CUDAHY, Circuit Judge.

On May 16, 1985, Judge Susan Getzen-danner sentenced Walter Barnes, who had pleaded guilty to distributing cocaine, to three years imprisonment and a subsequent five year special parole term. She suspended sentence on a conspiracy charge (to which Barnes had also pleaded guilty), but placed Barnes on five years probation to run concurrent with the special parole term. As part of the terms of probation, Judge Getzendanner required Barnes to participate in a drug and alcohol treatment program.

Barnes was released on May 20, 1988, to begin his special parole term. By December, however, the United States Probation Office reported to Judge Brian Barnett Duff 1 that Barnes had violated the terms of his parole. The Probation Office alleged that Barnes had failed to report to his probation officer, that he had failed to attend the required drug treatment program, that he had continued to use cocaine, that he had continued to distribute cocaine and that he had associated with others engaged in criminal activities. Based upon the Probation Office’s Report, the Parole Commission filed a warrant application and the U.S. Attorney’s Office filed a “Motion for a Rule to Show Cause Why Probation Should Not Be Revoked.”

On October 20,1989, the government and Barnes appeared before Judge Duff on the motion. Barnes told the court that he was willing to admit to continued drug use and to failure to attend drug rehabilitation sessions, but that he would not admit to distribution of cocaine or association with others engaged in criminal activities. At this point, Judge Duff cautioned the government; he remarked that any demonstration of Barnes’s activities relating to cocaine distribution (for purposes of his parole violation) might implicate the double jeopardy clause of the Constitution if the government attempted to pursue another indictment against Barnes for the same conduct. Judge Duff scheduled a hearing a week later to give the government time to determine how to proceed with the case.

At the second hearing, the government announced that, because of double jeopardy concerns, it would proceed only on the charges of continued drug use and the failure to attend rehabilitation sessions and probation meetings. As a result, the court dispensed with the need for further hearings. Barnes admitted in court that he had violated three of his parole conditions. Based upon this admission, Judge Duff revoked Barnes’s probation and sentenced him to five years incarceration to run consecutive to his sentence for the special parole violation.

The dispute at the center of this case relates solely to information relied upon by the district court to sentence Barnes for violating the terms of his probation and special parole. Before imposing sentence, the court said:

Well, Mr. Barnes ... the record reflects that you were a very, very heavy dealer with Mr. Mitchel [sic] and that you were a major participant in a major narcotics activity, and you were given really quite a break by Judge Getzendanner, and I don’t think that one can say just because one violates probation and that the offenses by which one violates probation are the offenses for which the person is being sentenced.

You’re going to be sentenced for what you were convicted of and for your behavior in not taking advantage of the break you were given, and I would be inclined to give even a more heavy sen *695 tence than either the probation officer or the Government or your lawyer suggests because you were a big player and you did get your break and you came in here and you got yourself a heck of a deal in the first place.

Transcript at 10-11 (Oct. 25, 1989).

In these comments before sentencing, Judge Duff described Barnes as “a big player” and noted Barnes’s relationship with Archiebald Mitchell, an alleged - cocaine distributor. The source of Judge Duffs information on -these points is somewhat unclear. Barnes did not mention Mitchell’s name during the parole revocation proceedings; he admitted only that he had failed to report to drug rehabilitation sessions, that he had failed to report to his probation officer and that he had continued to use cocaine. Nor did Mitchell’s name play a prominent role in Barnes’s underlying conviction, for Mitchell’s name did not even appear in the original indictment of Barnes for distribution and conspiracy. Mitchell’s name seems to have appeared only in an affidavit attached to the government’s Motion for a Rule to Show Cause. This affidavit, sworn to by IRS Special Agent Michael Priess, contains statements from a confidential informant suggesting that Barnes may have distributed cocaine with Mitchell during the summer of 1988, when Barnes was serving his probation. 2 See Affidavit of IRS Special Agent Michael Priess at 6-7 (Dec. 12, 1988) (reprinted in Appellant’s Brief at A-33 to A-34) (hereinafter “Priess Affidavit”).

Ordinarily, such a reference to a defendant’s past conduct would easily survive the scrutiny of an appellate court reviewing a sentence. A federal district judge may exercise wide discretion in determining an appropriate sentence for a convicted defendant, and, “before making that determination, a judge may appropriately conduct an inquiry broad in scope, largely unlimited either as to the kind of information he may consider, or the source from which it may come.” United. States v. Tucker, 404 U.S. 443, .446, 92 S.Ct. 589, 591, 30 L.Ed.2d 592, 596 (1972). Appellate review of such sentencing decisions is extremely limited. The scope of this review is not so narrow, however, as to prevent us from vacating a sentence where “the sentencing judge relied upon improper considerations or unreliable information in exercising his discretion or failed to exercise any discretion at all in imposing the sentence.” United States v. Harris, 761 F.2d 394, 403 (7th Cir.1985); see United States v. George, 891 F.2d 140, 143 (7th Cir.1989).

In this case, Barnes claims that the district court improperly consulted the Priess Affidavit to rely upon “unreliable information” — the hearsay statements of a confidential informant — in imposing his sentence. 3 Barnes questions the propriety of' the court’s reliance on this hearsay statement without affording him an opportunity to challenge it in court.

To support his allegation, Barnes points to our opinion in Harris: “where the sentencing judge relies upon prejudicial hearsay information, the accuracy of which is *696 contested, ‘fundamental fairness requires that a defendant be given at least some opportunity to rebut that information.’ ” 558 F.2d at 374; see, e.g., United States v. Nowicki, 870 F.2d 405, 407 (7th Cir.1989); United States ex rel. Welch v. Lane, 738 F.2d 863, 865 n. 3 (7th Cir.1984); United States v. Marshall, 719 F.2d 887, 891 (7th Cir.1982). Cf. United States v. Tucker, 404 U.S. 443, 92 S.Ct. 589, 30 L.Ed.2d 592 (1972) (affirming vacation of sentence based upon prior convictions obtained while defendant was unrepresented by counsel); Townsend v. Burke, 334 U.S. 736, 741, 68 S.Ct. 1252, 1255, 92 L.Ed. 1690, 1693 (1948) (reversing sentence violating due process because it relied upon “materially untrue” assumptions). Barnes claims that the district court violated this standard by relying upon the informant’s statements without affording him an opportunity to challenge their accuracy.

The parties do not dispute the propriety of the hearing procedure required by Harris or the relevant law supporting it. Indeed, the government acknowledges that Barnes challenged the accuracy of the hearsay statement and never had an opportunity to rebut the statement because of its own refusal to pursue the distribution charge. The only question separating the parties on appeal is whether the sentencing judge actually relied upon the hearsay information to impose his sentence upon Barnes. Of course, if the district court ignored the hearsay while sentencing, the sentence must be upheld. Johnson v. United States, 805 F.2d 1284, 1288-89 (7th Cir.1986).

In this light, Barnes has the burden of establishing that the sentencing judge relied upon the hearsay information in imposing his sentence. See United States v. Lewis, 880 F.2d 243, 246 (9th Cir.1989). Barnes attempts to satisfy this burden by pointing to Judge Duff’s remarks during sentencing: Judge Duff characterized Barnes as a “very, very heavy dealer with Mr. Mitchel [sic]” and as a “major participant in a major narcotics activity_” Transcript at 10. As suggested previously, the government has not shown that either of these characterizations finds support in any document in the record other than the Priess Affidavit, and the statements contained in that affidavit are hearsay statements made by a confidential informant not subject to cross-examination.

The government, in its brief, disputes this analysis. It argues that Judge Duff did not consider Barnes’s relationship with Mitchell in calculating the appropriate sentence; rather, Judge Duff’s statements about Mitchell and “major narcotics activity” were merely passing references: “Although the court initially referred to defendant’s involvement with Mr. Mitchell, the record shows that the court did not consider this factor in sentencing defendant.” Appellee’s Brief at 13. Further, the government contends that Judge Duff’s sentencing decisions were based upon the criminal activities underlying Barnes’s original conviction and, in addition, reflect the fact that Barnes received a “break” from the original sentencing court. Transcript at 10. Both the government’s view and Barnes’s view draw substantial support from the record.

This is an extremely close question, for it is impossible for appellate courts to know precisely what factors are relied upon by district judges during sentencing. We have no direct evidence — indeed, we have only the most indirect evidence (the transcript) — of the district judge’s thoughts during sentencing. But, for these purposes, that evidence may suffice:

The standard for determining whether the district court relied on improper information is a low one. A petitioner need only show it is “not improbable that the trial judge was influenced by improper factors in imposing sentence.”

Rizzo v. United States, 821 F.2d 1271, 1274 (7th Cir.1987) (quoting Harris, 558 F.2d at 375).

Given Judge Duff’s remarks about Mitchell and the amount of cocaine being distributed, we believe that Barnes has satisfied the low standard outlined by Rizzo and Harris. The district court’s comments at sentencing raise sufficient doubts in our *697 mind whether it imposed its sentence wholly without regard to the hearsay statements of the confidential informant. Granted, we cannot be sure that these statements influenced Judge Duffs sentencing determinations, but given the closeness of this question — and the relatively low standard, under the law of this circuit, by which Barnes must prove his case — we vacate and remand .so that Barnes may have an opportunity to challenge these statements. While Judge Duff is, of course, free to reimpose his original sentence upon Barnes after the hearing, we are confident that he will carefully consider Barnes’s claim. See Harris, 558 F.2d at 377.

II.

Spectators attending this case during oral argument might be surprised by this opinion’s rendering of the government’s position. They should be surprised. The government abandoned the analysis pursued in its brief and charted an entirely different course at oral argument. There, the government contended that Judge Duff had in fact relied upon the Priess Affidavit (and the statements contained in it) to impose sentence, but that Judge Duff relied only upon paragraphs that Barnes did not challenge in this appeal — paragraphs 8 and 11 — to determine the sentence.

When pressed by the bench about the novelty of this argument, the government repeatedly claimed that it was merely an extension of the argument contained in its brief. That is simply not true and is belied by the government’s own brief. There, the government wrote that “[wjhile the trial court did make a passing reference to defendant’s involvement with Mr. Mitchell, it is clear from the record that the court did not rely on that information in revoking defendant’s probation or in determining defendant’s sentence.” Appellee's Brief at 10 (emphasis supplied). To reinforce the point, the government added, “[ajlthough the court initially referred to defendant’s involvement with Mr. Mitchell, the record shows that the court did not consider this factor in sentencing defendant.” Id. at 13 (emphasis supplied). Nowhere do these statements even remotely suggest that Judge Duff consulted one part of the affidavit but ignored another part while imposing sentence. Indeed, these two statements are entirely inconsistent with the government’s position at oral argument. In any event, the government waived its novel analysis about the Priess Affidavit by not making that argument in its brief. Williams v. Chrans, 894 F.2d 928, 932 n. 1 (7th Cir.1990).

We are not sure why the government had a change of heart on the eve of oral argument. We are disturbed, however, by the government’s tactics and its less than candid responses to this tribunal. We acknowledge the government’s recent letter of apology for these errors in judgment and trust that such incidents will not recur in the future.

VACATED AND REMANDED with instructions.

. Barnes’s case was reassigned to Judge Duff on March 28, 1988.

. That these statements about Mitchell refer to Barnes's conduct during 1988 is supported by Judge Duff's characterization of Barnes as a "very, very heavy dealer.” It is highly unlikely that, when the district court made this characterization, it was alluding to Barnes’s 1985 conviction for distribution of cocaine. In 1985, Barnes was convicted of distributing only 7.15 grams of cocaine, hardly making him a "very, very heavy dealer.” It indeed is more likely that these references were to the informant’s suggestion (contained in the affidavit) that Barnes played a major role in cocaine distribution during his probation in 1988.

. Of course, Barnes does not claim that district courts may never rely upon any hearsay statements, nor can he. As we wrote in United States v. Harris, 558 F.2d 366 (7th Cir.1977):

"[O]nce the guilt of the accused has been properly established, the sentencing judge, in determining the kind and extent of punishment to be imposed, is not restricted to evidence derived from the examination and cross-examination of witnesses in open court but may, consistently with the Due Process Clause of the Fourteenth Amendment, consider responsible unsworn or ‘out-of-court’ information relative to the circumstances of the crime and to the convicted person’s life and characteristics.”

Id. at 373 (quoting Williams v. Oklahoma, 358 U.S. 576, 584, 79 S.Ct. 421, 426, 3 L.Ed.2d 516 (1959)).