
643 F.3d 1251 (2011)
UNITED STATES of America, Plaintiff-Appellee,
v.
Joshua Osmun KENNEDY, Defendant-Appellant.
No. 10-30065.
United States Court of Appeals, Ninth Circuit.
Argued and Submitted February 9, 2011.
Filed July 11, 2011.
*1252 Suzanne Lee Elliott, Law Offices of Suzanne Lee Elliott, Seattle, WA, for appellant Joshua Osmun Kennedy.
Helen J. Brunner, Assistant United States Attorney, Seattle, WA, for appellee United States of America.
Before: BETTY B. FLETCHER, RICHARD A. PAEZ, and SANDRA S. IKUTA, Circuit Judges.

OPINION
IKUTA, Circuit Judge:
Joshua Osmun Kennedy was convicted by a jury of possessing and transporting child pornography. He appeals his conviction, his sentence, and the district court's order directing him to pay $65,000 in restitution to two victims. We affirm Kennedy's conviction and sentence. Because the government failed to carry its burden of proving that Kennedy's offense conduct proximately caused the losses incurred by the victims, we vacate the restitution order.

I
On November 9, 2007, Kennedy arrived at Sea-Tac Airport from an overseas trip. After a secondary inspection of his baggage by U.S. Customs and Border Patrol revealed images of an "underage-looking" nude female "in different sexual positions" on his laptop computer, Kennedy's computer was seized. Forensic specialists later *1253 found 30 images of child pornography among the laptop's active files and approximately 5,000[1] images in the deleted cache files.[2] On November 5, 2008, a grand jury indicted Kennedy on one count of possession of child pornography, in violation of 18 U.S.C. § 2252A(a)(5)(B) (2006),[3] and one count of transportation of child pornography, in violation of 18 U.S.C. § 2252(a)(1) (2006).[4]
Prior to trial, Kennedy moved pursuant to Federal Rule of Evidence 403 to exclude the anticipated testimony of five government witnesses. Each witness was a law enforcement officer who had been involved in a sexual abuse investigation relating to one of the individuals depicted in the images found on Kennedy's computer. The government intended to use their testimony to prove that at least some of the individuals depicted were minors and real people, and that the images Kennedy possessed had moved in interstate commerce.[5] Kennedy argued that the testimony of the officers would be "cumulative," of "marginal relevance," and highly inflammatory, and stated that he did not "intend to argue that the images [we]re computer simulations" and would not contest that the *1254 "persons depicted in the images are minors." The district court denied Kennedy's motion. The five witnesses testified on the second day of trial. On August 27, 2009, after three days of trial, a jury convicted Kennedy on both counts.
Kennedy was sentenced on February 19, 2010. At the hearing, Kennedy's counsel argued that possession of child pornography is a lesser-included offense of transportation of child pornography and thus that Kennedy's dual convictions violated double jeopardy. The court agreed and stated that it would remedy the violation by exercising its discretion to vacate the possession conviction. It gave three reasons for this decision: first, "[t]he evidence clearly and unambiguously demonstrate[d] that [Kennedy] knowingly and intentionally transported a large number of pornographic images of children into this country"; second, it would be "paradoxical" to give Kennedy a shorter sentence just because the government chose also to charge him with a less serious offense;[6] and third, nothing in 18 U.S.C. § 2252(a)(1) suggested that Congress did not intend to apply that statute to someone in Kennedy's situation. The court therefore concluded that it "would be an improper use of its discretion" to vacate the greater offense. Proceeding to calculate a sentence based on the transportation conviction alone, the court calculated a guidelines range of 135 to 168 months. Kennedy, the government, and the Probation Office all agreed that a downward departure was warranted. The court imposed the statutory minimum sentence of 60 months. After discussing the seriousness of Kennedy's offense and his background and characteristics (including a difficult childhood, addictions to child pornography and illegal substances, and the fact that Kennedy had twice violated the conditions of his pre-trial supervision), the court also imposed a 15-year period of supervised release. The special conditions of release included active participation in "a certified sexual deviancy program," participation in plethysmograph testing "as determined by [the deviancy program] therapist," a ban on direct or indirect contact with minors "unless accompanied and supervised" by an adult approved by the therapist or probation officer, and a requirement that Kennedy have his place of residence "preapproved by the probation officer."
On April 21, 2010, the court held a hearing to consider what amount of restitution to award under 18 U.S.C. § 2259[7] to two *1255 victims whose images had been found on Kennedy's computer: "Amy" and "Vicky." Amy and Vicky claimed losses of $3 million and $227,000, respectively.
In support of her claim, Amy submitted a victim impact statement, a psychological evaluation, and a report from the Smith Economics Group calculating the value of "(1) the loss of wages and employee benefits; (2) the present value of future treatment and counseling costs; and (3) the reduction in value of life ... also known as loss of enjoyment of life." Amy's victim impact statement explained that she was "a real victim of child pornography" because the dissemination of the images meant that she was "being exploited and used every day and every night somewhere in the world by someone." Because of the pictures, "[w]hat happened to me hasn't gone away. It will never go away." The psychological evaluation confirmed that Amy had experienced "a resurgence of ... trauma" upon learning that the images were circulating on the Internet: "[Amy] feels that her privacy has been invaded on a fundamental level [and] fears the unknown and unnamed people who continued to be looking at these pictures of her for their own perverse interests or to `groom' other children into these acts."
Vicky submitted a forensic psychological examination and a printout from an online message board on which anonymous users of child pornography were discussing images of her in graphic detail. The psychological report included a lengthy analysis of the effects of Vicky's childhood sexual abuse. Vicky reported being "stunned" when she was learned that images of her abuse had been disseminated on the Internet, and told her psychologist that she began "obsessively ruminating about scenarios of males in her community having viewed the videos." She also started having night terrors and panic attacks. In an undated victim impact statement, Vicky wrote:
We now have in our house boxes full of victim notifications[8] from cases all around the country involving pornographic images of me. Practically every time I've went to get the mail, there have been two or three of these notifications. They are constant reminders of the horrors of my childhood.
The psychological report also included an estimate of the future costs of Vicky's therapy. None of Amy or Vicky's submitted materials mentioned the defendant Kennedy.
The government asked the court to order Kennedy to pay "full restitution" to Amy and Vicky on a theory of "joint and several liability" (i.e. $3 million for Amy and $227,000 for Vicky) or, in the alternative, $1000 per image. Kennedy responded that the government had failed to meet its burden of proving the victims' entitlement to any amount of restitution under 18 U.S.C. § 2259.
While agreeing with Kennedy that § 2259 "requires some degree of causal *1256 connection between the victims' losses and the defendant's conduct," the court found the detail Amy and Vicky provided in their communications sufficient to demonstrate that connection. In particular, Amy and Vicky had shown themselves to be "abundantly aware of [the] repeated violations they experience, knowing their images involving child pornography are played over and over again," and the expert evaluations confirmed that Amy and Vicky suffered psychological damage from the knowledge that people were viewing the images. Therefore, the court reasoned that Amy and Vicky were "victims" of Kennedy's offense and were accordingly entitled to compensation.
Noting that "[t]he government and the defendant have each posited recommended amounts of restitution ranging from zero by the defendant to the full amount suggested by the government," the court stated that it believed that the "amount of $1,000 per image" was "reasonable." The district court therefore entered an order directing Kennedy to pay $17,000 to Amy (for the 17 images of her Kennedy had on his computer) and $48,000 to Vicky (for the 48 images of her Kennedy had on his computer).

II
On appeal, Kennedy argues that the district court erred in admitting the testimony of the five law enforcement officers, abused its discretion in choosing to vacate his possession conviction rather than his transportation conviction,[9] abused its discretion in imposing a 15-year period of supervised release, and erred in ordering restitution under 18 U.S.C. § 2259. We address each argument in turn.

A
Kennedy's first claim is that the district court erred in admitting the testimony of the five law enforcement officers because the officers' testimony was unduly prejudicial, and the district court failed to consider less prejudicial evidentiary alternatives. The government sought to introduce these witnesses to testify to three elements: that the individuals depicted in the images were minors and real persons rather than computer simulations (elements of both the possession and transportation counts, see 18 U.S.C. §§ 2252A(a)(5)(B), 2252(a)(1), 2256 (2006)), and that the images "ha[d] been mailed, or shipped or transported in interstate or foreign commerce by any means" (an element of the possession count, see § 2252A(a)(5)(B) (2006)).
We review a trial court's evidentiary rulings for abuse of discretion. See United States v. Higuera-Llamos, 574 F.3d 1206, 1209 (9th Cir.2009). While a trial court is to be given "`wide latitude' when it balances the prejudicial effect of proffered evidence against its probative value" under Rule 403, id. (quoting United States v. Spencer, 1 F.3d 742, 744 (9th Cir.1993)), "sound judicial discretion" requires the court, in assessing the probative value of challenged evidence, to take into account the availability of alternative evidence with "substantially the same or greater probative value but a lower danger of unfair prejudice," Old Chief v. United States, 519 U.S. 172, 183, 117 S.Ct. 644, 136 L.Ed.2d 574 (1997).
Kennedy argues that the district court erred in admitting the officers' testimony, given Kennedy's offer not to dispute that *1257 the individuals depicted in the images were minors and real people, and the availability of other evidence such as the testimony from the government's expert witness (a nurse practitioner) as to the probable ages of the individuals depicted in the images, and the images themselves. Kennedy's argument fails because neither his offer not to dispute that the minors were real people, nor the other evidence provides "substantially the same or greater probative value" as the officers' testimony. Id. at 183, 117 S.Ct. 644. Kennedy's offer to refrain from disputing whether the individuals in the images were minors and real people is not equivalent to an affirmative stipulation that they were, and so his offer did not relieve the government of its burden of proving those facts beyond a reasonable doubt. United States v. Salcido, 506 F.3d 729, 733 (9th Cir.2007) (per curiam). Moreover, the testimony of officers who had met the victims and could authenticate their birth certificates was more probative on the question whether the victims were minors and real people than was the testimony of the nurse practitioner or the mere introduction of the images themselves. Further, neither Kennedy's offer to refrain from disputing the government's evidence, nor the testimony of the nurse practitioner, eliminated the government's burden to prove that the images had moved through interstate commerce, an element of the possession offense. Because Kennedy's offer not to dispute certain facts and the proposed alternative methods of proof were not true evidentiary alternatives to the law enforcement officers' testimony, the district court did not err in refusing to discount the probative value of that testimony. See United States v. Ganoe, 538 F.3d 1117, 1124 (9th Cir. 2008).
Kennedy also argues that the challenged testimony was unduly prejudicial because it "was offered to invoke an emotional response from the jury." Again, we disagree. The testimony of each witness followed the same pattern: First, the officer described his or her involvement in an investigation regarding the sexual abuse of a particular minor victim. Second, the witness identified the location where that abuse was determined to have taken place. Third, the witness stated whether he or she had met the victim and the victim's age at that time, with four of the five witnesses also authenticating the victim's birth certificate. Finally, the witness was shown one or more images from Kennedy's computer and identified each image as being of the minor child in question (either by recognizing the victim or by recognizing distinctive items in the image, such as a chair or bedspread). The prosecution then offered the relevant images in evidence. The testimony of each witness was brief and narrowly focused on relevant issues. While any evidence relating to the sexual abuse of children is likely to stir emotion, the testimony of these particular witnesses was hardly calculated to inflame the jury. Rule 403 does not require a trial court to "scrub the trial clean of all evidence that may have an emotional impact." Ganoe, 538 F.3d at 1123-24(quoting United States v. Morales-Aldahondo, 524 F.3d 115, 120 (1st Cir.2008)) (internal quotation mark omitted).[10]
*1258 We therefore hold that the district court did not abuse its discretion in admitting the testimony of the five law enforcement officers.

B
We next turn to Kennedy's argument that the district court abused its discretion in choosing to vacate his conviction for possession of child pornography, in violation of 18 U.S.C. § 2252A(a)(5)(B) (2006), rather than his conviction for transportation of child pornography, in violation of 18 U.S.C. § 2252(a)(1) (2006). At sentencing, the district court determined that the Double Jeopardy Clause prohibits a defendant from being convicted and sentenced for both possession and transportation of the same images of child pornography.[11] The district court was therefore obliged to exercise its discretion to vacate one of the two convictions. See United States v. Hector, 577 F.3d 1099, 1101, 1103 (9th Cir.2009). We have held that, in deciding whether to vacate a lesser-included offense or a greater offense, a district court should be guided by the sentencing factors set forth in 18 U.S.C. § 3553(a), and "`should' exercise its discretion to vacate the lesser-included offense, absent unusual circumstances and compelling reasons to vacate the greater offense." United States v. Maier, 639 F.3d 927, 932-33 (9th Cir.2011) (citing United States v. Jose, 425 F.3d 1237, 1247 (9th Cir.2005)).
The district court did not have the benefit of our decision in United States v. Maier when it imposed Kennedy's sentence. While the court did not explicitly mention the § 3553(a) sentencing factors in explaining its decision to vacate Kennedy's possession conviction, it nevertheless considered the relevant factors. The district court's reasoning that Kennedy was clearly guilty of the more serious transportation offense, that a court should not impose a shorter sentence solely because the government chose to charge both a graver and a less serious offense, and that Congress intended courts to impose the longer sentence for Kennedy's offense conduct, demonstrated "an acknowledgment of the need for a sentence `to reflect the seriousness of the offense' and `provide just punishment' for it," as well as "the need `to afford adequate deterrence to criminal conduct.'" Id. at 933 (quoting 18 U.S.C. § 3553(a)(2)(A), (B)). The district court's reasoning also reflected the court's recognition of the need for Kennedy's sentence to "promote respect for the law." 18 U.S.C. § 3553(a)(2)(A). On appeal, Kennedy has failed to present any "unusual circumstances and compelling reasons" why the district court should instead have vacated the transportation conviction. Id. at 932. In other words, the district court's *1259 exercise of discretion was consistent with our holding in Maier. We therefore affirm its decision to vacate Kennedy's possession conviction.

C
Kennedy next argues that the district court failed to adequately justify its imposition of a 15-year period of supervised release and that the length and conditions of release were substantively unreasonable.[12] We give considerable deference to a district court's decisions regarding supervised release because the district court has "at its disposal all of the evidence, [as well as] its own impression of [the] defendant," when making those decisions. United States v. Stoterau, 524 F.3d 988, 1002 (9th Cir.2008) (quoting United States v. Weber, 451 F.3d 552, 557 (9th Cir.2006)). In this case, the court had before it the pre-sentence report, the evidence adduced at trial, a report from a sexual deviancy evaluator, letters of support from Kennedy's family, and Kennedy's own statements at sentencing. In reaching its decision to impose a 15-year period of supervision, the court clearly considered this evidence in light of the relevant factors, including the serious nature of Kennedy's offense, his need for rehabilitation from his child pornography and substance addictions, and his background and characteristics (including the fact that he had twice violated the conditions of his pre-trial supervision). See 18 U.S.C. § 3553(a). We therefore hold that the district court adequately justified its decision, see United States v. Autery, 555 F.3d 864, 871-72 (9th Cir.2009), and affirm the duration of Kennedy's supervised release as substantively reasonable, see id. at 871.
We likewise conclude that the special conditions of release are reasonable. A court has wide discretion to fashion special conditions so long as they are "reasonably related" to the goals of supervised release, § 3553(a)(2)(B)-(D), and involve "no greater deprivation of liberty than is reasonably necessary," § 3583(d)(2). Kennedy argues that the special conditions imposed on him will result in a "greater deprivation of liberty" than is necessary to achieve those goals. Kennedy is mistaken; the court went to great lengths to tailor the special conditions of release to Kennedy's situation, such as limiting his participation in plethysmograph testing to only such times as are "determined by [his] therapist" to be necessary, and including an exception in the ban on contact with minors for when Kennedy is "accompanied and supervised" by an adult approved by the therapist or probation officer. Moreover, because the ban on contact with minors and the residency restrictions are reasonably related to the deterrent and public protection goals of supervised release, see § 3553(a)(2)(B)-(D), as well as the goal of rehabilitation, we see no error in the district court's decision to impose those conditions for the full term of supervised release rather than for just the duration of Kennedy's sexual deviancy treatment.

D
Kennedy's final argument is that the district court's restitution order was unlawful under 18 U.S.C. § 2259 because the government failed to prove, by a preponderance of the evidence, the measure of the losses to Amy and Vicky that were proximately caused by Kennedy's offense. This argument requires us to consider what constitutes a sufficient chain of causation *1260 between the defendant's offense conduct and the victims' losses to justify an award of restitution under § 2259. This difficult issue of statutory interpretation has been considered, but not satisfactorily resolved, by several of our sister circuits. See United States v. Monzel, 641 F.3d 528 (D.C.Cir.2011); In re Amy Unknown, 636 F.3d 190 (5th Cir.2011); United States v. McDaniel, 631 F.3d 1204 (11th Cir.2011).

1
We begin by outlining the relevant requirements of 18 U.S.C. § 2259. Because "[f]ederal courts have no inherent power to award restitution," we may order restitution only when and to the extent authorized by statute. United States v. Gossi, 608 F.3d 574, 577 (9th Cir.2010) (quoting United States v. Follet, 269 F.3d 996, 998 (9th Cir.2001)) (internal quotation marks omitted). Whether an order of restitution falls "within the bounds of the [relevant] statutory framework" is a question of law that we review de novo. United States v. Lazarenko, 624 F.3d 1247, 1249 (9th Cir.2010) (quoting United States v. Marks, 530 F.3d 799, 811 (9th Cir.2008)).
Section 2259 makes restitution "mandatory" for offenses involving sexual exploitation and other abuse of children. See § 2259(a), (b)(4). The order of restitution must "direct the defendant to pay the victim... the full amount of the victim's losses." § 2259(b)(1). The statute provides definitions of these key terms. It defines the word "victim" as "the individual harmed as a result of a commission of a crime under this chapter." § 2259(c). Because a victim must be "harmed as a result of" the defendant's criminal conduct, this language implies that the government must establish a causal connection between the defendant's offense and the harm to the victim. Courts have identified several ways in which the individuals depicted in images of child pornography are harmed by the circulation and viewing of those images. See New York v. Ferber, 458 U.S. 747, 759, 102 S.Ct. 3348, 73 L.Ed.2d 1113 (1982) (explaining that the suffering of victims of childhood sexual abuse is "exacerbated by [the] circulation" and viewing of images documenting that abuse). These injuries include "the emotional and psychic" pain of knowing that the images are being viewed, id. at 759 n. 10, 102 S.Ct. 3348, as well as the repeated violations of the individual's privacy interests, id. at 758 n. 9, 102 S.Ct. 3348. See also Ashcroft v. Free Speech Coal., 535 U.S. 234, 249, 122 S.Ct. 1389, 152 L.Ed.2d 403 (2002) ("Like a defamatory statement, each new publication [of the images] ... cause[s] new injury to the child's reputation and well-being."). In the sentencing context, we have likewise held that the minors depicted in child pornography should be considered the "victims" of a defendant who possesses or trades in images of their abuse. See United States v. Blinkinsop, 606 F.3d 1110, 1117-18 (9th Cir.2010); United States v. Boos, 127 F.3d 1207, 1210-11 (9th Cir.1997).
The statute also defines the "full amount of the victim's losses":
(3) Definition.For purposes of this subsection, the term "full amount of the victim's losses" includes 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.
*1261 18 U.S.C. § 2259(b)(3). We have interpreted this language as allowing restitution only for losses that were "proximately" caused by the defendant's conduct. See United States v. Laney, 189 F.3d 954, 965 (9th Cir.1999) (holding, based partly on the "proximate result" language in § 2259(b)(3)(F), that § 2259 "incorporates a requirement of proximate causation").[13]
Finally, § 2259 states that the order of restitution is to "be issued and enforced in accordance with section 3664 in the same manner as an order under section 3663A."[14] 18 U.S.C. § 2259(b)(2). Section 3664 provides that the burden of proving the amount of the victim's losses is on the government. 18 U.S.C. § 3664(e). In carrying this burden, we have held that the government must provide the court with enough evidence to allow the court to estimate the "full amount of the victim's losses" with "some reasonable certainty." United States v. Doe, 488 F.3d 1154, 1159-60 (9th Cir.2007). While this is not a requirement "approaching mathematical precision," id. at 1160, a restitutionary award under § 2259 will be improper if the district court must "engage in ... arbitrary calculations" to determine the amount of the victim's losses, Laney, 189 F.3d at 967 n. 14.
In ruling that courts may order restitution under § 2259 only for losses proximately caused by the defendant's offense, Laney did not provide a framework for determining the scope of this proximate cause limitation. However, we have frequently considered proximate causation in interpreting other statutory restitution schemes, specifically the Victim and Witness Protection Act of 1982 (VWPA), 18 U.S.C. §§ 1512-1515, 3663-3664, and the Mandatory Victims Restitution Act of 1996 (MVRA), 18 U.S.C. §§ 3663A, 3613A.[15] Because both statutes define a "victim" as "a person directly and proximately harmed as a result of the commission of an offense for which restitution may be ordered," §§ 3663(a)(2), 3663A(a)(2), we have held that restitution under these statutes may be awarded only for losses for which the defendant's conduct was an "actual and proximate cause." United States v. Peterson, 538 F.3d 1064, 1068 (9th Cir.2008); see also United States v. Hackett, 311 F.3d 989, 993 (9th Cir.2002) (stating that the loss must be "directly related to the defendant's conduct" (quoting United States v. Gamma Tech Indus., Inc., 265 F.3d 917, 928 (9th Cir.2001))). As noted above, we have similarly interpreted the text of § 2259 as making defendants liable only *1262 for losses sustained as a proximate result of their criminal offenses. See Laney, 189 F.3d at 965. Given that § 2259(b)(2) requires courts to look to the VWPA and the MVRA for direction in issuing and enforcing an order of restitution under § 2259, see 18 U.S.C. § 2259(b)(2), and in light of the similar restitutionary purpose of all three statutes, we conclude that our case law on proximate cause in the context of the VWPA and the MVRA may properly inform our analysis of the appropriate standard for awards of restitution under § 2259.
In determining what constitutes proximate cause under the VWPA and MVRA, we have attempted to steer a middle course, one that avoids imposing liability on defendants whose conduct is too remote from or too tangential to a victim's specific losses, while still ensuring that defendants pay restitution for the losses to which their offense conduct contributed. Thus, we have reversed restitutionary awards where there was an intervening cause, unrelated to the defendant's offense, between the defendant's offense and the victim's specific loss. In United States v. Meksian, 170 F.3d 1260 (9th Cir.1999), for example, we held that a defendant who had submitted a fraudulent loan application to a bank did not need to pay restitution for the bank's loss of the entire value of the property securing the loan, because that loss was attributable to an inaccurate environmental report, not to any misrepresentation by the defendant. See id. at 1263. Similarly, we have held that "[r]estitution is not available for consequential losses, or other losses too remote from the offense of conviction," such as attorney's fees, wages for trial witnesses, or profits from lost corporate opportunities. United States v. Rodrigues, 229 F.3d 842, 845 (9th Cir.2000) (citation omitted) (citing cases).
On the other hand, "we have approved restitution awards that included losses at least one step removed from the offense conduct itself," Gamma Tech, 265 F.3d at 928, so long as the intervening cause was "directly related to the offense conduct," Meksian, 170 F.3d at 1263. See also United States v. Keith, 754 F.2d 1388, 1393 (9th Cir.1985). Thus in Gamma Tech, we upheld a restitutionary award to a Navy contractor whose employee had obtained kickbacks from subcontractors hired for Navy jobs; we concluded that the contractor's lost profits due to inflated payments to those subcontractors were directly related to the offense conduct, the payment of the kickbacks. See 265 F.3d at 928. Moreover, even where factors other than the defendant's conduct contributed to a specific loss, a defendant may be held liable for restitution so long as the defendant's conduct was a "material and proximate cause" of the loss. See Peterson, 538 F.3d at 1076 (quoting United States v. Spicer, 57 F.3d 1152, 1159 (D.C.Cir.1995)); see also Doe, 488 F.3d at 1160 (stating that the government need not show "that a defendant's conduct was the sole and total cause of a victim's loss," so long as "the additional strain or trauma stemming from [the defendant's] actions was a substantial factor" in causing that loss (quoting Crandon, 173 F.3d at 126)).
Applying this guidance in the § 2259 context, we conclude that for purposes of determining proximate cause, a court must identify a causal connection between the defendant's offense conduct and the victim's specific losses. There may be "multiple links in the causal chain," Peterson, 538 F.3d at 1075, but the chain "may not extend so far, in terms of the facts or the time span, as to become unreasonable," Gamma Tech, 265 F.3d at 928. Although the "[d]efendant's conduct need not be the sole cause of the loss," Peterson, 538 F.3d at 1075 (quoting Gamma Tech, 265 F.3d at 928), it must be a "material *1263 and proximate cause," id. at 1076(quoting Spicer, 57 F.3d at 1159), and "any subsequent action that contributes to the loss, such as an intervening cause, must be directly related to the defendant's conduct," Gamma Tech, 265 F.3d at 928.
Accordingly, in order to award restitution under § 2259(b)(1), a district court must make three determinations: (1) that the individual seeking restitution is a "victim" of the defendant's offense, § 2259(b)(1), (c); (2) that the defendant's offense was a proximate cause of the victim's losses, see Laney, 189 F.3d at 965, pursuant to the guidance discussed above from our VWPA and MVRA precedents; and (3) that the losses so caused can be calculated with "some reasonable certainty," see Doe, 488 F.3d at 1160.

2
Applying this framework to the case at hand, we begin with the question whether Amy and Vicky are "victims" for purposes of § 2259. We agree with the district court that they are. Amy and Vicky presented ample evidence that the viewing of their images caused them emotional and psychic pain, violated their privacy interests, and injured their reputation and well-being. See Ferber, 458 U.S. at 759 & n. 10, 102 S.Ct. 3348; Free Speech Coal., 535 U.S. at 249, 122 S.Ct. 1389. Amy, for example, stated that her "privacy ha[d] been invaded" and that she felt like she was "being exploited and used every day and every night." Vicky described having night terrors and panic attacks due to the knowledge that her images were being viewed online. Even without evidence that Amy and Vicky knew about Kennedy's conduct, the district court could reasonably conclude that Amy and Vicky were "harmed as a result of" Kennedy's participation in the audience of individuals who viewed the images. See § 2259(c). We therefore hold that Amy and Vicky are "victims" of Kennedy's offense.
The second step of the restitution analysis demands a closer causal connection. At this step, the government must prove by a preponderance of the evidence that Kennedy's offenses proximately caused the losses incurred by Amy and Vicky. See Laney, 189 F.3d at 965. As explained above, although the government need not prove that Kennedy's conduct was the sole cause of the victims' losses, it must prove that his conduct was a "material and proximate cause" of those losses. Peterson, 538 F.3d at 1076 (quoting Spicer, 57 F.3d at 1159). Likewise, while there may be intervening links in the chain between Kennedy's conduct and the victims' losses, such links must be related to Kennedy's conduct. See Gamma Tech, 265 F.3d at 928.
The government has not carried its burden here, because it has not introduced any evidence establishing a causal chain between Kennedy's conduct and the specific losses incurred by Amy and Vicky. The government did not show how Kennedy's actions in transporting the images caused Amy's lost income and loss of enjoyment of life or Amy and Vicky's future counseling costs. Nor did the government introduce evidence that Amy and Vicky could have avoided certain losses had Kennedy not transported the images. Indeed, the government introduced no evidence that Amy and Vicky were even aware of Kennedy's conduct. By contrast, in United States v. McDaniel, 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, a cost recognized under § 2259(b)(3)(A). 631 F.3d at 1207, 1209.
Rather than proving a causal relationship between Kennedy's actions and the *1264 victims' losses, the government's evidence showed only that Kennedy participated in the audience of persons who viewed the images of Amy and Vicky. While this may be sufficient to establish that Kennedy's actions were one cause of the generalized harm Amy and Vicky suffered due to the circulation of their images on the internet, it is not sufficient to show that they were a proximate cause of any particular losses. Indeed, we have found no case in this circuit (and the government has cited none) in which a relationship as remote as that between Kennedy's conduct and the victims' losses in this case was held sufficient for an award of restitution.
Under certain circumstances, we have upheld restitutionary awards under the VWPA and MVRA when the defendant's offense was merely one part of a larger problem that caused the victim's losses. In those cases, however, we placed great weight on the evidence establishing that the defendant's conduct directly contributed to the claimed losses. Thus, in Peterson, we upheld a restitutionary award for the government's losses stemming from foreclosures on 43 homes that individual buyers had purchased with the help of the defendants' fraudulent real estate scheme. See Peterson, 538 F.3d at 1077. Although the defendants argued that the foreclosures (and thus the government's losses) were caused by the buyers' unrelated financial difficulties, we focused on evidence establishing that the government would not have incurred such losses if the defendants' fraud had not allowed the buyers to purchase the houses in the first place. See id. Similarly, in United States v. De La Fuente, 353 F.3d 766 (9th Cir.2003), we held a defendant liable for the full costs of evacuating and decontaminating a post office because the defendant's conduct of depositing two threatening letters containing a white powder into the mail "led directly to the possible anthrax exposure in a USPS mail processing center," which in turn led to the cleanup and decontamination effort. Id. at 773. The record before the district court in this case does not establish a similar causal relationship; indeed, the record did not include any evidence that Kennedy's conduct contributed to Amy and Vicky's claimed losses at all.
In short, the government here did not prove the existence of any causal connection between Kennedy's offense and Amy and Vicky's specific losses, let alone prove that Kennedy's offense was a "material and proximate cause" of those losses. Because the government failed to establish that the victims' losses were proximately caused by Kennedy's offense, it did not meet the second requirement of a restitution award under § 2259.
For much the same reason, the government also failed at the third step to offer a method for calculating the amount of Amy and Vicky's losses that were proximately caused by Kennedy's conduct. Here the government presented evidence of the total costs incurred (or to be incurred) by Amy and Vicky as a result of both the original sexual abuse and all subsequent circulation and viewing of the images. The government then asked the district court to award "$1000 per image," but provided no basis for such an award. Although the district court agreed that this seemed like a "reasonable" amount, it did not indicate how it determined that figure was the "full amount of [Amy and Vicky's] losses." The government contends that the court's attempt to make a "reasonable" guess should be enough, because the harm caused by one possessor of child pornography is not "easily divisible from the harm caused by [all] others." While we acknowledge the difficulties, they do not excuse non-compliance with the statute. Here, the district court's inability to calculate the loss attributable to Kennedy's offense is due to the government's failure to *1265 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, see United States v. Church, 701 F.Supp.2d 814, 833 (W.D.Va.2010)). Had the government provided such evidence, the district court could have calculated the amount of that loss without great difficulty. But picking a "reasonable" number without any explanation is precisely the kind of arbitrary calculation we rejected in Laney and Doe. Laney, 189 F.3d at 967 n. 14; see Doe, 488 F.3d at 1160(requiring that the amount of the loss attributable to the defendant be calculable with "some reasonable certainty"). Section 2259 does not authorize such arbitrary awards.
The government's alternative argument before the district court, that Kennedy should be directed to pay the total losses claimed by Amy and Vicky under the theory of "joint and several liability," founders on the same failure of proof. If the government had proven a causal relationship between Kennedy's conduct and the victims' losses, Kennedy could have been held liable (or jointly and severally liable) for all the losses he proximately caused, even if other defendants had also contributed to those same losses. See § 3664(h) (providing that a court may use joint and several liability to "make each defendant liable for payment of the full amount of restitution" where it has found that "more than 1 defendant has contributed to the loss of a victim"); see also Laney, 189 F.3d at 958. But the predicate for applying this rule is that the government has carried its burden of proving that the defendant's offense conduct proximately caused and contributed to the loss. For the reasons explained above, the government has not done so here. The doctrine of joint and several liability cannot be used to cure a failure of proof on the causal relation between a defendant's conduct and the victims' losses. Moreover, because Amy and Vicky have already recovered part of their claimed losses of $3 million and $227,000 from other defendants, ordering Kennedy to pay this entire amount would have been inconsistent with the goal of restitution, which is to allow a victim to recover for his or her "actual losses," not more. See United States v. Bussell, 504 F.3d 956, 964 (9th Cir.2007).

3
We acknowledge the difficulty the government may have in establishing the proper amount of restitution in cases involving offenses such as possession, receipt, or transportation. Although our sister circuits have also struggled with this issue, no court has yet developed a method for calculating a restitutionary award under § 2259 that comports with the statutory language. In McDaniel, for example, the record established that the victim "would need approximately $166,000 to $188,000 of future counseling or therapy" as a result of the original abuse and all past and future distribution and possession of the images, but did not provide a mechanism by which a court could allocate this loss among all potential defendants. 631 F.3d at 1207-09. Nevertheless, the Eleventh Circuit upheld the district court's award of $12,700 without explaining how the district court could have calculated this amount as the value of the losses proximately caused by the defendant's conduct. Similarly, in Monzel, the D.C. Circuit held that the district court erred in awarding restitution based on the evidence the government had provided and remanded to the district court to calculate "anew the amount of Amy's losses attributable to [the defendant's] offense." 641 F.3d at 540. But it provided no guidance regarding how this calculation was to be conducted, instead merely advising the district court that it might "order the government to suggest a formula for determining the *1266 proper amount of restitution" that would account for the fact that "the [total] number of offenders [is] impossible to pinpoint." Id.
As shown by these cases, identifying a method for imposing restitution on defendants convicted of possession, receipt, or transportation offenses is not easy. The underlying problem is the structure established by § 2259: it is a poor fit for these types of offenses. While direct evidence of a proximate loss, such as evidence that "after receiving notification of the [d]efendant's offense, the victim had to attend any additional therapy sessions [or] miss any days of work," Church, 701 F.Supp.2d at 833, would be sufficient, it is likely to be a rare case where the government can directly link one defendant's viewing of an image to a particular cost incurred by the victim. While we do not rule out the possibility that the government could devise a formula by which a victim's aggregate losses could be reasonably divided (for example, by developing a reasonable estimate of the number of defendants that will be prosecuted for similar offenses over the victim's lifetime, and dividing the total loss by that amount), we suspect that § 2259's proximate cause and reasonable calculation requirements will continue to present serious obstacles for victims seeking restitution in these sorts of cases. Nevertheless, the responsibility lies with Congress, not the courts, to develop a scheme to ensure that defendants such as Kennedy are held liable for the harms they cause through their participation in the market for child pornography. In the future, Congress may decide to reconsider whether § 2259 is the best system for compensating the victims of child pornography offenses, or whether statutory damages of a fixed amount per image or payments into a general fund for victims would achieve its policy goals more effectively.
Until Congress makes such a change, we remain bound by the language of the statute and our precedent. Because the district court's restitution order directed Kennedy to pay for losses that the government did not prove were proximately caused by his offense, the order was unlawful under § 2259 and must be vacated. On this issue, we remand to the district court for proceedings consistent with this opinion. In all other respects, Kennedy's conviction and sentence are affirmed.
AFFIRMED in part and VACATED in part.
NOTES
[1]  During sentencing, the government stated that 5,000 was a "very conservative" estimate and asserted that the actual number was closer to 16,000.
[2]  When a user views a web page online, the web browser automatically saves copies of the images on that page to the computer's internet "cache." These "cache files" improve browser performance by allowing the browser to quickly redisplay the same images if the user returns to the page. See United States v. Romm, 455 F.3d 990, 993 nn. 1, 3 (9th Cir. 2006). Cache files can be deleted: (1) automatically by the web browser, (2) at the request of the user from within the web browser, or (3) manually by the user. See id. at 995. Deleted cache files remain in the computer's "unallocated space" until they are overwritten by other material. United States v. Flyer, 633 F.3d 911, 918 (9th Cir.2011). While in "unallocated space," the files can be recovered using special software. Id.
[3]  At the time of Kennedy's offense, 18 U.S.C. § 2252A(a)(5)(B) (possession offense) prohibited:

knowingly possess[ing] any book, magazine, periodical, film, videotape, computer disk, or any other material that contains an image of child pornography that has been mailed, or shipped or transported in interstate or foreign commerce by any means, including by computer, or that was produced using materials that have been mailed, or shipped or transported in interstate or foreign commerce by any means, including by computer[.]
18 U.S.C. § 2252A(a)(5)(B) (2006).
[4]  At the time of Kennedy's offense, 18 U.S.C. § 2252(a)(1) (transportation offense) prohibited:

knowingly transport[ing] or ship[ping] in interstate or foreign commerce by any means including by computer or mails, any visual depiction, if
(A) the producing of such visual depiction involves the use of a minor engaging in sexually explicit conduct; and
(B) such visual depiction is of such conduct[.]
18 U.S.C. § 2252(a)(1) (2006).
[5]  To establish a basis for federal jurisdiction under the Commerce Clause, Congress included an interstate nexus as an element of both the possession and transportation offenses. To convict a defendant of the transportation offense, the government must prove beyond a reasonable doubt that the defendant himself "transport[ed]" or "ship[ped]" the images in "interstate or foreign commerce." 18 U.S.C. § 2252(a)(1) (2006). Here, the government proved this element through evidence that Kennedy was carrying his laptop computer when he entered the country at Sea-Tac airport. To convict a defendant of the possession offense, on the other hand, the government must prove beyond a reasonable doubt that the images possessed "ha[d] been mailed, or shipped or transported in interstate or foreign commerce by any means" or that the images were "produced using materials that have been mailed, or shipped or transported in interstate or foreign commerce by any means." 18 U.S.C. § 2252A(a)(5)(B) (2006).
[6]  A transportation conviction has a mandatory minimum sentence of 60 months. 18 U.S.C. § 2252(a)(1), (b)(1) (2006). In contrast, a possession conviction has no mandatory minimum sentence. 18 U.S.C. § 2252A(a)(5)(B) (2006).
[7]  Section 2259 states, in pertinent part:

(a) In General.Notwithstanding section 3663 or 3663A, and in addition to any other civil or criminal penalty authorized by law, the court shall order restitution for any offense under this chapter.
(b) Scope and Nature of Order.
(1) Directions.The order of restitution under this section shall direct the defendant to pay the victim (through the appropriate court mechanism) the full amount of the victim's losses as determined by the court pursuant to paragraph (2).
(2) Enforcement.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.
(3) Definition.For purposes of this subsection, the term "full amount of the victim's losses" includes 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.
...
(c) Definition.For purposes of this section, the term "victim" means the individual harmed as a result of a commission of a crime under this chapter....
18 U.S.C. § 2259.
[8]  The Child Victim Identification Program at the National Center for Missing & Exploited Children ("NCMEC") assists law enforcement in identifying the victims of child pornography. When a law enforcement agency submits copies of seized images, NCMEC returns a report identifying the child victims depicted in them. NCMEC also notifies an identified victim every time someone is arrested who is found to possess his or her image. See http://www.missingkids.com.
[9]  Although Kennedy also argues that the district court erred in applying a 5-level adjustment to his guidelines offense level pursuant to U.S.S.G. § 2G2.2(b)(7)(D), he has asked us to consider this argument only if we remand for re-sentencing on the ground that the district court erred in vacating the possession conviction rather than the transportation conviction. Because we hold that the district court did not err in choosing to vacate the possession conviction, see infra, we do not address Kennedy's guidelines argument.
[10]  Kennedy also complains that the law enforcement witnesses' testimony included two irrelevant facts: first, that NCMEC had a record of the photos of one of the victims, and second, that two of the abusers were the victims' fathers. Because Kennedy did not object to these statements on relevance grounds at trial, we review only for plain error. See United States v. Graf, 610 F.3d 1148, 1164 (9th Cir.2010). We hold that the district court did not plainly err in failing to sua sponte strike these remarks. See United States v. Mitchell, 502 F.3d 931, 968 (9th Cir.2007). Moreover, any error would be harmless, because it is not "highly probable" that such stray remarks, unrelated to Kennedy, "materially affected the [jury's] verdict." United States v. Chang, 207 F.3d 1169, 1175 (9th Cir.2000) (quoting United States v. Kessi, 868 F.2d 1097, 1103 (9th Cir.1989)); cf. United States v. Chu, 5 F.3d 1244, 1250 (9th Cir.1993) (holding that the admission of an irrelevant statement about a "crime syndicate" was harmless because the testimony did not suggest that the defendant was a "member of the syndicate").
[11]  In reaching this conclusion, the district court relied on United States v. Davenport, 519 F.3d 940 (9th Cir.2008), in which we held that possession of child pornography, § 2252A(a)(5)(B), is a lesser-included offense of receipt of child pornography, § 2252A(a)(2), because the crime of possession does not "require[ ] proof of an element that the crime of receipt does not." 519 F.3d at 945 (emphasis omitted). Although we have not held that possession of child pornography is a lesser-included offense of transportation of child pornography, the government has not appealed this determination, and therefore we need not reach the issue here; rather, we assume for purposes of this appeal that the district court's analysis was correct.
[12]  The district court was required to impose a period of supervision ranging from 60 months to life. See 18 U.S.C. § 3583(k) (2006).
[13]  Since our decision in Laney, a circuit split has developed as to whether § 2259 requires proximate cause as to all the types of losses described in § 2259(b)(3), or only as to "any other losses suffered by the victim as a proximate result of the offense," § 2259(b)(3)(F). The Third Circuit, Eleventh Circuit, and D.C. Circuit have agreed with us that restitution under § 2259 is limited to losses proximately caused by the defendant's offense. See United States v. Crandon, 173 F.3d 122, 125-26 (3d Cir.1999); McDaniel, 631 F.3d at 1208; Monzel, 641 F.3d at 534-35. Only the Fifth Circuit has limited the proximate cause requirement to the catchall category of "other losses" listed in § 2259(b)(3)(F). See In re Amy Unknown, 636 F.3d at 198. We are bound by Laney and thus have no occasion to revisit this question.
[14]  Section 3664 was enacted as part of the Victim and Witness Protection Act of 1982, Pub.L. No. 97-291, 96 Stat. 1248, and sets forth a procedure for ordering restitution. Section 3663A, which is part of the Mandatory Victims Restitution Act of 1996, Pub.L. No. 104-132, §§ 201-211, 110 Stat. 1227, also incorporates the procedural guidelines of § 3664.
[15]  The VWPA and the MVRA "are identical in all important respects, and courts interpreting the MVRA may look to and rely on cases interpreting the VWPA as precedent." United States v. Gordon, 393 F.3d 1044, 1048 (9th Cir.2004).
