
536 F.3d 746 (2008)
UNITED STATES of America, Plaintiff-Appellee,
v.
Jeffrey P. SCHROEDER, Defendant-Appellant.
No. 07-3773.
United States Court of Appeals, Seventh Circuit.
Argued May 29, 2008.
Decided August 5, 2008.
*748 Stephen Lee (argued), Office of the United States Attorney, Chicago, IL, for Plaintiff-Appellee.
Leonard C. Goodman (argued), Chicago, IL, for Defendant-Appellant.
Before CUDAHY, POSNER and TINDER, Circuit Judges.
CUDAHY, Circuit Judge.
Jeffrey Schroeder pleaded guilty to one count of tax preparer fraud and was sentenced to 36 months' imprisonment. Schroeder appeals, arguing that the district court deprived him of his due process right to a fair sentencing hearing, used the wrong burden of proof in applying a sentencing enhancement based on relevant conduct and did not give adequate consideration to his argument for a below-guideline sentence pursuant to 18 U.S.C. § 3553(a). Because Schroeder's sentencing hearing was marred by several serious errors, we vacate his sentence and remand, noting that this marks the second *749 time we have sent Schroeder's case back to the district court for resentencing.

I. Background
Schroeder operated a tax preparation business out of his Illinois home. In 2005 he was indicted and charged with 21 counts of tax preparer fraud in violation of 26 U.S.C. § 7206(2). In January 2006, he pleaded guilty to one count of tax preparer fraud that caused a tax loss to the United States Treasury of $6,556. In his plea agreement, Schroeder admitted to preparing tax returns that included information he knew to be false about his clients' itemized deductions. Itemized deductions are listed on federal income tax form Schedule A. By fraudulently overstating his clients' Schedule A deductions, Schroeder was able to decrease his clients' reported taxable income, yielding greater refunds than they were in fact entitled to. Schroeder also admitted that he had assisted in the preparation of at least 52 fraudulent tax returns for tax years 1999, 2000 and 2001. He also admitted that the false tax returns resulted in a total loss to the United States Treasury of at least $161,116. But at his change of plea hearing, he asserted that the $161,116 tax loss figure in the plea agreement was incorrect. The district court determined that the actual amount of loss would be determined at sentencing.
In April 2006, the United States Probation Office prepared a Pre-Sentence Investigation Report (PSR), which assigned Schroeder a criminal history category of III. The PSR based Schroeder's offense level in part on a tax loss of $428,555  the $161,116 that was included in the plea agreement as well as an additional loss of $267,439. This additional amount was based on Internal Revenue Service (IRS) correspondence audits of 25 of Schroeder's clients. The IRS concluded that Schroeder had overstated or misrepresented deductions on 52 tax returns prepared on behalf of the audited clients, resulting in a tax loss of $267,439. The results of the audits were summarized in a two-page spreadsheet that was provided to the Probation Office. The spreadsheet included the initials of the taxpayers, the year of the challenged return, the type of fraudulent itemized deduction, the amount of the fraudulently increased refund and the adjustment amount associated with the fraudulent itemized deductions.
The Tax Table in U.S.S.G. § 2T4.1 provides that a tax loss of over $400,000 corresponds to an offense level of 20. The $161,116 amount of loss that was originally included in the plea agreement corresponds to an offense level of 16. The PSR increased Schroeder's base offense level due to specific offense characteristics and reduced it due to Schroeder's cooperation and acceptance of responsibility. After accounting for these adjustments, the PSR concluded that the total offense level was 19. An offense level of 19 and a criminal history category of III yielded a sentencing range of 37 to 46 months.
Schroeder filed a sentencing memorandum challenging the inclusion of the additional $267,439 in the amount of loss calculation on the grounds that the materials supporting the additional tax loss had not been made available to the defense. He argued that his base offense level should be based on a tax loss of $161,116. After accounting for adjustments, using the lower tax loss amount would result in a total offense level of 15 and a sentencing range of 24 to 30 months. He also asserted that his family circumstances justified a sentence below the guideline sentencing range. In particular, he claimed that his incarceration would impose a hardship on his adopted daughter, who was born with various medical problems arising from her biological mother's drug abuse. The government *750 filed a response to Schroeder's sentencing memorandum in which it asserted that although the individual tax returns underlying the additional tax loss had not been provided to defense counsel, the spreadsheet had been given to the defense.
At Schroeder's sentencing, the district court heard testimony from IRS Special Agent Zagota, who testified that, in the course of the criminal investigation, he had interviewed witnesses to determine the original tax loss of $161,116. Although Special Agent Zagota was not involved in the civil audits, he explained that the additional tax loss was ascertained via correspondence audits, in which the IRS sent letters to Schroeder's clients inviting them to provide documentation supporting the itemized deductions on their tax returns. If they could not justify their deductions, their tax returns were adjusted accordingly. Special Agent Zagota also opined that the total amount of loss was a conservative figure given that Schroeder had prepared thousands of tax returns in 2000 and 2001 and that over 90 percent of those returns claimed refunds. The court accepted the additional $267,439 tax loss amount, declined to impose a below-guidelines sentence based on Schroeder's family situation and sentenced Schroeder to 42 months' imprisonment and one year supervised release.
Schroeder filed a notice of appeal on November 22, 2006, but a few months later, he and the government submitted a joint motion for summary reversal and remand based on the fact that the 42-month sentence imposed by the district court exceeded the statutory maximum of 36 months. The motion stated:
The parties agree that it would be a waste of their resources and of the Court's resources to proceed through briefing and consideration of an appeal for a case that must be reversed and remanded for resentencing.
Wherefore, it is respectfully requested that the judgment in this case be reversed and that the case be remanded to the district court for resentencing.
We granted the motion in a brief order, summarily reversing and remanding the case "for the limited purpose of resentencing." R. 60.
On remand, Schroeder moved for an order requiring the government to produce the tax returns on which the additional amount of loss was based and for an order authorizing defense counsel to obtain expert services for the resentencing hearing. These motions were granted and the defense reviewed the underlying documents with expert assistance. In June 2007, Schroeder submitted a supplemental sentencing memorandum challenging the $267,439 additional tax loss on the grounds that the civil audits did not represent a reliable method of assigning criminal liability. Schroeder argued that the purpose of the civil audit was to collect unpaid taxes and that it was not designed to attribute responsibility for the improper deductions. Some taxpayers did not respond to the IRS's invitation to defend their deductions and in those cases, he asserted, the IRS assumed that Schroeder was probably responsible for the false information. He pointed out that one taxpayer admitted that she had overstated her business expenses but that the government included the full amount of the liability resulting from her misrepresentation in calculating Schroeder's amount of loss. In addition, Schroeder again argued that his unique family circumstances weighed in favor of a lower sentence. He informed the court that he had been the primary caregiver for his adopted daughter while his wife worked outside the home. He stated that after his incarceration, his daughter had to *751 be placed in daycare. Being in daycare is a risky proposition for the child because she has a weak immune system that makes her more vulnerable to infection than most children. In support of this argument for leniency, Schroeder submitted a letter from his daughter's pediatrician stating that she should not be placed in daycare.
Schroeder's resentencing hearing was held on November 6, 2007 and his tax loss and family circumstances arguments did not fare any better the second time around. At the commencement of the hearing, the judge stated that he had reviewed the memoranda submitted by the parties and determined that "the $267,000 amount[ ] is based on information that is of sufficient reliability that can properly be considered under the guideline provisions" and thus, the total amount of loss was $428,555. Schroeder's attorney argued that many taxpayers who were the subject of a civil audit did not contest the deductions challenged by the IRS and that the audits were performed to collect unpaid taxes, not to assign liability for the improper deductions. The court rejected Schroeder's argument and maintained its initial tax loss determination. Addressing Schroeder's family circumstances argument, the court stated that because Schroeder's criminal acts caused his difficult family situation, family circumstances did not constitute a valid basis for imposing a lower sentence. After reviewing the factors set forth in 18 U.S.C. § 3553(a), the court imposed a sentence of 36 months' imprisonment.

II. Analysis
Before we reach the merits of Schroeder's challenges to his sentence, we briefly address the government's contention that the district court committed plain error in considering Schroeder's arguments on the amount of loss issue and his family circumstances on remand. It is the government's position that Schroeder waived these issues by moving to remand his original sentence based solely on the fact that it exceeded the statutory maximum. The government argues that the issues Schroeder now raises fall outside the scope of our remand and the district court should not have considered them. Because the government did not object to the district court's reopening of the sentencing issues on remand, it asserts that we should review the court's decision to revisit these issues for plain error. "Plain error exists when there is an error which is clear or obvious and which affects substantial rights." United States v. Otis, 107 F.3d 487, 489 (7th Cir.1997). An error affects substantial rights if it "seriously affect[s] the fairness, integrity or public reputation" of the proceedings. United States v. Van Allen, 524 F.3d 814, 819 (7th Cir.2008) (quoting United States v. Jumah, 493 F.3d 868, 875 (7th Cir.2007)) (alteration in original).
We are unpersuaded by the government's plain error argument. First, it was not erroneous for the court to consider the amount of loss or family circumstances issue on remand. Schroeder's case was not subject to either of the primary limitations on the scope of a remand. The first limitation is that "any issue that could have been but was not raised on appeal is waived and thus not remanded." United States v. Husband, 312 F.3d 247, 250 (7th Cir.2002). When we granted the parties' joint motion for summary reversal, Schroeder's first appeal was truncated before he had the opportunity to challenge the sentencing court's rulings on the issues he now raises. This fact distinguishes his case from others where we have declined to allow a defendant to "use the accident of a remand to raise in a second appeal an issue that he could just as well have raised *752 in the first appeal." United States v. Parker, 101 F.3d 527, 528 (7th Cir.1996); see also United States v. Swanson, 483 F.3d 509, 514 (7th Cir.), cert. denied, ___ U.S. ___, 128 S.Ct. 455, 169 L.Ed.2d 318 (2007) (same).
The second major limitation on the scope of a remand is that "any issue conclusively decided by this court on the first appeal is not remanded." Husband, 312 F.3d at 251. In their joint motion to this court, the parties stipulated that it would be a waste of the court's time to proceed through briefing and argument when the case obviously had to be remanded for resentencing. In response to the parties' motion, we issued a brief order that neither explicitly nor implicitly constrained the district court's authority to address contested issues at resentencing. In the absence of a mandate limiting the scope of resentencing, it was reasonable for the district court to revisit disputed sentencing issues. See Parker, 101 F.3d at 528 ("[T]he scope of the remand is determined not by formula, but by inference from the opinion as a whole."); cf. United States v. Polland, 56 F.3d 776, 778 (7th Cir.1995) (where our ruling specifically limited mandate on resentencing to particular sentencing issue, the district court was precluded from revisiting other sentencing issues on remand). Moreover, even if we assume that the district court erred in reopening sentencing issues following our remand, the government has not established that any substantial rights were affected. Thus, we take up Schroeder's arguments confident that considering them will not cast serious doubts on the integrity of these proceedings.

A. Amount of loss
Schroeder appeals the district court's determination that he was responsible for a total tax loss of $428,555, arguing that the court deprived him of his right to a fair sentencing hearing and did not hold the government to its burden of proof. Amount of loss must be determined "on the basis of the conduct of conviction and relevant conduct." United States v. Frith, 461 F.3d 914, 917 (7th Cir.2006). "[R]elevant conduct must be criminal or unlawful conduct...." Id. The government has the burden of proving losses attributable to relevant conduct by a preponderance of the evidence. Id. In this case, the government bore the burden of proving, by a preponderance of the evidence, that the unpaid taxes discovered through the civil audit were attributable to Schroeder's criminal or unlawful conduct. The amount of tax loss need not be determined with precision and a reasonable estimate will suffice. U.S.S.G. § 2T1.1 cmt. 1; see also Frith, 461 F.3d at 917 (discussing amount of loss determination for offenses involving fraud or deceit). In considering relevant conduct, a district court "may consider relevant information without regard to its admissibility under the rules of evidence applicable at trial, provided that the information has sufficient indicia of reliability to support its probable accuracy." U.S.S.G. § 6A1.3(a). This rule applies to facts asserted in the PSR as well, which a court may adopt "as support for its findings and conclusions" if they "bear sufficient indicia of reliability to support their probable accuracy." United States v. Taylor, 72 F.3d 533, 543 (7th Cir.1995) (quoting United States v. Salinas, 62 F.3d 855, 859 (7th Cir.1995)).
Schroeder's resentencing hearing was flawed from the outset. The district court commenced the hearing by announcing its findings on the amount of loss before allowing Schroeder's attorney to present any argument. The court stated that after reviewing the materials submitted by Schroeder and the government, including *753 the PSR and sentencing memoranda submitted by Schroeder's attorney:
I do find that the additional amount, the $267,000 amount, is based on information that is of sufficient reliability that can properly be considered under the guideline provisions.
So I find that the total amount of loss is as set out in the presentence investigation report.
Although a district court is not always required to provide an evidentiary hearing in which the parties can argue sentencing issues, United States v. Cantero, 995 F.2d 1407, 1413 (7th Cir.1993), the court is required to "allow the parties' attorneys to comment on the probation officer's determinations and other matters relating to an appropriate sentence." FED.R.CRIM.P. 32(i)(1)(C). The district court announced its findings as to the amount of tax loss  a critical sentencing determination  before Schroeder's attorney had an opportunity to comment on the issue. Schroeder's attorney had to interrupt the court's recitation of its loss findings to indicate that he intended to present testimony and argument disputing the amount of loss. Although the court allowed him to offer evidence in the form of an offer of proof, whether the court gave the defense's arguments the due consideration they deserved is questionable to say the least in light of the court's conclusive pronouncement of its findings at the beginning of the hearing. Indeed, the sentencing transcript suggests the court's impatience with the defense's challenges to the alleged tax loss. Although due process does not guarantee a perfect sentencing hearing, it does guarantee a fair one. United States v. Agyemang, 876 F.2d 1264, 1271 (7th Cir.1989). Here, the fact that the district court prejudged the tax loss issue undermined the fairness of Schroeder's hearing.
That the district judge appears to have commenced resentencing with his mind made up about the tax loss issue may explain his apparent unwillingness or inability to understand defense counsel's attempt to clear up serious confusion regarding the government's burden of proof. The court's statements at resentencing strongly suggest that it confused the standard for the admissibility of evidence at sentencing with that for proving relevant conduct, a very serious error. As we have already noted, it is well established that the government must prove amount of loss by a preponderance of the evidence. See United States v. Omole, 523 F.3d 691, 701 (7th Cir.2008). The preponderance of the evidence standard requires "that the factfinder believe that the existence of a fact is more probable than the non-existence of that fact." United States v. Smith, 267 F.3d 1154, 1161 (D.C.Cir.2001). In determining whether the government has met its burden of proof at sentencing, a court may consider information that would not have been admissible at trial if it has "sufficient indicia of reliability to support its probable accuracy." United States v. Artley, 489 F.3d 813, 821 (7th Cir.2007). But the presumed accuracy of information that has "sufficient indicia of reliability" does not relieve the court of its responsibility to weigh the proffered evidence and determine whether the government has proven that the existence of a disputed fact is more probable than not.
The court's apparent confusion is evident on review of the sentencing transcript. After Schroeder's attorney began to argue that the results of the civil audit did not establish Schroeder's culpability for the improper deductions, the court stated:
I agree with you, that if the standard here were proof by a preponderance of the evidence, that that might pose some problems. I also agree with you that *754 there might be a difficulty if strict Rules of Evidence were in effect. But I don't think the sentencing guidelines require that.
* * * *
I think the government bears the burden of proving certain components of the sentencing procedure by a preponderance of the evidence, but your focus is on the quality of the information the Court is using to determine the 
Schroeder's attorney then attempted to clarify that the government must prove by a preponderance of the evidence that the improper deductions are "attributable to Mr. Schroeder rather than to taxpayer A or taxpayer B, who may have submitted bogus claims or bogus deductions." At that point, the court stated:
Section 6A1.3 of the guidelines manual says as follows: "When any factor important to the sentencing determination is reasonably in dispute, the parties shall be given an adequate opportunity to present information to the Court regarding that factor.
In resolving any dispute concerning a factor important to the sentencing determination, the Court may consider relevant information without regard to its admissibility under Rules of Evidence applicable to trial, provided the information has sufficient indicia of reliability to support its probable accuracy." Now that's the standard that is in effect here. I have read everything that's been submitted, I looked at the spreadsheet, I looked at the information that I ordered produced to the defendant. I've consulted everything that is before me. And I find that the information provided has sufficient indicia of reliability to support its probable accuracy.
I understand your argument. I understand that in a civil trial under Rules of Evidence that this information might not be admissible, but for purposes of determining the total amount of loss as relevant conduct, I find that it does meet that standard.
Schroeder's attorney again attempted to clarify that the defense was not questioning the reliability of the information but was trying to question "whether the government has proven that it's attributable to Mr. Schroeder, which is the standard for relevant conduct." The court replied:
Okay. I find that based on that standard, that that information has sufficient indicia of reliability that it's properly considered in determining the relevant conduct and the total offense level.
In so stating, the court seemed to suggest that the government had met its burden of proof merely by submitting admissible evidence, as opposed to proving the amount of the tax loss by a preponderance of the evidence. Indeed, the court never found that the government proved the tax loss by a preponderance of the evidence.
The court's apparent confusion is particularly worrisome given that Schroeder has challenged persuasively the propriety of using the civil audit to attribute criminal liability. As he attempted to point out to the court, the civil audit revealed only that the clients of Schroeder who were audited had overstated their deductions. The audit did not purport to attribute responsibility for the improper deductions. Although a district court may rely on information in PSRs and in sentencing memoranda where the information is based on sufficiently reliable information, Taylor, 72 F.3d at 543, Schroeder was not attempting to argue that the summary of the civil audit results was inaccurate or misrepresented the amount of improper refunds that had been distributed to Schroeder's clients. The question Schroeder tried to raise was whether the PSR and the correspondence audit provided any proof of Schroeder's *755 culpability for the false information included on the tax returns that were the subject of the civil audit.
The government contends that the civil audit is circumstantial evidence of Schroeder's culpability for the additional tax loss amount, pointing out that almost all of the taxpayers who were audited told IRS agents that they could not provide documentation supporting the itemized deductions claimed on their tax returns. The government also notes that the inaccurate deductions on the audited returns were of the same type that Schroeder admitted to having falsified. But these facts do not make it more likely that the overstated deductions were due to Schroeder's criminal conduct than to a mistake or fraud on the part of his taxpayer clients. The district court treated the underpayments that were detected in the audit as frauds attributable to Schroeder without conducting any analysis as to what evidence proved that Schroeder's unlawful conduct caused the underpayments. We attribute this omission to the court's apparent belief that if evidence is admissible it proves the truth of the proposition for which it is being offered. The court's failure to hold the government to its burden of proof is a serious error  in this case, Schroeder's total offense level was increased from 15 to 19 due to the court's tax loss determination  and it requires us to remand for resentencing.

B. Family circumstances
Schroeder also contends that the court erred in giving short shrift to his argument that his incarceration imposed an extraordinary hardship on his family. We review a court's application of the Sentencing Guidelines de novo. United States v. Warren, 454 F.3d 752, 762 (7th Cir. 2006). After a sentencing court calculates the correct advisory guidelines range, it "must apply the factors set forth in 18 U.S.C. § 3553(a) in determining whether to apply a sentence within the advisory guidelines range." United States v. Miranda, 505 F.3d 785, 791 (7th Cir.2007) (citing United States v. Robinson, 435 F.3d 699, 700-01 (7th Cir.2006)). Section 3553(a) directs the court to include the history and characteristics of the defendant among the factors it weighs in determining the sentence it will impose. A district court may pass over in silence frivolous arguments for leniency, but where a defendant presents an argument that is "not so weak as not to merit discussion," a court is required to explain its reason for rejecting that argument. Id. at 792 (quoting United States v. Cunningham, 429 F.3d 673, 679 (7th Cir.2005)); see also Rita v. United States, ___ U.S. ___, 127 S.Ct. 2456, 2468-69, 168 L.Ed.2d 203 (2007) (by explaining reasons for rejecting arguments for higher or lower sentence, sentencing courts "assure[] reviewing courts (and the public) that the sentencing process is a reasoned process"). A short explanation will suffice where the context and record make clear the reasoning underlying the district court's conclusion. Id. at 2469. But "[a] judge who fails to mention a ground of recognized legal merit (provided it has a factual basis) is likely to have committed an error or oversight." Cunningham, 429 F.3d at 679.
A defendant's extraordinary family circumstances can constitute a legitimate basis for imposing a below-guidelines sentence. Sentencing Guideline 5H1.6 provides that "family ties and responsibilities are not ordinarily relevant in determining whether a departure may be warranted," but a district court may impose a below-guidelines sentence "once it finds that a defendant's family ties and responsibilities ... are so unusual that they may be characterized as extraordinary." *756 United States v. Canoy, 38 F.3d 893, 906 (7th Cir.1994); see also United States v. Jaderany, 221 F.3d 989, 996 (7th Cir.2000). Although "[t]he concept of departures has been rendered obsolete in post-Booker sentencing ... the district court may apply those departure guidelines by way of analogy in analyzing the section 3553(a) factors." Miranda, 505 F.3d at 792.
To support his extraordinary family circumstances argument, Schroeder's attorney elicited testimony from Mrs. Schroeder. Mrs. Schroeder testified that the couple's adopted daughter has a compromised immune system and that she is particularly vulnerable to illness, making daycare an implausible childcare option. Mrs. Schroeder then testified that because Schroeder worked from home before his incarceration, he had been able to look after their daughter. She described her attempts to find suitable care for their daughter since Schroeder's incarceration. After Mrs. Schroeder finished testifying, the court stated:
That's an unfortunate thing that's happened to your family, regarding the health of your daughter, but the fact that Mr. Schroeder is not there to assist is something based on conduct Mr. Schroeder chose to commit. And I expressly reject that as a consideration that should somehow mitigate his sentence.
The government asserts that Schroeder's family circumstances are not sufficiently unusual to warrant a below-guidelines sentence. This may or may not be what was actually driving the district court's rejection of Schroeder's argument. We do not know one way or the other because the district court did not address the merits of Schroeder's argument at all. Instead, the court brushed it aside and seemed to deem any discussion of the hardship to Schroeder's family as an irrelevant digression. In doing so, the court has left us "in serious doubt whether the judge connected the facts relating to the statutory factors to the sentence he imposed[.]" Cunningham, 429 F.3d at 676.
The court's observation that Schroeder's criminal conduct was the cause of the alleged hardship to his daughter is an obvious and not dispositive one, since the culpability of a defendant who appears for sentencing is a given. When a defendant presents an argument for a lower sentence based on extraordinary family circumstances, the relevant inquiry is the effect of the defendant's absence on his family members. See, e.g., United States v. Johnson, 964 F.2d 124, 129 (2d Cir.1992) ("The rationale for a downward departure here is not that [the defendant's] family circumstances decrease her culpability, but that we are reluctant to wreak extraordinary destruction on dependents who rely solely on the defendant for their upbringing."). The defendant's responsibility for the adverse effects of his incarceration on his family is not the determinative issue. If it were, there would never be an occasion on which the court would be justified in invoking family circumstances to impose a below-guide-lines sentence. The court was required to consider Schroeder's family circumstances argument and provide an adequate analysis of how much weight, if any, it should command. The fact that the consequences of incarceration are attributable to his own misconduct may be a factor in the analysis but it is not the sole factor nor is it dispositive. Thus, on remand the court should consider whether Schroeder's family circumstances are a mitigating factor.

III. Conclusion
For the foregoing reasons, we VACATE Schroeder's sentence and REMAND for resentencing. *757 Circuit Rule 36 shall apply on remand.
