OPINION

KEENAN, Circuit Judge:

Joseph Poole appeals from a judgment entered after a bench trial, in which the district court found him guilty of four counts of aiding in the preparation of false tax returns in violation of 26 U.S.C. § 7206(2). To obtain these convictions under § 7206(2), the government was required to prove that Poole willfully aided, assisted, or otherwise caused the preparation of a tax return that was fraudulent or false with regard to a material matter. See United States v. Hayes, 322 F.3d 792, 797 (4th Cir.2003); United States v. Aramony, 88 F.3d 1369, 1382 (4th Cir.1996).

Poole makes three principal arguments in his challenge to his convictions: 1) the district court unlawfully based its verdict on the guilty pleas of his co-defendants, which were not evidence in the case, thereby depriving Poole of his due process right to a fair trial; 2) the district court improperly credited testimony by the government's key witness that Poole contends was false; and 3) the evidence was insufficient to prove that Poole knew that the tax returns he prepared were fraudulent, and that he acted willfully in violation of § 7206(2). We conclude that these arguments are without merit, and we affirm the district court's judgment.

I.

Between 1998 and 2003, Poole worked as an accountant for Bay Area Accounting & Management, an accounting firm owned by Poole's wife. In his capacity as an accountant, Poole provided accounting services to Fidelity Home Mortgage Corporation (Fidelity Home), a mortgage brokerage firm with headquarters located in Baltimore, Maryland. Stilianos Mavroulis was the sole shareholder of Fidelity Home, and his son, Kyriakos Mavroulis, was the supervisor of the company's accounting department.

During the years in question, Poole prepared various financial documents for Fidelity Home. These documents included year-end financial statements and informational tax returns that were filed with the Internal Revenue Service (IRS). Poole also prepared personal financial statements for Stilianos Mavroulis, and joint income tax returns for Stilianos Mavroulis and his wife.

In 2003, the IRS received information from a former employee of Fidelity Home that Stilianos Mavroulis may have been filing fraudulent income tax returns. Following an investigation by the IRS, a federal grand jury returned an indictment charging Stilianos Mavroulis, Kyriakos Mavroulis (collectively, the Mavroulises), and Poole with conspiracy to defraud the United States, in violation of 18 U.S.C. § 371. The Mavroulises also were charged with filing fraudulent tax returns, in violation of 26 U.S.C. § 7206(1). The indictment separately charged Poole with four counts of aiding and assisting in the preparation of false tax returns, in violation of § 7206(2).

On the defendants' motion, the district court severed the trial of the Mavroulises from the proceedings against Poole. Stilianos Mavroulis later pleaded guilty to one count of filing false tax returns, and Kyriakos Mavroulis pleaded guilty under a superseding indictment to one count of willfully failing to file an income tax return, in violation of 26 U.S.C. § 7203. In a separate proceeding before the same district court judge, Poole entered a plea of not guilty.

The case proceeded to a bench trial, in which the government sought to prove that Poole conspired with Stilianos Mavroulis (Mavroulis) in preparing false income tax returns that materially underreported Mavroulis's taxable income from Fidelity Home. The district court acquitted Poole of the conspiracy charge, but found him guilty of the four counts of aiding in the preparation of false tax returns. Poole later filed motions for a judgment of acquittal under Federal Rule of Criminal Procedure 29, and for a new trial pursuant to Rule 33. The district court denied these motions, and sentenced Poole on the four counts to a total term of two years' imprisonment and to one year of supervised release.

II.

A.

We first consider Poole's argument that the district court erroneously based its judgment on the Mavroulises' guilty pleas, which were not admitted into evidence. Poole relies primarily on three references that the district court made to the pleas during the trial and in the court's memorandum opinion.

We consider these statements by the district court in the context in which they were made. During the government's case-in-chief, the district court interrupted Poole's counsel as he cross-examined the government's key witness, IRS Agent Richard Wallace. The district court suggested that Poole's counsel spend less time eliciting from Agent Wallace the details of the tax returns at issue, stating:

The second statement by the court occurred after the government rested its case and Poole filed a motion for a judgment of acquittal. During a hearing on that motion, counsel for Poole argued that although the government had presented a strong case against Mavroulis, there was insufficient evidence to support a conviction of Poole. The district court interjected, suggesting that counsel did not need to evaluate the strength of the government's case against Mavroulis because he had "already pled guilty."

The third reference relied on by Poole occurred in footnote three of the district court's memorandum opinion. There, the district court stated that to the extent Poole claimed that the tax returns at issue did not violate federal tax laws, that defense was "belied" by Agent Wallace's testimony and "by the simple fact that Stilianos Mavroulis had pled guilty on the advice of his attorneys to four separate counts of filing tax returns in violation of 26 U.S.C. § 7206(1)." The district court made a substantially similar statement when it announced its verdict in open court.1

Poole asserts that these statements by the district court demonstrate that the district court "prejudged" the issue of the falsity of the tax returns and the willful nature of Poole's conduct, thereby effectively relieving the government of its burden to prove these elements of § 7206(2) beyond a reasonable doubt and denying Poole his due process right to a fair trial.2 As an example of this alleged bias, Poole argues that the district court mistakenly thought that Mavroulis pleaded guilty to four counts of tax fraud, and was influenced by that factual error in finding Poole guilty "by association" of four counts of aiding in the preparation of false tax returns.

B.

Before we consider the merits of these arguments, we must resolve the parties' dispute regarding the proper standard for our appellate review. At the outset, we observe that the district court's comments constitute error, because the court apparently gave some consideration to matters that were not in evidence in the case. When a district court commits an evidentiary error, even one that implicates a defendant's constitutional rights, we ordinarily review that error for harmlessness. Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 17 L.Ed.2d 705 (1967); Bauberger v. Haynes, 632 F.3d 100, 104 (4th Cir.2011). Poole maintains, however, that harmless error review is inappropriate in this case. He contends that the district court's repeated references to the Mavroulises' guilty pleas demonstrate that the error in this case affected the entire course of his trial, from beginning to end, depriving him of his due process right to a fair trial. Thus, Poole asserts, the district court's alleged reliance on the pleas, which were not admitted into evidence, constitutes "structural" error, requiring automatic reversal of his conviction of the four charges.

To warrant such automatic reversal as "structural" error, however, an error must implicate a defendant's constitutional rights and affect the very framework in which a trial proceeds. See Neder v. United States, 527 U.S. 1, 8-9, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999); Arizona v. Fulminante, 499 U.S. 279, 310, 111 S.Ct. 1246, 113 L.Ed.2d 302 (1991); United States v. Higgs, 353 F.3d 281, 304-305 (4th Cir. 2003); United States v. Curbelo, 343 F.3d 273, 278, 285 (4th Cir.2003); United States v. Blevins, 960 F.2d 1252, 1262 (4th Cir. 1992). The Supreme Court has described structural errors as "defects in the constitution of the trial mechanism [that] defy analysis by `harmless-error' standards," because those defects are "necessarily unquantifiable and indeterminate." Sullivan v. Louisiana, 508 U.S. 275, 281-82, 113 S.Ct. 2078, 124 L.Ed.2d 182 (1993) (quoting Fulminante, 499 U.S. at 309, 111 S.Ct. 1246); Curbelo, 343 F.3d at 278, 285.

The Supreme Court has found structural error in only a very limited class of cases involving errors that affect constitutional rights and that are so intrinsically harmful to the proceeding that the errors render the trial an unreliable vehicle for determining innocence or guilt. See Tumey v. Ohio, 273 U.S. 510, 535, 47 S.Ct. 437, 71 L.Ed. 749 (1927) (biased trial judge); Vasquez v. Hillery, 474 U.S. 254, 263-64, 106 S.Ct. 617, 88 L.Ed.2d 598 (1986) (racial discrimination in selection of grand jury); McKaskle v. Wiggins, 465 U.S. 168, 177, 104 S.Ct. 944, 79 L.Ed.2d 122 (1984) (denial of self-representation at trial); Sullivan, 508 U.S. at 281-82, 113 S.Ct. 2078 (defective reasonable-doubt instruction). Thus, the vast majority of constitutional errors are not structural, but are "trial errors" that are subject to harmless error review.

We explained the difference between structural and trial errors in United States v. Blevins, a case involving a jury trial. In that case, the district court allowed the introduction into evidence of guilty pleas entered by non-testifying codefendants. 960 F.2d at 1261. On appeal, this Court assumed that the defendant timely objected to this evidence and concluded that although the district court's error was of "constitutional dimension," it was not a structural error, because the error did not affect the "entire conduct of the trial from beginning to end." Id. at 1262. We held instead that the district court's improper admission of the guilty pleas into evidence was a "trial error" subject to harmless error review, because the error "occurred during the presentation of the case to the jury [and could] be quantitatively assessed in the context of other evidence presented in order to determine whether [the admission] was harmless beyond a reasonable doubt." Id.

We will assume, without deciding, that the district court's repeated references to the guilty pleas implicated Poole's constitutional rights. Nevertheless, we hold that the district court's error is a classic example of a trial error subject to harmless error review. The effect of the district court's statements about the Mavroulises' guilty pleas, which occurred at various discrete times during the presentation of the case and in the court's memorandum opinion, can be measured against the other evidence that was properly admitted during the trial. See id. at 1262. Thus, we disagree with Poole's contentions that the cumulative effect of the error renders it "structural" in nature, and that the repetition of the remarks demonstrates bias on the part of the district court. The repetition of this error does not change its fundamental nature, because we can still evaluate its impact along with the properly admitted evidence in the case. See id. at 1263.

C.

Having rejected Poole's claim of structural error, we apply harmless error review to the district court's several references to the guilty pleas by the Mavroulises. Id. at 1262. Accordingly, we determine whether the impact of the references to those pleas was harmless beyond a reasonable doubt, such that it is clear that a rational fact finder would have found Poole guilty absent the error. Chapman, 386 U.S. at 24, 87 S.Ct. 824.

Relying on our prior decisions involving jury trials, Poole contends that the district court's references to the Mavroulises' guilty pleas were presumptively prejudicial. These decisions, however, are inapposite to the consideration of such error in the context of a bench trial. As we explained in Blevins, when a defendant is tried by a jury, there are two dangers inherent in a jury being permitted to consider a guilty plea of a co-defendant who does not testify. A defendant in such a case cannot probe the motivations for the guilty plea through cross-examination, and a reviewing court cannot be certain that a conviction rests on an assessment of the defendant's own culpability rather than the disposition of the charges against the co-defendant. Blevins, 960 F.2d at 1260.

In contrast, when a district court finds a defendant guilty after a bench trial, appellate courts generally presume that the conviction rested only on admissible evidence. See United States v. Smith, 390 F.2d 420, 422 n. 2 (4th Cir.1968) (ordinary presumption in non-jury cases is that trial judge relied only on the evidence that properly was admitted); United States v. Cardenas, 9 F.3d 1139, 1156 (5th Cir.1993) (judge, sitting as a trier of fact, is presumed to have rested decision only on admissible evidence and to have disregarded inadmissible evidence). Moreover, in the present case, the district court addressed Poole's challenge to footnote three of the memorandum opinion during post-trial proceedings and expressly disavowed any reliance on Mavroulis's plea, stating that "[the] plea had no effect at all" on the court's judgment. (emphasis added.)

In the post-trial proceedings, the district court further explained that its judgment rested solely on the government's "overwhelming" evidence that the tax forms prepared by Poole failed to report portions of Mavroulis's taxable income. The district court also stated that it rejected Poole's defense that he was merely incompetent, rather than willful, and found that Poole deliberately avoided learning that the financial records provided to him by the Mavroulises were materially false.

We agree with the district court's characterization of the evidence in this case as "overwhelming." As discussed in greater detail below, the various financial documents prepared by Poole contained large discrepancies, supporting a conclusion that Poole deliberately avoided learning of materially false representations in Mavroulis's income tax returns. For example, in 2001, Poole prepared both an informational tax return for Fidelity Home indicating that the business paid $120,000 in compensation to Mavroulis, and a personal income tax return for Mavroulis that resulted in Mavroulis receiving an "earned income tax credit," a credit available to individuals earning low wages. Therefore, based on our review of the record, including the district court's explanation of the bases for its judgment, we conclude that the district court's various references to the Mavroulises' guilty pleas did not contribute to the verdict and, accordingly, were harmless beyond a reasonable doubt. See Chapman, 386 U.S. at 24, 87 S.Ct. 824; see also ePlus Tech., Inc. v. Aboud, 313 F.3d 166, 178 (4th Cir.2002).

III.

We next address Poole's claim that the district court erred in crediting certain "objectively flawed" testimony by Agent Wallace, who was qualified by the district court as an expert witness in forensic accounting and income tax computation. Poole maintains that Agent Wallace did not understand "passive-loss" rules for related-party transactions, and that he failed to consider the impact of certain deductions on the calculation of Mavroulis's personal tax liability. Based on these alleged mistakes, Poole contends that Agent Wallace's testimony was too unreliable to be considered as evidence of Poole's guilt.

Practically speaking, however, Poole asks us to accept his own calculations of Mavroulis's tax liability in place of the numbers and conclusions drawn by Agent Wallace. We decline Poole's request, because it effectively asks us to reverse a credibility determination by the district court and to reweigh the evidence presented at trial.

In qualifying Agent Wallace as an expert witness, the district court determined that his testimony was the product of reliable principles and methods. See Fed. R.Evid. 702. Moreover, any purported factual mistakes by Agent Wallace or alleged weaknesses in his methodology were subject to cross-examination at trial. Based on the questions posed by Poole's counsel on cross-examination, the district court was aware of the asserted problems regarding Agent Wallace's testimony, and nonetheless accepted that testimony as credible and reliable. We defer to this credibility determination by the district court. United States v. Abu Ali, 528 F.3d 210, 232 (4th Cir.2008); see United States v. Murray, 65 F.3d 1161, 1169 (4th Cir. 1995).

IV.

Poole next argues that the government failed to prove that he acted willfully, and with knowledge that the tax forms he prepared contained materially false information.3 We consider this argument under an established standard of review. In examining the sufficiency of the evidence supporting a conviction, an appellate court must affirm the district court's judgment if "any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt." United States v. Madrigal-Valadez, 561 F.3d 370, 374 (4th Cir.2009) (quoting Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979)). This standard is met when there is "substantial evidence" in the record, viewed in the light most favorable to the government, to support the district court's judgment. Id.

Generally, in criminal prosecutions, the government elects to establish a defendant's guilty knowledge by either of two different means. The government may show that a defendant actually was aware of a particular fact or circumstance, or that the defendant knew of a high probability that a fact or circumstance existed and deliberately sought to avoid confirming that suspicion. See United States v. Ruhe, 191 F.3d 376, 384 (4th Cir.1999); United States v. Abbas, 74 F.3d 506, 513 (4th Cir.1996).

Under this second method of proof, in a criminal tax prosecution, evidence establishing a defendant's "willful blindness" constitutes proof of his subjective state of mind, thereby satisfying the scienter requirement of knowledge. United States v. Stadtmauer, 620 F.3d 238, 245 (3d Cir.2010); United States v. Bussey, 942 F.2d 1241, 1246 (8th Cir.1991). Thus, in a criminal tax prosecution, when the evidence supports an inference that a defendant was subjectively aware of a high probability of the existence of a tax liability, and purposefully avoided learning the facts pointing to such liability, the trier of fact may find that the defendant exhibited "willful blindness" satisfying the scienter requirement of knowledge. United States v. Anthony, 545 F.3d 60, 64 (1st Cir.2008); United States v. Wisenbaker, 14 F.3d 1022, 1027 (5th Cir.1994); United States v. Hauert, 40 F.3d 197, 203 n. 7 (7th Cir.1994); Bussey, 942 F.2d at 1246.

The rationale supporting the principle of "willful blindness" is that intentional ignorance and actual knowledge are equally culpable under the law. See Stadtmauer, 620 F.3d at 255; United States v. Jewell, 532 F.2d 697, 700 (9th Cir.1976). Intentional ignorance reflects a state of mind that extends beyond any form of mere negligence, or even recklessness. United States v. Guay, 108 F.3d 545, 551 (4th Cir.1997) ("This circuit approves willful blindness instructions when the jury is not permitted to infer guilty knowledge from a mere showing of careless disregard or mistake."); see also United States v. Wert-Ruiz, 228 F.3d 250, 255 (3d Cir. 2000); Wisenbaker, 14 F.3d at 1027.

Poole asserts that the government failed to prove either actual or constructive knowledge in the present case. He maintains that the evidence demonstrated that he reasonably relied on the financial data provided by the Mavroulises, and that this reliance insulated him from criminal liability for the four offenses at issue. After reviewing the extensive evidence before us, we disagree with Poole's arguments.

The evidence at trial showed that in 1998, Poole instructed the Mavroulises regarding the proper use of a computer software program for bookkeeping. Poole testified that, as part of this training, he advised the Mavroulises to classify any withdrawals of funds from Fidelity Home for personal use as "shareholder draws," to indicate funds withdrawn by an owner with accumulated equity in the business.4 Thereafter, according to Poole, he generally relied on the Mavroulises and their staff to record and properly classify the income and expenses of the business, including any withdrawals of funds from Fidelity Home by the Mavroulises for their personal use.

Agent Wallace testified, however, that in practice, the tax returns prepared by Poole from 1999 to 2003 so greatly underreported Mavroulis's taxable income that Mavroulis owed an additional $476,000 in federal income taxes for those tax years. According to Agent Wallace, over the tax years in question, Mavroulis withdrew more than $1,000,000 from Fidelity Home accounts for personal use. These funds were used to make school tuition payments for Mavroulis's children, to pay company credit card bills that included personal purchases, to make personal insurance payments, and to fund a personal "E-Trade" account. Agent Wallace also testified that in 2003, the sum of $98,000 was transferred to Fidelity Promotions, LLC, a music production company owned by Mavroulis and his wife. Agent Wallace explained that this payment to Fidelity Promotions was not an expense incurred by Fidelity Home and, therefore, the amount of the transferred funds should have been listed on Mavroulis's tax return as taxable personal income. Additionally, Agent Wallace testified that the "shareholder draws" shown on the ledger account included withdrawals to compensate certain of Mavroulis's children who performed work for Fidelity Home but were not listed on the company's payroll.

Agent Wallace testified that in calculating Fidelity Home's tax liability, Poole incorrectly deducted many of these personal expenses as business expenses. These improper deductions had the effect of reducing the income of Fidelity Home, which otherwise would have "passed through" to Mavroulis creating a personal tax liability. Further, according to Agent Wallace, Poole was required to treat personal expenses, which were classified by Fidelity Home as "shareholder draws," as taxable capital gains distributions to Mavroulis who at that time had a negative tax basis in the corporation.

Poole conceded that he knew when he prepared Fidelity Home's informational tax returns that Mavroulis was using company credit cards for personal expenses and taking "shareholder draws" from Fidelity Home. However, Poole also testified that he mistakenly thought that in taking "shareholder draws" Mavroulis merely was recouping $20,000,000 in equity, which Poole thought Mavroulis retained in Fidelity Home because Mavroulis personally had guaranteed certain lines of credit for Fidelity Home in that amount. Poole further stated that he did not review Fidelity Home's ledger accounts to ensure that its accounting staff were properly classifying individual transactions.

Poole also testified that at the time he prepared the tax returns in question, he applied partnership rules for calculating Mavroulis's tax basis in Fidelity Home when he should have applied rules applicable to Subchapter S corporations. According to Poole, he did not learn of this mistake until an IRS agent brought the matter to his attention.

Agent Wallace testified that on the 2001 income tax return prepared by Poole for Fidelity Home, the company claimed a deduction of $120,000 for officer compensation. This deduction reduced the corporation's profits by $120,000 and, in turn, reduced the corporate income that should have "passed through" to Mavroulis. As noted above, however, in that same year, Poole prepared a Form 1040 personal income tax return for Mavroulis that failed to include the $120,000 as income and that, in fact, resulted in Mavroulis receiving an "earned income tax credit." Poole testified at trial that the omission of the $120,000 from Mavroulis's personal income tax return was a "regrettabl[e] mistake."

Poole prepared all the financial forms at issue. He had access to all the financial data recorded and classified by the Fidelity Home staff. Considering the evidence described above, we hold that a reasonable trier of fact could conclude that Poole purposefully "closed his eyes to" large accounting discrepancies, which strongly indicated that the tax forms he prepared during the years in question contained materially false financial information. Accordingly, we affirm Poole's convictions on the four counts of tax fraud because they are supported by substantial evidence. See Madrigal-Valadez, 561 F.3d at 374.

V.

In conclusion, we hold that the district court's erroneous references to the unadmitted guilty pleas of his co-defendants constituted harmless error. We also hold that the district court did not err in its consideration of Agent Wallace's testimony, and that the evidence was sufficient to support Poole's conviction. Accordingly, we affirm the district court's judgment.

AFFIRMED