Bracy and Collins were convicted in 1981 by a jury in an Illinois state court of three gangster-style murders committed the previous year, and they were sentenced to death upon the jury’s recommendation, which under Illinois law bound the judge. After exhausting their state remedies, see People v. Collins, 106 Ill.2d 237, 87 Ill.Dec. 910, 478 N.E.2d 267 (1985), 153 Ill.2d 130, 180 Ill.Dec. 60, 606 N.E.2d 1137 (1992), they sought federal habeas corpus, which was denied; and we affirmed the denial in Bracy v. Gramley, 81 F.3d 684 (7th Cir.1996). (The facts relating to the crimes, which are not germane to this appeal, are summarized in that opinion.) The Supreme Court reversed, 520 U.S. 899, 117 S.Ct. 1793, 138 L.Ed.2d 97 (1997), holding that Bracy had made a sufficient showing under Rule 6(a) of the Rules Governing Section 2254 Cases in the United States District Courts to entitle him to conduct discovery concerning his claim that the judge who had presided at the petitioners’ trial, Thomas Maloney, had been biased. The Court remanded Collins’s case for reconsideration in light of its opinion in Bracy’s case. Collins v. Welborn, 520 U.S. 1272, 117 S.Ct. 2450, 138 L.Ed.2d 209 (1997) (per curiam). The cases were again consolidated in the district court, which after the discovery ordered by the Supreme Court issued an opinion denying the two petitioners a new trial on the issue of guilt but holding that they were entitled to a new sentencing hearing. United States ex rel. Collins v. Welborn, 79 F.Supp.2d 898 (N.D.Ill.1999). The parties have cross-appealed.
Judge Maloney was convicted in a federal court in 1993 of various offenses relating to his having taken bribes from criminal defendants during a period that included the year of the petitioners’ trial. See United States v. Maloney, 71 F.3d 645 (7th Cir.1995). He had not solicited or received bribes from these petitioners, but they argue that he habitually came down harder on defendants who had not bribed him than he would have done had he not been taking bribes. He , did this, they argue, both to deflect any suspicion that might arise, in the cases in which he had accepted bribes and as a result acquitted or gone easy on the defendants, that he was “soft” on criminals (which might endanger his reelection), and to increase the size and frequency of the bribes offered him. The Supreme Court held that, “if it could be proved, such compensatory, camouflaging bias on Maloney’s part in petitioner’s own case would violate the Due Process Clause of the Fourteenth Amendment.” 520 U.S. at 905, 117 S.Ct. 1793 (emphasis added). In concluding that Bracy had presented enough evidence of such bias to entitle him to seek additional evidence through discovery, the Court focused on the contention that Bracy’s trial counsel, Robert McDonnell, who had been appointed by Maloney to represent Bracy, had practiced law with Maloney before the latter had become a judge, and that McDonnell “might have been appointed with the understanding that he would not object to, or interfere with, a prompt trial, so that petitioner’s case could be tried before, and camouflage the bribe negotiations in,” a contemporaneous case before Maloney. Id. at 908, 117 S.Ct. 1793. The Court pointed out that “this is, of course, only a theory at this point; it is not supported by any solid *606evidence of petitioner’s trial lawyer’s participation in any such plan.” Id. But if substantiated, this theory that Bracy’s “trial attorney, a former associate of Malo-ney’s in a law practice that was familiar and comfortable with corruption, may have agreed to take this capital ease to trial quickly so that petitioner’s conviction would deflect any suspicion the rigged ... cases might attract,” id. at 909, 117 S.Ct. 1793, would support “his claim that Malo-ney was actually biased in petitioner’s oum case.” Id. (emphasis in original). The Court rejected the view of the judge who had dissented in our court that “petitioner was entitled to relief whether or not he could prove that Maloney’s corruption had any impact on his trial. The latter conclusion, of course, would render irrelevant the discovery-related question presented in this case.” Id. at 903 n. 4, 117 S.Ct. 1793 (citation omitted). Regarding “the correctness of the various discretionary rulings cited by petitioner as evidence of Ma-loney’s bias,” the Court remarked that “many of these rulings have been twice upheld, and that petitioner’s convictions and sentence have been twice affirmed, by the Illinois Supreme Court.” Id. at 906 n. 6, 117 S.Ct. 1793.
Twice the Supreme Court said that compensatory bias must, to provide a basis for relief for Bracy (and hence for Collins), be shown “in petitioner’s own case.” This means that even if Maloney engaged in compensatory bias in some cases, this would not be enough to justify a conclusion that the petitioners had been convicted and sentenced in violation of the due process clause; the petitioners would have to prove that Maloney had been biased (“actually biased,” as the Court said) at their trial. A further straw in the wind is the Court’s approving reference to our description of the theory of compensatory bias as “speculative”: “The Court of Appeals, in its opinion, pointed out that this theory is quite speculative; after all, it might be equally likely that a judge who was ‘on the take’ in some criminal cases would be careful to at least appear to favor all criminal defendants, so as to avoid apparently wild and unexplainable swings in decisions and judicial philosophy.” Id. at 906, 117 S.Ct. 1793, citing 81 F.3d at 689-90.
Sometimes the temptation to bias is so great that proof of bias is not required. This is true when the judge has a substantial pecuniary stake in the outcome of the case or when he is bribed by one of the parties. See, e.g., Aetna Life Ins. Co. v. Lavoie, 475 U.S. 813, 825, 106 S.Ct. 1580, 89 L.Ed.2d 823 (1986); Del Vecchio v. Illinois Dept. of Corrections, 31 F.3d 1363, 1370-80 (7th Cir.1994) (en banc); Cartalino v. Washington, 122 F.3d 8, 11 (7th Cir.1997). Given the difficulty of peering into a judge’s mind, in the absence of confession a high probability of bias is the most that can ever be proved and sometimes the objective circumstances alone are enough to establish the requisite probability. But it is apparent from the passages that we have quoted from the Bracy opinion that the Supreme Court does not regard the temptation to engage in compensatory bias as falling into the per se category, where proof of the temptation is enough to entitle a defendant to a new trial because the likelihood that the judge succumbed (perhaps quite unconsciously) is great. If it did fall into the per se category, as our dissenting colleague had argued it should, there would have been no occasion to conduct discovery, since the existence of the temptation was conceded and the only question was whether Malo-ney had yielded to it, either generally or in Bracy and Collins’s case. The Court obviously thought it important to inquire whether Judge Maloney had succumbed. Later we decided a case, Cartalino, in *607which the requisite proof was supplied: the bribery scheme included convicting Cartalino. There is no evidence that Ma-loney’s bribery scheme involved convicting Bracy and Collins. It is not irrelevant to note that if the possibility of compensatory bias is alone enough to establish actual bias, all decisions by a judge who accepts bribes would be invalidated — in the case of Judge Maloney, literally thousands.
The evidence establishing the existence of compensatory bias in a particular case need not be case specific. Had Maloney, who was deposed as part of the discovery conducted on remand, testified that he had practiced compensatory bias in all the cases in which he had not been bribed, and his testimony had been believed, it would be irrelevant that there was no evidence about the motive for his rulings in the trial of these petitioners, or even that he had no recollection of that trial. All that would have to be established, all that had to be established in the remand proceedings, was a factual basis for inferring that Malo-ney probably did harbor an actual bias against the defendant. That could not be inferred from the fact that Maloney took bribes, or even, as we have noted, from the fact, if it was a fact, that he practiced compensatory bias, for he may not have done so in every case. We do not know whether he practiced it in any case; and he would have been unlikely to practice it in every case. If he thought that a defendant was certain to be convicted and receive a severe sentence, he would have no incentive to lean in favor of the prosecution and by doing so jeopardize the conviction or sentence by making it more vulnerable to reversal on appeal.
But we must consider more closely the findings of the district judge on remand and the evidence on which they are based. To begin with, the judge found that McDonnell had never practiced law with Maloney and had pulled no punches in his defense of Bracy. This finding is not clearly erroneous, and so it binds us and wipes out the theory of bias that was the focus of the Supreme Court’s discussion of the need for discovery.
The district judge noted that during his allocution before being sentenced, Maloney had spoken of the convictions and sentences of Bracy and Collins as “a credit to his record as a judge and evidence that he was not corrupt.” 79 F.Supp.2d at 907. This led up to the judge’s last and critical finding in the part of his opinion captioned “findings of fact,” which was that (id. at 908)
during the same time petitioners’ case was pending, other cases were pending in which Maloney took bribes, particularly the close in time Chow and Rosario cases. Before and after this time, Malo-ney was engaged in a pattern of receiving money. Based on the evidence in the record, it is a possible and reasonable inference in this case that Thomas Maloney was motivated, at least in part, to maintain a prosecution-oriented attitude and to make pro-prosecution rulings by a desire to deflect suspicion from cases in which he accepted bribes. Other documented instances of Maloney so acting to deflect suspicion from his corrupt conduct are reported in the Hawkins and Titone cases.
This finding has no factual support; it is merely conjecture. It was natural for Ma-loney, at his sentencing for accepting bribes from criminal defendants, including defendants in murder cases, to point to a case before him in which the murderers had been convicted and sentenced to death, though the jury, not he, had convicted them and had made a recommendation for death that bound him. It does not follow that when he presided at the defendants’ trial he was thinking of how their *608convictions and sentences might stave off future accusations of bribe taking, or even how they might dispel suspicions of it if he was even aware at that time, early in his bribe-taking career, that there were any suspicions — probably he was not, or he would not have continued taking bribes for nine more years. The two cases to which the district judge referred as examples of Maloney’s “acting to deflect suspicion from his corrupt conduct” are cases in which Maloney accepted bribes, but in one he returned the bribe because he realized that he was under investigation and in the other he convicted the defendant anyway. Neither case had anything to do with compensatory bias.
The district judge seems to have based his conclusion about Maloney’s motivation largely on the “Government’s Official Version of the Offense” submitted in Malo-ney’s criminal trial. This document, which the parties refer to as the sentencing recommendation or sentencing memorandum, is also the cornerstone of the petitioners’ appeal. In it the Justice Department accused Maloney (whom it called “degenerate” and “a mafia factotum”) of practicing compensatory bias. The document consists, however, of 57 single-spaced pages, and the allegation of compensatory bias appears on just one of them. It is colorful (“Thomas Maloney far surpassed the category of corrupt jurist to chart a new territory of defilement”), vivid, even plausible. But no substantiation or elaboration is offered. No cases in which Maloney may have engaged in compensatory bias are cited; no evidence, direct or circumstantial, admissible or inadmissible, that he ever engaged in the practice is offered. The allegation is sheer conjecture. The Justice Department understandably was pressing for a very long sentence (more than 20 years), and it pulled out all the stops.
The district judge may have had misgivings about the theory of compensatory bias that he had just embraced, because in the section of his opinion captioned “conclusions of law” he made a finding (a finding of fact, not a conclusion of law) that “the evidence does not establish that an interest in covering up wrongdoing or motivating larger bribe payments pervaded every action taken by Maloney as a judge. Malo-ney’s bribe-taking has not been shown to have been so pervasive a part of his judicial practices that it can be assumed he was always, or even usually, motivated by his pecuniary and/or penal interests when exhibiting his prosecution-oriented tendencies.” Id. at 909 (emphasis added). The judge then examined Maloney’s rulings at the trial of Bracy and Collins to see whether this might have been one of those unusual cases and found no rulings at the guilt phase of the trial that were suggestive of bias. He concluded that the convictions were untainted. The conclusion is correct, and compelled by the evidence, which was even weaker than the judge thought. For all that appeal’s, Maloney was a prosecution-minded judge for reasons unrelated to his taking bribes. That he would accept payment to acquit criminals does not imply any affection for criminal defendants or their lawyers such that he must have been acting against character when he x-uled in favor of the prosecution in cases in which he was not bribed. That is a possibility, but no more than a possibility. Maloney’s conduct was appalling, his character depraved, but the bridge to the trial of Bracy and Collins is missing. He was certainly capable of dreaming up and acting on compensatory bias, but there is no evidence that he did.
But examining Maloney’s rulings at the sentencing phase of the petitioners’ trial, the district judge found the taint of compensatory bias. The only ruling (or pair of rulings) he mentioned was Maloney’s re-*609fúsal to sever Collins’s sentencing hearing from Bracy’s and conduct it first in order to give Bracy’s lawyer more time to prepare for his client’s hearing. The ruling is said to have harmed Collins because it meant that the jury would hear evidence about additional murders that Bracy had committed in Arizona, murders in which Collins had not been implicated. Collins had not raised the issue of severance in his state-court appeal, and as a result it was treated as forfeited in the federal habeas corpus proceeding. It is not surprising that he didn’t raise the issue, because it is very difficult to see how he would have been harmed, rather than helped, by evidence that Bracy was a worse murderer than he. And so it is difficult to see how the ruling could be thought evidence of bias. More important, the ruling on the motion to sever was not new evidence obtained through discovery after the remand by the Supreme Court. That in itself would not disqualify the ruling as a basis for finding compensatory bias; but in fact the discovery brought nothing to light that made the inference of compensatory bias any more compelling than it had been before the discovery was conducted. • Much of the discovery consisted of a wild goose chase after McDonnell’s relationship to Maloney. The chase did uncover ugly evidence of criminality and mob ties of both McDonnell and Maloney, but nothing that bore on the issue of compensatory bias— except to dispel the suspicion that Maloney had appointed McDonnell to make sure that Bracy would be convicted, or that McDonnell had tried to throw the case in order to curry favor with Maloney.
The petitioners note the district judge’s failure to mention that their trial was a high-profile case sandwiched in between two cases in which Maloney had accepted bribes and that Maloney may have believed that if he gave the prosecutor what he wanted in this high-profile case, a case of particular importance to the Cook County state’s attorney’s office, the prosecutor would be less likely to act on any suspicion that he might have had of Maloney’s bribe-taking. These points do not nudge the possibility that Maloney was biased in the petitioners’ case to a probability. After lengthy discovery the theory that Ma-loney practiced compensatory bias, either generally or in this case, remains hopelessly speculative. The petitioners have failed to show that they were denied due process of law either at trial or in sentencing. To reverse their convictions or sentences would merely compound Maloney’s wrongdoing.
The denial of relief with respect to the petitioners’ convictions is affirmed. The grant of such relief with respect to their sentences is reversed with instructions to enter judgment for the respondents.