Opinion by Chief Judge KOZINSKI; Dissent by Judge IKUTA.
*1026OPINION
KOZINSKI, Chief Judge:This is a strange case. Its resolution hinges on the absence, as a factual matter, of something we must accept as a legal matter. There are unlikely to be many more like it, so this opinion’s precedential value is probably limited. We nevertheless publish pursuant to General Order 4.3. While we’re at it, we offer some advice to lawyers: Don’t apologize unless you’re sure you did something wrong. And there’s also a lesson for district judges: Don’t accept too readily lawyers’ confessions of error or rely on your own memory of what happened. Trials are complicated and we sometimes misremember details. That’s why we have transcripts.
Facts
This case arises from a lawsuit filed by Philip Miller’s family against the City of Los Angeles, its police department, police chief and Sergeant Mata. Philip died after Mata shot him, and plaintiffs claimed that Mata was not justified in using deadly force. The district court issued an in limine order precluding defendants from arguing that the decedent was armed when he was shot. In his summation, defense counsel Richard Arias argued that Mata thought Miller failed to surrender because he had shot Bean just moments earlier. Plaintiffs’ counsel objected, apparently based on the in limine order. The court sustained the objection and instructed the jury to ignore Arias’s statement.
The jury was unable to reach a verdict and the district court declared a mistrial. The case was eventually retried and a second jury returned a defense verdict.
Plaintiffs moved for sanctions against Arias for his statement during the first trial’s summation. Defendants conceded that Arias had violated the in limine order but opposed sanctions on the grounds that the transgression was inadvertent, fleeting and harmless. Arias attached a declaration admitting fault and apologizing. Exercising its inherent power, see Chambers v. NASCO, Inc., 501 U.S. 32, 43-46, 111 S.Ct. 2123, 115 L.Ed.2d 27 (1991), the district court granted the motion and sanctioned defendants $63,687.50. They appeal.
1. We must first determine whether and to what extent Arias violated the district court’s in limine order. This might seem superfluous, given that defendants conceded Arias violated the order and Arias even apologized for it. But defendants never conceded that Arias’s conduct “ ‘constituted or was tantamount to bad faith,’ ” as it had to have been in order to be sanctionable under the court’s inherent power. Primus Auto. Fin. Servs., Inc. v. Batarse, 115 F.3d 644, 648 (9th Cir.1997) (quoting Roadway Express, Inc. v. Piper, 447 U.S. 752, 767, 100 S.Ct. 2455, 65 L.Ed.2d 488 (1980)). A clear-cut or egregious violation is more likely to support a finding of bad faith than a minor or ambiguous transgression. To determine whether we can sustain the district court’s finding that Arias acted in bad faith, we must know what line he crossed and how far he crossed it. Defendants do contest before us the district court’s finding that there was a violation, so we have the benefit of briefing and argument on this issue. We review for substantial evidence.
The in limine order precluded Arias from arguing “that the decedent Philip Miller ... possessed a weapon when shot by Defendant Sergeant Mata.” But Arias’s summation was about how Sergeant Mata perceived the situation. This is Arias’s entire argument, with immaterial and repetitive portions omitted:
*1027Anyway, he sees the decedent coming out. Now, he thinks this is happening in a split second.... And that’s the time he’s got to compute to see what happens. And he sees, oh, my God, he’s going to do the kid.
Well, what’s he base that upon? Does he have a reasonable — would an objectively reasonable officer conclude that there’s a fair probability that the decedent posed an imminent threat of death or serious bodily injury? What is it that Sergeant Mata bases that opinion upon to make it reasonable? Does Sergeant Mata have a fair probability? He thinks he’s going to do it.
Well, this is what Sergeant Mata testified to. This is the evidence.... The plaintiff has not brought in anybody to contradict what Sergeant Mata said what he saw that night....
All right. He’s coming out the door and he’s thinking, oh, my God. He’s going to do him. Well, what does he see that leads him to believe that the decedent is going to kill him?
... These are the things that he sees. And your body — your mind is a computer, and you’re computing all of this. And he’s got less than a second to compute all of this and come to the reasonable conclusion this guy is going to shoot the kid. He’s an imminent threat of killing that citizen down there.
But it doesn’t stop right there. He’s dressed like this. He’s got his hand over here, and everybody else is acting differently. When he walks up to him— and he’s walking deliberate and erect at that time and focused on him. And what does Sergeant Mata say, when he believes that this guy has a gun in his hand — and the reason that he believes that he’s got a gun in the hand is the physical position. There’s no reason for him to be handing like this. He’s just got his hand in his pocket. You’re walking with the left hand out. He’s got the left hand over the right.
... Sergeant Mata thinks, my God, he’s going to kill him; so he stops says, “Get down.” And he says, “Get the fuck down. Get the fuck down.”
All of a sudden he starts to use a swear word. He gives more emphasis to his command. The situation has changed. Now it’s imminent threat of death....
... What does he do? He’s got his hand in the pocket. Does he take his hand out so the officer can see it? The testimony is — and they’re taught it’s the hands that kill you.... That’s why Sergeant Mata is trying to see where their hands are. Do they present a threat? Did he bring his hands up? ...
Did he do that? No. He keeps his hand in his pocket, other hand over. It’s uncontroverted. There’s no evidence to the contrary. Then he turns to the right and starts walking, and Sergeant Mata describes it as if he is trying to hide the gun from Sergeant Mata.
He’s got his hand in here and he’s walking up like this, and all of a sudden he stops. Sergeant Mata says I don’t know why....
Okay. He stops. He comes up.... And then there’s that moment where he makes a decision. And he turns to the left, and he faces Sergeants [sic] Mata.
Now, here’s where the toxicologist does come in. We know that he was drunk. He had .12 or 14 — yeah. .12, 14 alcohol. We know he had marijuana in his system.
Maybe that clouded his mind. I don’t know. Maybe he was some a rage be*1028cause he just fought with this kid and, you know, the blood was pumping. I don’t know. But whatever happened, he’s got a police officer in front of him when he saw Sergeant Silva, oh, I’m trapped. I can’t go anywhere. My God, man. Get down on the ground. End it right there.
He can’t because he had shot Bean inside.
(Emphases added.)
The tale Arias narrates is consistently from Sergeant Mata’s perspective. He tries to get the jury to see the situation from the policeman’s point of view, which makes perfect sense given that the jury had to decide whether Mata acted like a reasonable officer. See Graham v. Connor, 490 U.S. 386, 396-97, 109 S.Ct. 1865, 104 L.Ed.2d 443 (1989). Twenty times Arias makes it clear he’s looking through Sergeant Mata’s eyes and explaining what was going on in Sergeant Mata’s head. Not every one of his sentences is so qualified, but there was no need to do that; it was enough, for ordinary understanding, to punctuate the narrative with occasional indications that it was a tale told through Mata’s eyes.
The last sentence — the one for which the district court imposed over $60,000 in sanctions — says nothing about Miller being armed when he confronted Mata. For it to violate the district court’s order, it would have to carry the clear implication that Miller still had the gun with which he’d shot Bean. It plainly did not; Miller could have shot Bean and dropped the gun or passed it to a confederate before coming out into the open.
To the extent there may have been some confusion, plaintiffs moved to instruct the jury, which the district court did and with which Arias immediately agreed:
MR. BURTON: Objection, your honor. Move to strike. I mean—
THE COURT: The objection is sustained, and that statement is ordered stricken.
MR. BURTON: Can the jury be admonished, your honor?
THE COURT: Yes.
Ladies and gentlemen, remember that argument is not evidence in the case. There’s no evidence to support that last statement.
MR. ARIAS: I stand corrected. There is absolutely no evidence that he had a gun in his hand. Sergeant Mata even admits that.
For Arias to accept the district court’s ruling and get on with the trial was entirely understandable. But for him to admit to a transgression he never committed when the opposing party was seeking sanctions was foolish and unhelpful. His after-the-fact apology locked him and his clients into a violation he never committed. And he misled the district judge by doing so.
At the same time, the district judge should have troubled to explain just how Arias violated the order before imposing sanctions. If the district court didn’t want Arias to argue that Miller might have shot Bean, it should have entered an order saying so. But having issued the order it did, the district court, like the parties, was bound by its terms. Orders can constrain conduct only to the extent their words give clear notice of what is prohibited. See Robert Bolt, A Man for All Seasons 72 (Vintage Books 1960) (“It will mean what the words say!”). The district court’s in limine order did not provide adequate notice that the statement Arias made was offside. See Foster v. Wilson, 504 F.3d 1046, 1052-53 (9th Cir.2007).
Clarity and precision are particularly important when limiting what lawyers may argue to the jury. The lawyer’s job in *1029summation is not merely to rehash the evidence but also to suggest inferences the jury should draw from the proven facts. A lawyer is a poor advocate if he fails to nudge the jurors to use their common sense and experience in filling gaps left by the proof.
This is precisely what Arias was doing here, and he was certainly entitled to argue that Sergeant Mata reasonably believed that Miller posed a threat because he had just shot Bean. Indeed, it’s hard to imagine how a lawyer defending police in these circumstances can avoid arguing that the officer reasonably believed the suspect was dangerous. Were we to hold that such an argument is sanctionable, the bar would notice and it would doubtless chill zealous advocacy.
2. The fact remains that Arias conceded that he violated the order, and defendants didn’t argue otherwise to the district court, which is probably why the district court saw no reason to belabor the point. And this concession amounts to a waiver so that we must deem a violation established for purposes of this appeal even though Arias didn’t actually violate the order.
Arias and the city did not, however, concede that the violation was made in bad faith; they vigorously dispute it here and below. This raises the unusual question of how we treat a finding of bad faith for a transgression that didn’t actually occur. We conclude that Arias couldn’t have acted in bad faith if he did not, in fact, violate the district court’s order. You can’t have chicken parmesan without chicken; you can’t have an amazing technicolor dream-coat without a coat; you can’t have ham and eggs if you’re short of ham or eggs. And you can’t have a bad faith violation without a violation.
Had Arias actually crossed the line drawn in the sand by the district court, it would have been permissible to infer bad faith from his action plus the surrounding circumstances. For any such determination we’d owe the district judge considerable deference. United States v. Hinkson, 585 F.3d 1247, 1251 (9th Cir.2009) (en banc). But no inference about Arias’s state of mind in committing the violation can be drawn when he committed no violation. The waiver establishes the violation as a legal matter, but any inference that Arias had an evil state of mind in doing something he didn’t do is “illogical, implausible, [and] without support in inferences that may be drawn from facts in the record.” Id.
3. There’s an additional reason we can’t sustain the sanctions in the amount imposed by the district court. It’s clear the district court meant the sanctions to be compensatory. The amount sanctioned ($63,678.50) was precisely what the Millers claimed as attorney’s fees for the first trial. The Millers asked for “sanctions which reflect the costs and attorney’s fees for the first trial,” and that’s what the district court said it was granting: “The Court awards to Plaintiffs their reasonable attorneys’ fees and costs incurred for the conduct of the trial.”
But if the sanctions were compensatory, the district court had to link Arias’s statement to the harm suffered by the Millers. In other words, the amount awarded had to compensate the Millers for the damage actually caused by Arias’s eight-word sentence. The district court made no finding of causation, and without a finding that Arias’s eight words caused the first jury to hang, the district court had no authority to order defendants to compensate plaintiffs for the attorneys’ fees and costs they spent on the first trial. See In re Dyer, 322 F.3d 1178, 1195 (9th *1030Cir.2003) (remanding “for a determination of the Trustee’s actual damages flowing from the automatic stay violation alone” if compensatory sanctions were to be upheld); B.K.B. v. Maui Police Dep’t, 276 F.3d 1091, 1109 (9th Cir.2002) (upholding compensatory sanctions because “the amount the court imposed reflected its assessment of the actual harm incurred by Plaintiff’).
To the extent the district court tacitly found that Arias’s eight-word sentence caused the jury to hang, any such finding is unsupported by the record. Immediately after Arias spoke those eight words, (1) opposing counsel objected, (2) the judge cautioned the jury and (3) Arias himself said, “I stand corrected. There is absolutely no evidence that he had a gun in his hand. Sergeant Mata even admits that.” See p. 1028 supra. We have a strong presumption that jurors follow instructions. See Richardson v. Marsh, 481 U.S. 200, 206-07, 107 S.Ct. 1702, 95 L.Ed.2d 176 (1987). And Arias himself helped neutralize whatever impression his sentence may have left on the jury by devouring his words and conceding there’s “absolutely no evidence” that Miller was armed.
In the opposition to sanctions, Arias noted that he and the Millers’ attorney spoke at some length with the jurors after the first trial. If any of the jurors had said anything suggesting that Arias’s single sentence caused the jury to hang, the Millers would no doubt have proffered a declaration to that effect. Federal Rule of Evidence 606(b) bars only “an inquiry into the validity of a verdict or indictment.” Juror statements would have been admissible because the jury here returned no verdict. That the Millers failed to proffer any juror statements strongly suggests that the jurors said nothing supporting them claim.
The district court may also have meant the sanctions to vindicate the court’s authority and deter future misconduct, but $63,687.50 is an extraordinary amount for such non-compensatory sanctions. We’ve held that non-compensatory sanctions of that magnitude are akin to criminal contempt and may be imposed only by following the procedures applicable to criminal cases, including appointment of an independent prosecutor, proof beyond a reasonable doubt and a jury trial. F.J. Hanshaw Enters., Inc. v. Emerald River Dev., Inc., 244 F.3d 1128, 1136-42 (9th Cir.2001) (citing Blanton v. City of N. Las Vegas, Nev., 489 U.S. 538, 544, 109 S.Ct. 1289, 103 L.Ed.2d 550 (1989), as “implying that $5,000, at least in 1989 dollars, is the cutoff for a serious fine warranting a jury trial”). Other circuits are in accord. See Mackler Prods., Inc. v. Cohen, 225 F.3d 136, 139 (2d Cir.2000) (reversing $2000 in non-compensatory sanctions because “the District Court erred in imposing [the sanctions] without providing the procedural protections employed in the criminal process”); see also Plaintiffs’ Baycol Steering Comm. v. Bayer Corp., 419 F.3d 794, 808-09 (8th Cir.2005) (requiring the protections of the criminal process before imposing $50,000 in non-compensatory sanctions); Crowe v. Smith, 151 F.3d 217, 227-29 (5th Cir.1998) (citing Int’l Union, United Mine Workers of Am. v. Bagwell, 512 U.S. 821, 833, 114 S.Ct. 2552, 129 L.Ed.2d 642 (1994), as suggesting that, for individuals, non-compensatory sanctions of $5000 or more require a jury trial); Buffington v. Balt. Cnty., Md., 913 F.2d 113, 133-35 (4th Cir.1990) (requiring the protections of the criminal process before imposing $7000 in non-compensatory sanctions). None of these procedures was employed here, and we therefore cannot sustain the sanctions as intended to vindicate the court’s authority and deter future misconduct.
*1031We reverse the district court’s order imposing sanctions. On remand, the district court may, if it chooses, hold further proceedings, consistent with our opinion, to determine whether any sanction is warranted for Arias’s conceded violation.
REVERSED AND REMANDED.