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I'm curious as to how the US legal system determines who should present evidence and how much evidence is required by them to prove one side of an argument against a counterargument.

Here's an example I thought of when stuck behind a bus this morning - I'm developing the idea as I go so I apologize if it's wordy.

This was a modern school bus that had a sign on the back which would first say "preparing to stop" when the driver begins slowing down and, presumably, pushes a button to initiate the system. Once stopped, the sign switched to a reminder that there is a camera that enforces the no passing law.

Let's imagine that the people who developed the system had the camera begin taking still images of passing cars when the system was initiated rather than when the bus actually stopped moving. Let's also imagine that the applicable law uses wording that only makes it illegal to pass the bus when it is at a complete stop.

If someone was charged with passing this bus, the prosecutors would present the image of the defendant's license plate and explain that it was taken by the bus's automated system that was engaged when the bus is stopped. The defendant doesn't know how the system works but they know that they passed a still-moving bus that was in the process of slowing and they suspect that the system is faulty.

Is it up to the prosecution to present full and complete evidence that the system only takes pictures when the bus is stopped (presumably reviewing source code or conducting tests) or is there some kind of legal concept of "good enough at a glance" evidence where they've met some minimum burden of proof that the picture is taken when the system is turned on and it's only on when the bus is stopped, therefore it must be functioning as expected?

I'm curious as to how the US legal system determines who should
present evidence and how much evidence is required by them to prove
one side of an argument against a counterargument.

In General

In both criminal and civil cases in common law legal systems (legal systems derived from the English legal system, basically, the U.S., U.K., Ireland, Canada, Australia, New Zealand, India, Pakistan and Bangladesh), the burden of proof is on the party seeking to have a court do something. So, if the absence of evidence, the party seeking relief loses.

Proof Of The Elements Of The Charge Or Cause Of Action

Presentation of Evidence and the Prima Facie Case

The party seeking court action presents their evidence first.

If at the close of their opening case that party has not presented enough evidence to meet their burden of proof with respect to every "element" of the list of legal elements that they must prove to prevail in court, that party has not established a "prima facie case" and the case is dismissed without granting relief.

If the prosecution or party bringing a civil case establishes a prima facie case, or if the defense does ask to have the case dismissed for failing to establish a prima facie case at the close of the evidence of the party asking the court to do something, then the defense presents their evidence if the defense wishes to do so (this is optional).

(If the defense does present evidence, the prosecution or civil party seeking relief can then present a rebuttal case to disprove the new points of evidence in the defense case, and so on, back and forth until all evidence is taken.)

Evaluating The Evidence In Light Of The Burden Of Proof

Once both the party asking the court to do something and the defense have presented all of their evidence, the trier of fact (i.e. the jury in a jury trial, or a judge in a bench trial) decides if every element of the case of the party asking the court to do something has been established by the relevant burden of proof.

In a civil case, the burden of proof is usually a "preponderance of the evidence" (i.e. that the evidence more strongly favors that the element was established than that it was not established); some elements on some claims in civil cases must be established by the higher standard of "clear and convincing evidence."

In a criminal case, the burden of proof is "proof beyond a reasonable doubt".

Affirmative Defenses

In addition to elements of a case that must be established to make a prima facie case, there are also "affirmative defenses" to a request that a court do something. Examples of affirmative defenses include self-defense, statute of limitations, immunity from suit, a pardon in a criminal case, etc.

A defendant can win ether by showing that the party asking the court to do something has failed to meet their burden of proof with respect to one or more elements of the case, or by showing that an affirmative defense bars the request.

In both criminal cases and civil cases, the burden is on the defense to show that there is at least some evidence that justifies consideration of an affirmative defense. This is called a "burden of production." In a civil case (and in some criminal cases in some jurisdictions), the burden of proof is on the defendant to prove an affirmative defense by preponderance of the evidence. In some criminal cases in some jurisdictions, once the defense has met a burden of production with regard to an affirmative defense, the prosecution must rule out the affirmative defense beyond a reasonable doubt to prevail.

Deciding Who Wins

Once both the party asking the court to do something and the defense have presented all of their evidence, the trier of fact (i.e. the jury in a jury trial, or a judge in a bench trial) decides if an affirmative defenses prohibit the party asking the court to do something from prevailing.

The party asking the court to do something wins unless the defense can show that this party did not meet the burden of proof as to any one element of a particular criminal charge or civil cause of action (for each charge or cause of action), or that an affirmative defense bars that particular charge or cause of action.

Often cases have conflicting testimony regarding what happened. The jury (or judge in a bench trial) can choose to belief that one person is telling the truth and that the other statement is either a lie or is unintentionally inaccurate for some reason. If the jury (or judge in a bench trial) isn't at all sure whose statement is true and whose is not, this favors the defendant if one is not more credible than the other.

Complex Cases

In a simple case, there is just one charge or cause of action, and there is just one defendant. But, often, there are multiple charges or causes of action, and each one must be evaluated as to each defendant of the multiple defendants in a single trial.

In a civil case, sometimes there are counterclaims that defendants are trying to prove against plaintiffs, or cross-claims that defendants or counterclaim defendants are trying to prove against each other that have to be evaluated. Also, in civil cases, sometimes one or more of the defendants is also prosecuting one or more separate causes of action against someone other than the original plaintiffs or co-defendants. In that case, that defendant is also a third-party plaintiff, and someone other than the original plaintiff and defendants is a third-party defendant (third-party defendants can also bring third-party counterclaims against the third-party plaintiff, third-party crossclaims against third-party codefendants, or their own claims against new parties or against the original plaintiffs).

Other Rules

Special Statutes Regarding Proof Of Facts

Sometimes, there are particular kind of facts for which a statute says that a "prima facie case" is established automatically if a certain kind of evidence is presented.

For example, it is common for the law to say that a prima facie case regarding ownership of real estate, or the status of a bus as a school bus, is established by presenting a copy of an official document that says so.

Usually, when a statute says something like that, the prima facie case can still be overcome, for example, by presenting a subsequent document that shows that the real estate was then sold to someone else, or that the school bus status of the bus was later revoked. But, when a statute like that is present, the plaintiff or prosecutor doesn't have a duty to prove the negative that there was no subsequent sale of the real estate or that the school bus status certificate was still in force on the date of the incident.

Rules of Evidence

There are also "rules of evidence" that govern what kind of facts can be presented at a trial to prove a case.

For example, in a U.S. criminal trial a fact cannot be established with evidence that is hearsay, such as an affidavit or a statement that a witness heard someone else say and is retelling to the court.

A very important rule of evidence in U.S. criminal trials that flows from the United States Constitution, is the evidence obtained by law enforcement illegally may not be presented by the prosecution, even if it definitively shows that a defendant is guilty. This is called the "exclusionary rule."

Application To Facts

Is it up to the prosecution to present full and complete evidence that
the system only takes pictures when the bus is stopped (presumably
reviewing source code or conducting tests) or is there some kind of
legal concept of "good enough at a glance" evidence where they've met
some minimum burden of proof that the picture is taken when the system
is turned on and it's only on when the bus is stopped, therefore it
must be functioning as expected?

The prosecution has to convince the jury (or the judge in a bench trial) that every element of the crime as define in the statute has been proved beyond a reasonable doubt and that any affirmative defense upon which the defense meets a burden of production has been overcome by the relevant burden of proof.

Usually, this is a broad legal standard, and the jury (or judge in a bench trial) has to decide if the burden of proof has been met by the facts presented which were legally admissible as evidence.

It wouldn't be uncommon for a defendant to present no new evidence in a defense case (other than having cross-examined the prosecution's witnesses) and merely argue at the completion of the prosecution's case that the evidence presented didn't establish a particular element of the prosecution's case beyond a reasonable doubt.

For example, the defense might argue that the picture presented by the prosecution was not taken when the bus was at a complete stop, and if the prosecution didn't present some convincing evidence that the bus was at a complete stop when the picture was taken (e.g. the testimony of the bus driver and other witnesses), the defense should win.

But, it is almost always up to the jury (or the judge in a bench trial) to decide if the prosecution's evidence is good enough to prove beyond a reasonable doubt that the prosecution proved the case.

Often a defendant will not want to call any witnesses beyond the witnesses presented in the prosecution case, because a defendant's witness might cause the jury to overcome its doubt that a fact only weakly proved by the prosecution was actually true, for example, when only one not very credible prosecution witness had testified regarding the same fact.

If the identical case were presented to two different juries, one jury could decide to believe the bus driver who said that the bus was at a stop when the picture was taken, and a different jury could decide not to belief the bus driver, and both decisions would be valid.

Consequences Of A Verdict

If the judge or jury acquits the defendant in a criminal case, the case is over and there is no appeal.

If the jury is hung (there is no unanimous ruling to convict or acquit (but see endnote)), in a criminal case, there is a mistrial and the defendant can be tried again.

If the jury convicts, one of the grounds for an appeal by the defendant is that the evidence was insufficient to prove some element of the charge beyond a reasonable doubt, and if the appellate court agrees than the conviction is overturned and there can be a retrial (or in some cases, the defendant is acquitted).

Appellate Review Of The Sufficiency Of The Proof

The law recognizes that different juries could interpret exactly the same facts in different ways and will reverse a conviction because the burden of proof was not met only if "no reasonable jury" could have interpreted the evidence in a manner consistent with a conviction.

For example, on appeal, an appellate court will always assume that the jury thought that every pro-defendant witness, whose credibility was questioned in any way by the prosecution, was lying and that the jury believed that every pro-prosecution witness was telling the truth, even if the defense presented evidence that could have caused a reasonable juror to question the truthfulness of a prosecution witness.

Appeals for failure to prove something beyond a reasonable doubt can be easier in a bench trial than in a jury trial because following a bench trial the judge will often publicly state the actual reasons in terms of findings of fact and law that the judge used to reach a conclusion. So, the defendant need only show that a key fact actually found by the judge was not supported by the requisite proof.

END NOTE

Oregon State, and prior to 2019, Louisiana, did not require juries in all criminal cases to be unanimous.

Under US law, the prosecution has the obligation to prove all of the elements of the crime "beyond a reasonable doubt" (the exact explanation in the jury instructions varies from state to state, in aid of avoiding the inference that the defense has an obligation to create a doubt). The prosecution may have the advantage that the camera evidence has already been accepted in court as reliable evidence, so the pictures almost speak for themselves (someone will testify what the picture shows). The defense still can rebut this evidence with technical findings (this happens), but that burden is on him. If we suppose that you're talking about the first time someone thinks of a school bus stop arm and employs it (without special legislation), the prosecution would have the burden of proving that the technology is reliable.

In evaluating the reliability of such evidence, the court will follow what is known as the Daubert standard. There are rules of evidence esp. title 7 where the court is required to assess the relevance and reliability of testimony (pictures don't really speak for themselves), so the prosecution being the side that wants this testimony introduced will have the burden of persuading the court that the evidence and testimony is admissible. The prosecution would not have to prove that the device is absolutely reliable, in fact some kind of evidence (voiceprints for example) have been admissible but are not scientifically reliable. Still, they may be admitted, until successfully challenged.

In a criminal case, the prosecution must prove each element of the offense beyond a reasonable doubt, to the satisfaction of the trier of fact (jury, or judge if a jury has been waived). How much proof is required to do this is pretty much up to the trier of fact.

If in your hypothetical buss case, the prosecutor presented the image and said that this showed the car passing the stopped bus, and the defendant did not challenge this, the jury (or judge) might accept this as proved, or might decide that it wasn't proved and acquit on that basis.

If the defendant does challenge the point, the jury would, in theory at least, listen to the evidence presented by both sides on the point, and decide weather the element was proved beyond a reasonable doubt. The jury can decide that the evidence for one side is not credible, and totally believe the evidence for the other side, if it thinks that proper.

In a civil case the statute under which the case is brought, or perhaps case law interpreting that statute, will decide who has the burden of proof. It may require some particular element to be proved by one side or the other, and may require "clear and convincing evidence" (or some other particular level of evidence) of some of the elements of the case. The default position, if the statute or case law is silent, is that the plaintiff must prove every element by a 'preponderance of the evidence" that is, just one bit better than 50/50.

In your hypothetical, if the driver was acquitted, it is probable that either the law would be amended to also prohibit passing the slowing bus, or else the system would be modified to visually indicate the moment when the bus stopped. Prosecutors don't enjoy losing cases, and they often can effectively lobby for changes in the law.

Note that if the bus driver testified that the car passed after the bus had stopped, the jury could choose to accept the bus driver's testimony, and ignore the video and the car driver's statement. Courts are not perfect devices to find the truth.

So to be convicted of a Crime in the United States, the prosecution must show evidence "Beyond a Reasonable Doubt". This means that when all the evidence is taken into account, it is impossible for any logical progression of events to occur in a way other than the one presented by the prosecutor (Here: That the Driver Passed a stopped bus, which is a crime."). The Burden of Proof rests with the accuser, not the denier.

The Defense is presumed innocent and thus does not even need to mount a defense against the prosecutor's evidence (he can sit there and hope the Judge or Jury see the main fault. This doesn't usually happen though). The Defense may examine all evidence against him and show why the prosecutor is incorrect. It's important to note they need not prove that they didn't do it, but rather that it's plausible that the Prosecution did not get his story right. For example, if the car was stolen, this is defense against the crime, because obviously the owner of the car wasn't driving the car at that moment. To get to your technical matter, there would be ways to show this (such as comparing the distance traveled between two pictures (the first and the movent of the second and some math that I'm not good at and you can prove speed, of motion of the camera and motion of the car. If the picture was taken and your car is closer in the first than the second, it means the camera was moving away from you, and that would imply forward motion of the bus. If there's no second picture, you can say the prosecution failed to show actual motion... you are clearly in the picture, but it's not shown that you were moving at the time at all.).

It doesn't have to be what really happen, at all... but that the prosecutor didn't prove that it happened 100% his way.

First, the hypothetical scenario describes traffic infraction, not a crime or subsequent criminal prosecution. There would be no "prosecution" and "prosecutors" and no jury involved. The case would be initially heard and decided by a single magistrate or municipal judge.

The defendant doesn't know how the system works but they know that
they passed a still-moving bus that was in the process of slowing and
they suspect that the system is faulty.

Although there is no expectation of privacy in the public, the first item that would examine is whether notice was provided to the public that images were being taken of vehicles and people.

Once stopped, the sign switched to a reminder that there is a camera
that enforces the no passing law.

That appears to be backwards, the constructive notice that images would be taken of vehicles should be posted before a "reminder".

There is no "beyond a reasonable doubt" standard for a traffic violation. There is the official who wrote the citation (if they appear) and the defendant and the judge in the courtroom.

There would be a reasonable time between the citation being issued and the hearing at the court. It is the duty of the defendant to perform their own research into the system developed and implemented by the state. The "prosecution" as it were, in this case the state's implementation of the policy, or the state agent who wrote the citation, is not under any obligation to advise the public that their system does not work as intended.

The "prosecution" does not help the defense to undermine their own case, whether in civil or criminal cases (which the hypothetical scenario is not). The defendant should have ample time to perform research into the technology used by the system; who the contractors are that installed the system in state owned and operated vehicles; internal reports of the companies which developed the system which could include documents mentioning the errors the system had during testing; how the system determines "movement"; etc.

If the defendant can prove to the judge presiding over the case that the state's system, as implemented, takes the images while the bus is moving, then the judge might side with the defendant. If not, the defendant can appeal the case, up to the U.S. Supreme Court if they decide to, to challenge the actual practice that the state's system engages in, contrary to the language in the actual statutes designed to enforce a state policy.