is unable to halt these processes.

Joyanna Silberg, Update on Psychological Consultation 3^4 (Oct. 21, 2010) [hereinafter 2010 Report].

Amy returned to Dr. Silberg for a second re-evaluation on December 20, 2010. During this interview, Dr. Silberg and Amy discussed Amy’s reasons for seeking restitution from defendants convicted of possessing her images. Amy explained that “she believes that it is important for those people who are continuing to victimize her to pay in some way, so that they have some knowledge of the harm they are causing.” Joyanna Silberg, Update on Psychological Consultation 3 (Jan.23, 2011) [hereinafter 2011 Report], During that same session, Amy also “briefly discussed her incarcerated uncle and her fear that he will be released from prison soon as one of the things holding her back in her life.” Id. Based on this evaluation, Dr. Silberg concluded:

Amy’s inability to move forward is ... inhibited by a sense of pervasive fear. She describes fear of her uncle ... as well as the pervasive fear of multitudes of men out there who could recognize her from a picture and have already abused her in their fantasies....
It is clear that Amy continues to suffer from the ongoing effects of her victimization from child abuse and from the continued use of her image by child pornography viewers, traders, and abusers.

Id. at 4.

B. Proceedings Below

On April 14, 2011, Amy’s attorney submitted, on her behalf, a request for restitution from Lundquist. Attached to the letter request were Amy’s Victim Impact Statement, Dr. Silberg’s 2008, 2010, and 2011 Reports, and a 2008 expert economic report prepared by Stan Smith, Ph.D., which calculated Amy’s lost future income, the cost of her future treatment, and the value of her decreased enjoyment of life (the “Smith Report”).

The government presented Amy’s restitution request to the district court in its sentencing memorandum. In addition to Amy’s materials, the government enclosed the NCMEC’s identification report and a table of other defendants convicted of possessing Amy’s images. At the June 17, 2011 sentencing hearing, the district court indicated its intent to order restitution in the amount of $37,126.50. This amount was based on the restitution order issued in a recent case involving another defendant who had been convicted of possessing Amy’s images. See United States v. Aumais, No. 08-CR-711, 2010 WL 3033821 (N.D.N.Y. Jan. 13, 2010), report and recommendation adopted by, 2010 WL 3034730 (N.D.N.Y Aug. 3, 2010), aff'd in part and rev’d in part, 656 F.3d 147 (2d Cir.2011). The district court deferred decision on the issue of restitution, however, to allow the government to consult with Amy and to await this Court’s decision in Aumais.

After the sentencing hearing, the government submitted a letter confirming that Amy was willing to accept the proposed amount of restitution, but Lundquist submitted letters renewing both his objection to any order of restitution and his request for a hearing. Lundquist argued that: (1) he was not a proximate cause of Amy’s losses; (2) he should be able to depose Amy or alternatively learn her identity so he could investigate her claims; (3) the district court committed mathematical errors in its calculations during the sentencing hearing; and (4) the district court should not rely on any of the factual And-ings made in Aumais. Soon after these letters were submitted to the district court, this Court reversed the order of restitution in Aumais. See Aumais, 656 F.3d 147.

In a memorandum decision and order filed December 14, 2011, the district court denied Lundquist’s request for a hearing and ordered him to make restitution, on a joint and several basis, in the amount of $3,381,159. United States v. Lundquist, 847 F.Supp.2d 364, 383 (N.D.N.Y.2011). The district court concluded that Lund-quist had proximately caused Amy’s losses because Lundquist possessed her image between February 14 and March 5, 2010, and during that time “Amy has sustained, and continues to sustain, significant psychological damage as a result of her knowledge that unidentified individuals have downloaded pornographic images of her from the Internet.” Id. at 371. Although the court did not believe it was necessary that Amy know of Lundquist’s possession of her image, it nonetheless found that she had such knowledge because the government notified her of the NCMEC identification and, soon afterwards, Amy visited Dr. Silberg twice to obtain new expert reports, which she then used in her restitution request. See id. at 373-74.

In calculating the amount of loss, the district court concluded that “all of the losses under consideration were due to the victim’s re-victimization” caused by learning about people downloading her images from the Internet. See id. at 374-75 & n. 13. The court reasoned that Amy was deemed to be “back to normal” in 1999 and the only reason that her symptoms reemerged later was because she began to receive notifications about new possessors of her images. See id. To the extent it was necessary to quantify Lundquist’s personal share of Amy’s losses, the court estimated that Lundquist proximately caused one one-hundred-thirteenth (1/113), or 0.88 percent, of her total losses because 113 defendants had been convicted of possessing Amy’s images as of the date of the government’s restitution request. Id. at 375. After finding that nearly all of Amy’s claimed losses were reasonable, the court calculated Lundquist’s personal share as $29,754.19. Id. at 376-78. But because it construed 18 U.S.C. § 3664(h) as authorizing joint and several liability in this instance, the district court ordered that Lundquist be held jointly and severally liable for the full amount of $3,381,159. See id. at 381-83.

Lundquist appeals, challenging only the order of restitution.

DISCUSSION

A. Applicable Law

We review an order of restitution for abuse of discretion and will reverse only if the “ ‘challenged ruling rests on an error of law, a clearly erroneous finding of fact, or otherwise cannot be located within the range of permissible decisions.’ ” Aumais, 656 F.3d at 151 (quoting United States v. Pearson, 570 F.3d 480, 486 (2d Cir.2009)). Under this standard, we will review the factual findings underlying the district court’s finding of proximate cause for clear error, but we “review de novo [the] ‘district court’s application of th[e] facts to draw conclusions of law, including a finding of liability.’ ” Id. at 154 (quoting Travellers Int’l, A.G. v. Trans World Airlines, 41 F.3d 1570, 1575 (2d Cir.1994)).

1. Mandatory Restitution Under Section 2259

Originally enacted as part of the Violence Against Women Act of 1994, Pub.L. No. 103-322, tit. IV, 108 Stat.1902, 18 U.S.C. § 2259 requires courts to “order restitution for any offense under this chapter,” including the crimes of receiving and possessing child pornography. 18 U.S.C. § 2259(a), (b)(4); see also id. § 2252A(a)(2)(A), (a)(5)(B). The offender must make restitution for “the full amount of the victim’s losses.” Id. § 2259(b)(1). As used in the statute, “victim” means “the individual harmed as a result of a commission of a crime under this chapter.” Id. § 2259(c). In Aumais, we held that the children depicted in child pornography generally — and Amy specifically — are victims within the meaning of this section. Aumais, 656 F.3d at 152.

The “full amount of the victim’s losses” is defined as:

any costs incurred by the victim for—
(A) medical services relating to physical, psychiatric, or psychological care;
(B) physical and occupational therapy or rehabilitation;
(C) necessary transportation, temporary housing, and child care expenses;
(D) lost income;
(E) attorneys’ fees, as well as other costs incurred; and
(F) any other losses suffered by the victim as a proximate result of the offense.

Id. § 2259(b)(3). We have held that the “victim’s losses must be proximately caused by the defendant’s offense” even if they fall within one of the enumerated categories in subparagraphs (A) through (E) because “proximate cause is a deeply rooted principle in both tort and criminal law that Congress did not abrogate when it drafted § 2259.” Aumais, 656 F.3d at 153 (citing United States v. Monzel, 641 F.3d 528, 535-36 (D.C.Cir.2011)). For this standard to be met, there must be “ ‘some direct relation between the injury asserted and the injurious conduct alleged.’” Id. (quoting Hemi Grp., LLC v. City of New York, 559 U.S. 1, 9, 130 S.Ct. 983, 175 L.Ed.2d 943 (2010)).

To the extent there is “[a]ny dispute as to the proper amount or type of restitution!,] [it] shall be resolved by the court by the preponderance of the evidence,” with the government bearing the burden of proof. 18 U.S.C. § 3664(e); see id. § 2259(b)(2) (“An order of restitution under this section shall be issued and enforced in accordance with section 3664 in the same manner as an order under section 3663A.”). The district court need not hold a “full-blown evidentiary hearing” on these issues, but it must offer each defendant “an adequate opportunity to present his position.” United States v. Sabhnani, 599 F.3d 215, 258 (2d Cir.2010) (internal quotation marks omitted) (quoting United States v. Maurer, 226 F.3d 150, 151-52 (2d Cir.2000) (per curiam)). Even if the district court holds an evidentiary hearing, however, “[n]o victim shall be required to participate in any phase of a restitution order.” 18 U.S.C. § 3664(g)(1).

In setting the amount of restitution, the court need only make a “reasonable estimate” of the victim’s loss. See Pearson, 570 F.3d at 486-87. “‘We will uphold an award of restitution under Section 2259 if the district court is able to estimate, based upon facts in the record, the amount of [a] victim’s loss with some reasonable certainty.’ ” Id. (quoting United States v. Doe, 488 F.3d 1154, 1160 (9th Cir.2007)); see also United States v. Burgess, 684 F.3d 445, 460 (4th Cir.2012) (“[T]he district court is not required to justify any award with absolute precision-”); Monzel, 641 F.3d at 540 (noting that, under section 2259, “some degree of approximation” is acceptable and “mathematical precision” is not required (quotation omitted)).

Section 3664(h) also authorizes the district court to impose joint and several liability in certain circumstances:

If the court finds more than 1 defendant has contributed to the loss of a victim, the court may make each defendant liable for payment of the full amount of restitution or may apportion liability among the defendants to reflect the level of contribution to the victim’s loss and economic circumstances of each defendant.

18 U.S.C. § 3664(h).

In Awmais, it was unnecessary for the Court to address the issue of joint and several liability because there was no evidence that the defendant in that case had proximately caused any of Amy’s losses. See Aumais, 656 F.3d at 155. Nevertheless, the Court suggested that holding a criminal defendant jointly and severally liable with others who were not before the sentencing court would be problematic and inconsistent with the text of section 3664(h), which “implies that joint and several liability may be imposed only when a single district judge is dealing with multiple defendants in a single case (or indictment).” Id. at 155-56.

2. The Evidence of Proximate Cause in Aumais

In Aumais, defendant Gerald Aumais had pled guilty to transporting and possessing child pornography, including images of Amy, and the district court ordered him to make restitution to Amy, pursuant to section 2259. See Aumais, 656 F.3d at 149-51. We reversed because “[pjroximate cause demands ‘some direct relation between the injury asserted and the injurious conduct alleged/” and there was no evidence that such a relationship existed between Aumais’s possession and Amy’s losses. See id. at 154-55 (citation omitted).

The district court had relied on Amy’s victim impact statement and the testimony of Dr. Silberg, see id. at 149-50, but Amy’s statement was prepared and Dr. Silberg’s evaluations took place before Aumais was arrested on November 16, 2008. Id. at 154. This evidence suffered from several deficiencies: (1) it did not show that Amy had “direct contact with Aumais” or that she even knew of “his existence”; (2) the victim impact statement did not mention Aumais; and (3) Dr. Silberg could not “speak to the impact on Amy caused by this defendant” because her evaluations were all performed before Aumais’s arrest. Id. Accordingly, there was a complete “absence of evidence linking Aumais’ possession to any loss suffered by Amy.” Id. at 155.

We did not decide the circumstances in which a victim of child pornography can recover restitution, or the type of proof that would suffice to show causation. We did quote with approval a portion of the Ninth Circuit’s decision in United States v. Kennedy, 643 F.3d 1251 (9th Cir.2011):

The government’s evidence showed only that the defendant participated in the audience of persons who viewed the images of Amy. While this may be sufficient to establish that the defendant’s actions were one cause of the generalized harm Amy suffered due to the circulation of her images on the internet, it is not sufficient to show that they were a proximate cause of any particular losses.

Aumais, 656 F.3d at 154-55 (internal quotation marks and alterations omitted) (quoting Kennedy, 643 F.3d at 1264). We carefully noted, however, that our “opinion does not categorically foreclose payment of restitution to victims of child pornography from a defendant who possesses their pornographic images.” Id. at 155. Indeed, we acknowledged that the evidence of harm suffered by Amy was credible and well-established. See id. We only narrowly held that “where the Victim Impact Statement and the psychological evaluation were drafted before the defendant was even arrested' — or might as well have been — ... the victim’s loss was not proximately caused by a defendant’s possession of the victim’s image.” Id.

In Aumais and Kennedy, both courts were concerned that “the record did not include any evidence that [the defendant’s] conduct contributed to [the victim’s] claimed losses at all.” Kennedy, 643 F.3d at 1264 (emphasis added); see Aumais, 656 F.3d at 155. In both cases, the government relied on essentially the same evidence: victim impact statements and psychological evaluations demonstrating that the victim generally suffered harm from her awareness that unidentified persons were able to look at images of her childhood abuse. See, e.g., Aumais, 656 F.3d at 149-50; Kennedy, 643 F.3d at 1255-56. In neither case did the government present evidence that the victim continued to suffer this harm after the defendant’s possession of her images, at a time when the victim would have learned — and in fact did learn — about the specific defendant’s offense. Thus, in both cases, the government proved only that the victims suffered harm from their general fear that others were likely in possession of their images — harm that would have existed whether or not the specific defendants had ever actually possessed the victims’ images. See United States v. Fast, 709 F.3d 712, 722 (8th Cir.2013) (explaining that defendant “could not have caused — and thus could not be liable for — losses before” the date he possessed images of the victim); United States v. Gamble, 709 F.3d 541, 554 (6th Cir.2013) (“As a logical matter, a defendant generally cannot cause harm prior to the date of his offense.”).

The Ninth Circuit’s decision in Kennedy does suggest, in a portion not cited in Aumais, that the government must prove that the defendant directly caused some discrete, measurable aggravation of the injury. See Kennedy, 643 F.3d at 1264-65 (“[T]he district court’s inability to calculate the loss attributable to Kennedy’s offense is due to the government’s failure to introduce evidence of such a loss (such as evidence that Kennedy’s conduct led to Amy and Vicky needing additional therapy sessions or missing days at work).”). We decline to follow this dicta, for several reasons.

First, a recent Ninth Circuit per curiam opinion appears to cast doubt on this aspect of Kennedy. See In re Amy, 710 F.3d 985, 987 (9th Cir.2013) (per curiam). In that case, the district court concluded that Amy’s and another victim’s evidence did not satisfy the standard established in Kennedy, but the Ninth Circuit issued a writ of mandamus ordering the lower court to award restitution because there was “sufficient evidence to establish a causal connection between defendant’s offense and petitioners’ losses.” Id. Although the restitution requests are sealed, the publicly available briefs submitted to the Ninth Circuit suggest that the victims did not present particularized evidence of the sort described in Kennedy; indeed, the evidence in that case appears to be no different than the evidence now before us. See Petition for a Writ of Mandamus at 12-14, 21-22, In re Amy, No. 13-70858 (9th Cir. Mar. 8, 2013) (ECF No. 2); see also Response by the United States at 9 n. 5, In re Amy, No. 13-70858 (9th Cir. Mar. 11, 2013) (ECF No. 8) (advocating for the same method of calculating restitution used in this case). In sum, the Ninth Circuit has found non-particularized evidence sufficient to support an award of restitution. To the extent that Kennedy imposes a stricter requirement for proof of causation, we decline to adopt that approach.

Second, while nearly every circuit has recognized a proximate cause requirement in section 2259, see supra note 9, most of these courts have also acknowledged the inherent difficulty of applying that concept in this context. Accordingly, none has required, as a matter of law, more particularized proof than what has been presented in this case.

Finally, a legal rule demanding more particularized proof would be inconsistent with the principles underlying the concept of proximate cause, which “Congress did not abrogate when it drafted § 2259.” Aumais, 656 F.3d at 153. Proximate cause refers generally to the concept that “[i]njuries have countless causes, [but because] not all should give rise to legal liability,” the law will “ ‘decline[ ] to trace a series of events beyond a certain point.’ ” CSX Transp., Inc. v. McBride, — U.S. -, 131 S.Ct. 2630, 2637, 180 L.Ed.2d 637 (2011) (quoting Palsgraf v. Long Island R.R. Co., 248 N.Y. 339, 352, 162 N.E. 99 (1928) (Andrews, J., dissenting)). “At bottom, the notion of proximate cause reflects ‘ideas of what justice demands, or of what is administratively possible and convenient.’ ” Aumais, 656 F.3d at 154 (quoting Holmes v. Sec. Investor Prot. Corp., 503 U.S. 258, 268, 112 S.Ct. 1311, 117 L.Ed.2d 532 (1992)).

Of course, what justice demands is heavily dependent on the circumstances. Hence, there can be “no bright line demar-eating a legally sufficient proximate cause from one that is too remote.” People v. Roberts, 2 Cal.4th 271, 6 Cal.Rptr.2d 276, 826 P.2d 274, 300 n. 11 (1992). Accordingly, we decline to adopt a rule mandating the type of proof victims of child pornography must present before they can obtain restitution. It is sufficient if the evidence shows that there is more likely than not “some direct relation between the injury asserted and the injurious conduct alleged.” Aumais, 656 F.3d at 154 (quotation omitted); see also 18 U.S.C. § 3664(e) (“Any dispute ... shall be resolved by the court by the preponderance of the evidence.”).

B. Application

First, we consider whether Aumais precludes a finding of proximate cause in this case and conclude it does not. Next, we determine that the government’s evidence reasonably supports a finding of proximate cause. We then conclude that the district court’s method of estimating the amount of restitution was reasonable, although we also conclude that the district court erred in including damages for harm that Lund-quist could not have caused. Finally, we hold that the district court erred by holding Lundquist jointly and severally liable for Amy’s total losses.

1. Aumais Does Not Preclude Restitution

Lundquist argues primarily that Aumais forecloses a finding of proximate cause as a matter of law in this case. He points out that Amy’s victim impact statement and the 2008 Report are the same documents submitted in Aumais, and that neither the 2010 nor the 2011 Report identifies Lundquist specifically as a cause of Amy’s harm. Thus, according to Lundquist, both “the Victim Impact Statement and the psychological evaluation[s] were drafted before the defendant was even arrested — or might as well have been ” — and thus cannot support a finding of proximate cause as a matter of law. Aumais, 656 F.8d at 155 (emphasis added).

We conclude that Lundquist takes the phrase “or might as well have been” in Aumais out of context. As Lundquist acknowledges, Dr. Silberg’s testimony in Au-mais was based on the 2008 Report, which was prepared on November 21, 2008, a few days after Aumais’s arrest on November 16, 2008. But the 2008 Report was based on evaluations conducted on June 11-12, July 29, and November 10, 2008, days and even months before Aumais’s arrest. See Aumais, 656 F.3d at 154; 2008 Report at 1. In context, then, the phrase “or might as well have been” undoubtedly refers to reports — like the one in Aumais — that are technically prepared after the defendant’s arrest, but which could not possibly contain any information concerning the victim’s condition as a result of the defendant’s possession. These types of reports “cannot speak to the impact on [the victim] caused by this defendant.” Id. at 154; accord United States v. McGarity, 669 F.3d 1218, 1269-70 (11th Cir.2012) (citing Aumais, 656 F.3d at 154) (holding proximate cause cannot be based on testimony of psychological expert who had not evaluated Amy after the defendant’s arrest).

Here, in contrast, the new psychological reports were prepared on October 21, 2010 and January 23, 2011, and were based on evaluations conducted on August 17 and December 20, 2010, respectively. Hence, both new reports were prepared and based on evaluations conducted months after Lundquist’s arrest on March 5, 2010. Although the reports do not mention Lundquist specifically, they show that Amy continued to suffer harm as a result of learning about new possessors of images of her abuse during the time period when she would have learned about Lundquist. Accordingly, the new reports could potentially “speak to the impact on Amy caused by this defendant.” Aumais, 656 F.3d at 154.

Additionally, the district court in Au-mais found that Amy did not even know of Aumais’s existence. See id. at 154. Here, in contrast, the district court found that Amy learned about Lundquist’s possession of her image, and we conclude that this finding was not clearly erroneous. Although the record is unclear as to precisely when Amy received the notification, the district court noted that the government had a duty to notify Amy and, between the dates of Lundquist’s arrest and his guilty plea, Amy visited Dr. Silberg twice for reevaluations “to determine to what extent she was continuing to be psychologically re-victimized due to her knowledge that individuals were exchanging and viewing pornographic photographs of her on the Internet.” Lundquist, 847 F.Supp.2d at 372. During one of these evaluations, Amy disclosed that she was personally involved in seeking restitution from these individuals. The two evaluations provided the basis for the 2010 and 2011 Reports, which Amy then used to support her request for restitution from Lundquist. This sequence of events supports the reasonable inference that Amy went for the reevaluations at least in part because she had learned about Lundquist and intended to seek restitution from him.

Accordingly, this case is distinguishable from Aumais because the government has presented evidence that: (1) Amy continued, to suffer harm after Lundquist’s arrest as a result of learning about individuals — like Lundquist — who had come into possession of images of her abuse; and (2) Lundquist was one of the possessors that Amy learned about during that time. These facts support an inference that the harm documented in the 2010 and 2011 Reports was caused at least in part by learning about Lundquist’s offense. Hence, the record before the district court was sufficient to support a finding that Lundquist’s conduct was a proximate cause of some of Amy’s harm. See United States v. McDaniel, 631 F.3d 1204, 1209 (11th Cir.2011).

We acknowledge that it may appear anomalous to tie a defendant’s liability for restitution directly (and often solely) to action taken by the government. A victim of a child pornography crime ordinarily learns of a defendant’s offense (and thereby can make a showing of proximate cause) only after receiving notice from the government. The government’s role in the chain of causation, however, is an unavoidable, practical consequence of Congress’s decision that child pornography victims must be given notice of the crime committed against them and an opportunity to assert their rights. We conclude that this role does not provide a basis for denying restitution to a victim harmed by a defendant’s actions.

There is no doubt that victims of child pornography suffer harm from the circulation of their images, see New York v. Ferber, 458 U.S. 747, 759, 102 S.Ct. 3348, 73 L.Ed.2d 1113 (1982) (“[T]he harm to the child is exacerbated by the[] circulation [of these images].”); Aumais, 656 F.3d at 152, and it is also indisputable that Congress mandated restitution for victims of these offenses, see 18 U.S.C. §§ 2259(a), 2252A. While it is clear that Congress did not mean for section 2259 “to serve as strict liability against any defendant possessing such admittedly repugnant images or videos,” McGarity, 669 F.3d at 1270, the requirement of proximate cause strikes a balance, limiting the defendant’s liability to those losses that can be proved to have “some direct relation” to his offense. Aumais, 656 F.3d at 154 (quotation omitted). Perhaps it is not ideal, but using the victim’s after-the-fact knowledge of the defendant’s conduct is an “ ‘administratively possible and convenient’ ” manner of proving this relationship, which is all that proximate cause requires here. Id. (quoting Holmes, 503 U.S. at 268, 112 S.Ct. 1311). Even though the government ordinarily helps to create this evidentiary link between the defendant’s crime and the victim’s harm, the defendant’s conduct remains the proximate cause of the victim’s resulting harm because one of the foreseeable risks of possessing child pornography is that the victim may eventually learn about the crime in some manner. See Dobbs, supra, § 204 (“[I]f the intervening cause itself is part of the risk negligently created by the defendant, or if it is reasonably foreseeable at the time of the defendant’s negligent conduct, ... the defendant is not relieved of liability merely because some other person or force triggered the injury.”).

2. Speciñc Evidence of Proximate Cause

We conclude that the district court did not abuse its discretion in finding that Lundquist proximately caused some portion of Amy’s losses. The original 2008 Report explains that notifications about new possessors like Lundquist are “ongoing triggers” for Amy, which cause her problems in the areas of “[m]ood regulation, cognitive distortions, feelings of shame, self-blame, and guilt, self-esteem, alcohol abuse, dissociation, academic progress, interpersonal relationships, and vocational success.” 2008 Report at 8-9. The 2010 Report confirms that after Lund-quist’s arrest, “many of the symptoms that Amy evidenced in [the 2008 Report] remain, and some have worsened.” 2010 Report at 3. The 2011 Report specifies that: (1) her “poor interpersonal choices are seen as direct effects of the previous and ongoing abuse of Amy on the Internet and as a child”; (2) her inability to set or fulfill career goals “is exacerbated by her awareness of this pervasiveness of her image on the Internet which makes her fearful of interacting with many people outside the comfort of her familiar surroundings”; and (3) her “inability to move forward is also inhibited by a sense of pervasive fear ... of her uncle ... as well as the pervasive fear of multitudes of men out there who could recognize her from a picture and have already abused her in their fantasies.” 2011 Report at 4.

In short, “[i]t is clear that Amy continues to suffer from the ongoing effects of her victimization from child abuse and the continued use of her image by child pornography viewers.” Id. Because the sequence of events following Lundquist’s arrest supports the reasonable inference that Amy learned about Lundquist before visiting Dr. Silberg for the re-evaluations, the district court reasonably determined that Lundquist caused part of the harm described in the 2010 and 2011 Reports and thus was more likely than not a proximate cause of some of Amy’s losses. Moreover, Lundquist submitted a letter in connection with sentencing, in which he admitted having reviewed the victim impact statements of the children depicted in the images in his possession and acknowledged that he had “made them my vict[i]ms by my actions, that they relive it 24/7.” Accordingly, Lundquist has admitted that his actions harmed Amy. Although such evidence is not necessary to show proximate cause, it provides additional support for the district court’s finding in this instance.

Lundquist argues that the district court abused its discretion by making its findings without holding an evidentiary hearing. We disagree. It was well within the district court’s discretion to decline to hold a “full-blown evidentiary hearing” because Lundquist had “an adequate opportunity to present his position.” Sabhnani 599 F.3d at 258 (quotation omitted). Lund-quist had ample opportunity to raise objections to Amy’s supporting materials, but failed to do so. Instead, he made primarily legal arguments that did not require a hearing to resolve. The only “evidentiary” relief that Lundquist sought was the opportunity to depose Amy or learn her true identity, but “[n]o victim shall be required to participate in any phase of a restitution order.” 18 U.S.C. § 3664(g)(1). Accordingly, the district court did not abuse its discretion by relying only on the parties’ submissions.

3. The Amount of Restitution

While the record supports a finding that Lundquist proximately caused some of Amy’s harm, we recognize that trying to determine which specific losses are attributable to him is an arduous task. As discussed above, we reject any suggestion in Kennedy that Amy must show specific losses caused by her knowledge of Lund-quist, such as “additional therapy sessions or miss[ed] days at work.” Kennedy, 643 F.3d at 1265. Such an approach misconstrues the nature of Amy’s psychological injury — an ongoing trauma that manifests itself in every phase of her life — and would likely underestimate the amount of loss for which Lundquist is responsible. See 18 U.S.C. § 2259(b)(1) (requiring that restitution be ordered in “the full amount of the victim’s losses” (emphasis added)). Instead, as with all restitution orders under section 2259, the district court need only make a “reasonable estimate” of the victim’s loss. Pearson, 570 F.3d at 487.

Here, the district court calculated Lund-quist’s share of Amy’s loss as the total loss caused by Amy’s knowledge that individuals possessed images of her abuse, divided by the total number of persons convicted of possessing these images at the time of the restitution request. See Lundquist, 847 F.Supp.2d at 375. Although, as discussed below, the district court committed certain errors in setting the numerator in this equation — that is, the total loss caused by Amy’s re-victimization — we conclude that this approach leads to a reasonable estimate of the losses caused by Lund-quist. See Pearson, 570 F.3d at 487. In so holding, we join those courts that have concluded that section 2259 does not require a more precise measure of losses. See Gamble, 709 F.3d at 554 (finding per capita method to be reasonable and consistent with Congress’s desire to provide compensation to victims while also assigning some value to the social harm); see also United States v. Kearney, 672 F.3d 81, 100 (1st Cir.2012) (affirming an alternative calculation based on an average of the amount of restitution ordered in similar cases).

We find no reason to impose a more exacting standard for calculating loss here when none is required in many comparable areas of the law. For example, when making loss valuations under the U.S. Sentencing Guidelines, “[a] reasonable estimate of the loss is all that is necessary.” United States v. Singh, 390 F.3d 168, 192 (2d Cir.2004); accord United States v. Coppola, 671 F.3d 220, 250 (2d Cir.2012). Of course, the district court’s determination must be grounded in evidence and not derived from mere speculation, see United States v. Deutsch, 987 F.2d 878, 886 (2d Cir.1993), but the court need not establish a victim’s loss with exactitude, see United States v. Markle, 628 F.3d 58, 64 (2d Cir.2010); United States v. Bryant, 128 F.3d 74, 75-76 (2d Cir.1997) (per curiam). The district court may estimate the losses based on a calculation that is “appropriate and practicable under the circumstances.” U.S. Sentencing Guidelines Manual § 2B1.1 cmt. n.3(C).

Similarly, juries are instructed to approximate damage awards and not to require a mathematically precise connection between the defendant’s actions and the victim’s harm. As the Supreme Court has noted, “juries [historically] were accorded broad discretion to award damages as they saw fit ... [and] ‘in cases where the amount of damages was uncertain, their assessment was a matter so peculiarly within the province of the jury that the Court should not alter it.’ ” Atl. Sounding Co. v. Townsend, 557 U.S. 404, 409, 129 S.Ct. 2561, 174 L.Ed.2d 382 (2009) (quoting Feltner v. Columbia Pictures Television, Inc., 523 U.S. 340, 353, 118 S.Ct. 1279, 140 L.Ed.2d 438 (1998)); see also TXO Prod. Corp. v. Alliance Res. Corp., 509 U.S. 443, 457, 113 S.Ct. 2711, 125 L.Ed.2d 366 (1993) (“[Punitive damages] awards are the product of numerous, and sometimes intangible, factors; a jury imposing a punitive damages award must make a quantitative assessment based on a host of facts and circumstances unique to the particular case before it.”). As the district court is the fact finder in disputes over the amount of restitution, its method of calculating loss is also entitled to deference, unless it “cannot be located within the range of permissible decisions.” Aumais, 656 F.3d at 151 (quotation omitted).

Finally, allowing a reasonable approximation of the loss, rather than requiring a precise appraisal, furthers Congress’s intent to compensate victims of child pornography crimes. See Kearney, 672 F.3d at 99; see also Gamble, 709 F.3d at 550 (“The harm endured by the subject of child pornography upon realizing that others are viewing her image is part of what the child pornography prohibitions are designed to deter.”). Congress made restitution under section 2259 “mandatory” for these offenses, demonstrating its intent to allow these victims to recover for their losses. See 18 U.S.C. § 2259(a), (b)(4)(A); Kearney, 672 F.3d at 99; see also 18 U.S.C. § 2252A(a)(2)(A), (a)(5)(B). But Congress could not have intended that these victims be required to prove their losses to such an exact degree that they would effectively have to testify in every restitution hearing to explain how each defendant’s “conduct led to [them] needing additional therapy sessions or missing days at work.” Kennedy, 643 F.3d at 1264-65; see 18 U.S.C. § 3664(g)(1) (“No victim shall be required to participate in any phase of a restitution order.”); Kearney, 672 F.3d at 99 (“Congress was attempting to compensate the victims of child pornography, not to intensify the harm they have already suffered as a condition of obtaining restitution.”). Accordingly, we endorse the district court’s use of a per capita calculation as reasonable and consistent with the purpose of the statute.

The district court did commit clear error, however, in calculating the numerator because it included amounts for harm that Lundquist could not have proximately caused. First, the district court found that “all of the losses under consideration were due to the victim’s re-victimization, caused by all of the individuals downloading pornographic images of her from the Internet,” because Amy was “back to normal” in 1999 and her relapse was triggered only by learning that others had downloaded images of her abuse. Lundquist, 847 F.Supp.2d at 374 & n. 13. This finding was clearly erroneous. The record unambiguously demonstrates that Amy’s uncle, the original abuser who created and disseminated Amy’s images, continued to be a proximate cause of some of Amy’s losses.

The district court relied heavily on the 1999 prognosis that Amy was “back to normal,” but in the 2008 Report, Dr. Sil-berg characterized that prognosis as “optimistic.” 2008 Report at 3. She explained that Amy’s symptoms predictably “escalated in prominence” during her adolescence because she was “faced with deeision-mak-ing involving issues of trust and intimacy and future planning about her life.” Id. This “conformed] to the expected trajectory of victims like [Amy] who experience early sexual abuse.” Id. at 8. Accordingly, the evidence clearly demonstrates that Amy would have suffered some of her current difficulties because of the original sexual abuse, even if she had never learned that others were looking at her images online.

Indeed, the 2010 and 2011 Reports make clear that Amy’s uncle continues to be an independent cause of her ongoing suffering. See 2010 Report at 3 (indicating that Amy is “triggered by the basement door at her uncle’s house”); 2011 Report at 3 (describing Amy’s fear of her uncle being released from prison); id. at 4 (“It is clear that Amy continues to suffer from the ongoing effects of her victimization from child abuse and from the continued use of her image by child pornography viewers, traders, and abusers.” (emphasis added)). Moreover, even if her uncle were not independently causing Amy’s current symptoms, he would still be a proximate cause of the aggravation of her injuries because that was the foreseeable result of disseminating her images on the Internet. See Restatement (Second) of Torts § 433A cmt. (c) (explaining that “the original wrongdoer is hable for the additional harm caused by the intervening negligence of the later one, while the latter is liable only for what he has himself caused”); 74 Am. Jur.2d Torts § 67 (“The original tortfeasor is responsible for both injuries because it is foreseeable as a matter of law that the original injury may lead to a causally distinct additional injury.”).

Lundquist cannot be ordered to make restitution for harm that Amy’s uncle’s conduct proximately caused. See Aumais, 656 F.3d at 155 (“If Amy’s future counseling costs are thus partly caused by her uncle’s abuse, then Aumais cannot be responsible for all of those losses — a problem under the wording of § 2259, which mandates that Aumais make restitution for the full amount of Amy’s losses caused as a result of Aumais’ possession.”). Accordingly, the district court abused its discretion by including all of Amy’s losses in calculating Lundquist’s share. On remand, the district court shall apportion some of Amy’s total losses to her uncle before determining the loss caused by Amy’s knowledge that individuals had downloaded images of the abuse she suffered. See 18 U.S.C. § 3664(h) (providing that the court “may apportion liability among the defendants” when “more than 1 defendant has contributed to the loss of a victim”).

Second, the district court apparently included losses that Amy suffered before Lundquist’s offense in 2010. For example, the district court seems to have awarded “future” counseling costs estimated to begin in 2009. Lundquist, 847 F.Supp.2d at 376. Lundquist did not obtain Amy’s image until 2010, however, and he cannot be held responsible for therapy costs, or any other losses, that Amy incurred in 2009 or earlier. See Fast, 709 F.3d at 722; Gamble, 709 F.3d at 554; Aumais, 656 F.3d at 154-55. Accordingly, on remand, the district court shall exclude any losses that were incurred before Lundquist’s arrest.

4. Joint and Several Liability

Finally, we conclude that the district court abused its discretion in making Lundquist jointly and severally liable for all of Amy’s losses. We understand, as a policy matter, why joint and several liability is an appealing option in this type of case. Joint and several liability would permit the victims of child pornography to collect their full losses from any well-heeled defendant, rather than- require them to pursue defendants who may be, or later become, insolvent. Such an approach would also place the onus on guilty defendants to seek contribution from each other, rather than require the innocent victims to request restitution from each defendant.

We sympathize with these policy arguments and acknowledge that joint and several liability might be appropriate if Amy had brought a civil tort action against those who downloaded images of her abuse. But this is not a civil action. Instead, Amy has requested mandatory restitution under section 2259, and her request is subject to the restrictions set by that statute. On appeal, the government concedes that there is no legal justification for holding Lundquist jointly and severally liable for losses caused by other defendants who are not before the court. We agree.

If more than one defendant is responsible for a victim’s loss, section 3664(h) permits the court to either “make each defendant hable for payment of the full amount of restitution or ... apportion liability among the defendants.” 18 U.S.C. § 3664(h) (emphasis added); see also id. § 2259(b)(2) (cross-referencing section 3664). If the district court lacks the power to “make each defendant liable for payment of the full amount” — because it does not have jurisdiction over all of the responsible parties — the plain language of the statute leaves the court only one option: to “apportion liability among the defendants.” Id. (emphasis added). Accordingly, section 3664(h) does not authorize ordering restitution on a joint and several basis with other individuals who are not before the court (or included in the same indictment). See Fast, 709 F.3d at 723 n. 6; United States v. Laraneta, 700 F.3d 983, 992-93 (7th Cir.2012); Aumais, 656 F.3d at 156 (noting that “[sjection 3664(h) implies” this result).

Not only is joint and several liability in these circumstances inconsistent with the text of section 3664(h), but it also contravenes the proximate cause requirement of section 2259. The evidence shows that Lundquist contributed to Amy’s losses, but there is no evidence that he has caused all of her losses. Indeed, Amy’s losses exceed $3 million primarily because there are so many people viewing her images on the Internet. These circumstances make it unclear when, or if, she will ever obtain the necessary “sense of safety that the trauma is over and that the past will not be replayed in the present,” which is essential to recovery. 2008 Report at 9. If Lund-quist were the only person who had downloaded images of the abuse Amy suffered, his arrest might provide her with that feeling of safety. Unfortunately, he is not alone.

CONCLUSION

We conclude that Aumais is distinguishable and that the district court did not abuse its discretion in finding that Lund-quist proximately caused part of Amy’s losses or in calculating his share of the loss on a per capita basis. The district court did err, however, in failing to apportion some of Amy’s losses to her uncle, including losses incurred before Lundquist’s arrest when calculating his share of Amy’s losses, and holding Lundquist jointly and severally liable with other defendants who were not before the court. Accordingly, the order of restitution is AFFIRMED IN PART and VACATED IN PART, and the case is REMANDED for recalculation of the amount of restitution. 
      
      . See United States v. Aumais, No. 08 Cr. 711, 2010 WL 3033821, at *5 (N.D.N.Y. Jan. 13, 2010), report and recommendation adopted by, 2010 WL 3034730 (N.D.N.Y. Aug. 3, 2010), aff'd in part and rev’d in part, 656 F.3d 147 (2d Cir.2011).
     
      
      . Amy’s attempts to obtain restitution from defendants convicted of possessing her images have generated public interest. See generally Mary Margaret Giannini, Slow Acid Drips and Evidentiary Nightmares: Smoothing Out the Rough Justice of Child Pornography Restitution With a Presumed Damages 'Theory, 49 Am.Crim. L.Rev. 1723 (2012); Melanie Reid & Curtis L. Collier, W hen Does Restitution Become Retribution?, 64 Okla. L.Rev. 653 (2012); Emily Bazelon, Money Is No Cure: The Price of a Stolen Childhood, N.Y. Times Mag., Jan.27, 2013, at MM22; John Schwartz, Child Pornography, and an Issue of Restitution, N.Y. Times, Feb. 3, 2010, at A19.
     
      
      . The National Center for Missing & Exploited Children ("NCMEC”) is a nonprofit organization that works with law enforcement, pursuant to Congressional authorization, to "build a coordinated, national response to the problem of missing and sexually exploited children, establish a missing children hotline and serve as the national clearinghouse for information related to these issues.” Nat’l Ctr. for Missing & Exploited Children, http:// www.missingkids.com/NCMEC (last visited Sept. 9, 2013); see also Missing Children's Assistance Act, Pub.L. No. 98-473, div. II, § 660, 98 Stat. 2125 (1984) (codified as amended at 42 U.S.C. §§ 5771-80a).
     
      
      .It is unclear exactly when the government notified Amy about Lundquist, but the district court found that this occurred sometime between March 5, 2010, when Lundquist was arrested, and at the latest April 14, 2011, when Amy submitted her request for restitution. See United States v. Lundquist, 847 F.Supp.2d 364, 374 (N.D.N.Y.2011).
     
      
      . These losses were as follows: $512,681 for future counseling expenses (consisting of $7,800 per year for counseling costs from 2009 through the estimated remainder of Amy’s life, and $120,000 for three expected institutionalizations throughout her life); $2,855,173 for lost wages and benefits for the remainder of her life; $3,500 in attorney’s fees; and $13,305 in costs for all of Dr. Sil-berg’s and Dr. Smith's expert reports. Lundquist, 847 F.Supp.2d at 376-78. The district court excluded fees for an expert report that was not submitted to the court and unsubstantiated costs ambiguously labeled "Miscellaneous Expenses, Copying, [and] Records.” Id. at 378 (alteration in original and internal quotation marks omitted).
     
      
      . The district court concluded, in the alternative, that 18 U.S.C. § 2255(a) creates a presumption that the minimum value of Amy’s losses is $150,000, and that $5,000 in nominal damages was reasonable under section 2259. Lundquist, 847 F.Supp.2d at 378-79. On appeal, the government concedes that the section 2255(a) presumption should not apply to restitution orders under section 2259 and argues that it is not necessary to consider nominal damages because the district court correctly found that Lundquist proximately caused a greater amount of actual losses. The notion that section 2259 authorizes presumed statutory or nominal damages is inconsistent with our holding in Aumais that the losses must be proximately caused by the offense. See United States v. Aumais, 656 F.3d 147, 153 (2d Cir.2011).
     
      
      . The district court neglected to include in its order the repayment schedule required by 18 U.S.C. § 3664(f)(2) and has sua sponte issued an application for leave to correct this clerical mistake. Because we remand for recalculation of the amount of restitution, we deny that application as moot.
     
      
      . We reject Lundquist's argument that Amy was not his victim because he did not know that the child in the image was Amy. Although the term " 'knowingly' ... applies to every element” of Lundquist’s crimes, United States v. Williams, 553 U.S. 285, 294, 128 S.Ct. 1830, 170 L.Ed.2d 650 (2008), the identity of the child is not an element of those crimes, see 18 U.S.C. § 2252A(a)(2)(A), (a)(5)(B). Lundquist needed to know only that the image was "child pornography,” id., defined as "sexually explicit visual portrayals that feature children,” Williams, 553 U.S. at 288, 128 S.Ct. 1830. By pleading guilty, Lundquist admitted he had such knowledge and therefore knew that each child in those images was a victim of his offense. See Aumais, 656 F.3d at 152.
     
      
      . We note that as of the date of this opinion, every circuit except the Fifth Circuit has concluded, based on either the text or background common law principles, that section 2259 contains a proximate cause requirement. See United States v. Benoit, 713 F.3d 1, 20 (10th Cir.2013); United States v. Fast, 709 F.3d 712, 721-22 (8th Cir.2013); United States v. Gamble, 709 F.3d 541, 546-47 (6th Cir.2013); United States v. Laraneta, 700 F.3d 983, 990 (7th Cir.2012); United States v. Burgess, 684 F.3d 445, 459 (4th Cir.2012); United States v. Kearney, 672 F.3d 81, 95-96 (1st Cir.2012); United States v. Kennedy, 643 F.3d 1251, 1261 (9th Cir.2011); United States v. Monzel, 641 F.3d 528, 535-36 (D.C.Cir.2011); United States v. McDaniel, 631 F.3d 1204, 1208-09 (11th Cir.2011); United States v. Crandon, 173 F.3d 122, 125 (3d Cir.1999). But see In re Amy Unknown, 701 F.3d 749, 774 (5th Cir.2012) (en banc) (holding that there is no proximate cause requirement for the five enumerated categories of compensa-ble losses). The Supreme Court recently granted certiorari to resolve this circuit split. See Paroline v. United States, - U.S. -, 133 S.Ct. 2886, - L.Ed.2d - (2013) (mem.).
     
      
      . See, e.g., Gamble, 709 F.3d at 549-50 (“Generally if the injury is the type that the statute was intended to prohibit, it is more likely to be proximately caused.”); Kearney, 672 F.3d at 99 (rejecting "the theory that the victim of child pornography could only show causation if she focused on a specific defendant's viewing and redistribution of her images and then attributed specific losses to that defendant's actions”); Burgess, 684 F.3d at 460 ("While the district court is not required to justify any award with absolute precision, the amount of the award must have a sufficient factual predicate.”); McDaniel, 631 F.3d at 1207 (considering proximate cause to be a factual finding reviewed for clear error); see also Kennedy, 643 F.3d at 1263 (citing McDaniel with approval and noting that in that case, "the government established proximate cause through evidence that; (1) NCMEC had notified the victim that the defendant possessed her image, (2) the victim suffered when she received such notices, and (3) this suffering necessitated further therapy”)-
     
      
      . See also Palsgraf v. Long Island R.R. Co., 248 N.Y. 339, 353-54, 162 N.E. 99 (1928) (Andrews, J., dissenting) ("[T]he problem of proximate cause is not to be solved by any one consideration.... There are no fixed rules to govern our judgment.”); Dan B. Dobbs et al., The Law of Torts § 199 (2d ed. supp.2013) ("The [proximate cause] rules call for judgments, not juggernauts of logic.... [N]o version of the rules can be expected to assure any given answer in a particular case....”). This inherent difficulty has led many to criticize the various articulations and applications of the proximate cause standard as "arbitrar[y],” CSX Transp., Inc. v. McBride, - U.S. -, 131 S.Ct. 2630, 2637, 180 L.Ed.2d 637 (2011) (quotation omitted), and "confusing],” Exxon Co., U.S.A. v. Sofec, Inc., 517 U.S. 830, 838, 116 S.Ct. 1813, 135 L.Ed.2d 113 (1996). Despite its shortcomings, however, proximate cause remains an "enduring common law concept that is useful despite its imprecision.” CSX Transp., Inc., 131 S.Ct. at 2645 (Roberts, C.J., dissenting).
     
      
      . See, e.g., Kearney, 672 F.3d at 100 (considering fact that "[the victim’s] lawyer received a victim notification letter, and [the victim] affirmatively requested restitution” as evidence of the victim’s knowledge); McGarity, 669 F.3d at 1269 (characterizing Aumais as holding that "proximate cause cannot exist without a showing that a victim of sexual abuse learns of a defendant's harmful possession”); McDaniel, 631 F.3d at 1209 (affirming finding of proximate cause based on evidence that victim received notice of the defendant’s arrest and these notices typically aggravated victim’s injuries).
     
      
      . Pursuant to 18 U.S.C. § 3771(a), a crime victim has specified rights, including the "right to be reasonably heard at any public proceeding in the district court involving ... plea [or] sentencing,” the "reasonable right to confer with the attorney for the Government in the case,” and the "right to full and timely restitution as provided in law.” 18 U.S.C. § 3771(a)(4)-(6). The government "shall make [its] best efforts to see that crime victims are notified of, and accorded, the rights described” above. Id. § 3771(c)(1). Generally, the DOJ satisfies its obligations under section 3771 through an automated program, the Victim Notification System ("VNS”). See Victim Notification Program (VNS), U.S. Dep’t of Justice, http://www.justice.gov/criminal/ vns/ (last visited Sept. 9, 2013); see also Kearney, 672 F.3d at 85 n. 4 (describing the VNS). With respect to child pornography cases, the Child Pornography Victims Assistance Program of the FBI contacts victims to determine whether they would like to continue receiving notification letters or to opt out of such notices in the future. See Office for Victim Assistance, Fed. Bureau of Investigation, Child Pornography Victim Assistance, available at http://www.fbi.gov/stats-services/ victim_assistance/brochures-handouts/ cpva.pdf.
     
      
      . But see Benoit, 713 F.3d at 22 & n. 8 (rejecting the "implicit” use of a per capita calculation, but noting that this method would be permissible if the court made "factual findings as to whether the number of judgments was approximately equal to the number of end-users [of the pornography] [and] whether Benoit caused approximately the same amount of damages as other end-users”); Kennedy, 643 F.3d at 1266 (declining to rule out the possibility that a per capita calculation could satisfy the proximate cause standard, but noting that the standard would "continue to present serious obstacles for victims seeking restitution in these sorts of cases”).
     
      
      . See 4 Leonard B. Sand et al., Modern Federal Jury Instructions (Civil) ¶ 77.01, al Instruction 77-3 (2011) ("[Y]ou may award compensatory damages only for injuries that a plaintiff proves were proximately caused by a defendant's allegedly wrongful conduct.... [T]he law does not require a plaintiff to prove the amount of his or her losses with mathematical precision, but only with as much definiteness and accuracy as the circumstances permit.”).
     
      
      . We acknowledge that the denominator will increase as more possessors are convicted, creating a potential anomaly where the defendants convicted earlier will pay a larger share. The numerator, however, also has the potential to increase if the victim can show additional damages attributable to re-victimization.
     
      
      . These amounts were based on the estimates in the 2008 Smith Report. Although that report was prepared before Lundquist's arrest, the district court may rely upon it if it finds that Dr. Smith's calculations still provide a reasonable estimate of Amy’s future losses.
     
      
      . We note that there is a difference of opinion on this issue. See In re Amy Unknown, 701 F.3d at 770 (concluding that “nothing in § 3664 forbids it, either expressly or through implication”); see also United States v. Hargrove, 714 F.3d 371, 377-78 (6th Cir.2013) (Clay, concurring in part and in judgment) (arguing that Fast, Laraneta, and Aumais “overread the statute”); Fast, 709 F.3d at 727 n. 8 (Shepherd, concurring in part and dissenting in part) (same).
     
      
      . See Gamble, 709 F.3d at 551-52 ("Each individual defendant is not necessary to cause the aggregate harm — it would have happened without him.”); Laraneta, 700 F.3d at 992 ("If the defendant in this case is not responsible for the viewing of the images ... by even one person besides himself, joint liability would be inappropriate.”); Burgess, 684 F.3d at 459 (“[Tjhose individuals cannot have proximately caused a victim the same injury.”); Monzel, 641 F.3d at 539 ("Because the record does not show that Monzel proximately caused all of Amy’s injuries, the district court did not clearly and indisputably err by declining to impose joint and several liability on him for the full [losses] she seeks.”).
     