What determines a suspect's guilt based purely on circumstantial evidence?

The decision of the jury. But if the evidence is so shaky the prosecutor may not decide to bring the case unless he is sure he can win.

Circumstantial evidence is how grave injustices may occur, and why a defendant needs a good attorney. The prosecutor is convincing a jury that that the only explanation for the circumstantial evidence is that the defendant committed the crime.

But weak juries are easily swayed by a strong prosecutor and a less than impartial judge. Add an overworked public defender and someone is going to jail. And many on the criminal justice system figure, “if he’s not guilty of this crime, I am sure he is guilty of others.”

Do not forget that there is an entire industry that specialises in spying on jury members so that the prosecution can dismiss those who are not likely to vote in their favour.
That is why I am not a fan of juries. They are little children swayed by candy.

Circumstantial evidence is not necessarily weak evidence. Fingerprints on a smoking gun? That’s circumstantial! Wikipedia has an entry that gets it right.
There may be other explanations for such evidence, but don’t think for a moment that it carrys little weight.

“Proof” is what the prosecutors are able to put in front of a jury (based on what they can discover during investigation, the rules of evidence, and the opinion of the trial judge as it relates to admissibility, and despite any objections from the defense) ... and what the jury believes.

If the only evidence is a person saying “I saw him do it,” and if the jury believes that person, then that’s all the proof that’s required. Naturally, a prosecutor won’t want to stake a major case on the say-so of one person, since a good defense attorney can make a single witness look bad in various ways. But when ten people say more or less the same thing, then it’s harder to discredit them all.

Edit: I meant to continue.

Circumstantial evidence has to present a believable story to the jury. Fingerprints on a gun. Gun used in murder. Ammunition in the suspect’s home matches the ammunition used in the crime. Evidence of bad feeling between victim and suspect (or a chance to gain from the victim’s death), no other alibi for the suspect.

When means, motive and method have been satisfactorily demonstrated to the jury, that can be all that’s required.

A circumstantial case can be very strong. It shows the likelihood that someone comitted a crime and no one else did.

DNA evidence is circumstantial evidence.

I agree with @marinelife , many cases are purely circumstantial, but with so much evidence that guilt can definitely be determined. Case in point, Casey Anthony. No one saw her kill her child. However, she had the child last, she lied over and over to the police, including the lie that a fictional babysitter stole the child. Her car reeked of dead body and chloroform, a search of “how to make chloroform” was done on Casey’s computer when no one else was home, post-mortem hair was found in her truck from the child, and no other explanation for finding the child’s body dumped in the woods even makes any sense.

Another case in point; Josh Powell. He takes off in the middle of the night in a blizzard in December with two toddlers, comes back Monday night when he was supposed to have started a new job Monday morning. Police discovered a big wet spot on his living room carpet, and he had floor fans going on the spot, and his wife is missing. Hmmm.

Anyway, neither one of these yahoos were even convicted. If we did not allow circumstantial evidence to be used in trials, you might as well declare open season on crime.

It depends primarily on two things. The jury and the skill of the defense attorney being able to convince them that reasonable doubt is the most important reason why they should not return a guilty verdict.

And whether someone has the money to hire attorneys with that level of skill and experience. Case in point: the difference between the financial means of the average low to middle class black man and OJ. Simpson. Nuff said.

Circumstantial evidence can be very strong evidence and is no different than direct evidence. The myth of circumstanial evidence not being strong evidence or not being enough to convict someone of a crime is the thing that movies are made up. It isn’t real. Circumstantial evidence is not inferior to other evidence.