The body of Ms O’Neill, a 15-year-old student at Portobello High School from Willowbrae, was discovered around 6.30 on Saturday morning by a passer-by.
The teen had been out on Friday with a friend in the city centre.
The pair had been walking in the vicinity around Waverley Station before making their way to Calton Hill.
The friends parted ways some time afterwards, with Ms O’Neill found dead near a bench several hours later after being reported missing by her family.
Speaking at an official conference on Wednesday at the city centre landmark close to where Ms O’Neill was discovered, Detective Inspector Susan Balfour from the Major Investigation Team says the death is not being treated as suspicious.
“Just after 1am, the family had been carrying out their own investigations to try and find her through friends and went out looking for her themselves,” Inspector Balfour said.
“She travelled to Edinburgh city centre to meet a friend during the day and we know they made their way up to Calton Hill and her friend later returned home.
“We have a period of a good few hours which are unaccounted for. It’s not being treated suspicious at the moment.
“In terms of the cause of death, we just don’t know at this time. It’s unexplained and we don’t believe it’s suicide, however, we’re at the early stages.
“The cause of death depends on a lot of results.”
A number of heartfelt tributes have been left around the Calton Hill bench close to where Mhari’s body was found.
In a statement issued through Police Scotland on Wednesday morning, the family of Mhari O’Neill said: “Mhari was much loved by all that knew her and we know that her passing will have a profound impact on her closest friends, as it has ourselves.
“Mhari was considering applying to attend university and one of her main passions was music; she loved going to gigs.
“What we need now is to find out exactly what happened to Mhari and we want anyone who thinks they may have information to get in touch with the police.
“We thank everyone for their support and ask that we be left in privacy to grieve at this time.”
A family member has also taken to social media pleading for anyone with information to come forward.
The tweet reads: “Please if anyone knows what happened please contact the police on 101. My family needs to know what happened to Mhari at Calton Hill.”
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