Mr Yousaf's statement to the Scottish Parliament came after it was confirmed that no police officers would be prosecuted over the death of Mr Bayoh in 2015, a matter contested by his family and their solicitor, Aamer Anwar.
Mr Bayoh died after being restrained by up to nine officers called to a disturbance in Kirkcaldy. He had taken drugs at the time and was said to have been armed with a knife. He died from suspected asphyxia and was found to have extensive injuries which the family said he had not previously sustained. The police are accused of having failed to cooperate with a subsequent investigation.
The Crown decided a year ago that no criminal proceedings would be brought, but the family exercised their right to ask for a review. The decision has now been confirmed, on the basis that the evidence currently available would not justify criminal proceedings. Mr Aamer claimed the family "felt totally betrayed by the Lord Advocate, for not holding power to account".
It has still to be decided who will chair the inquiry, which will examine the circumstances leading up to and following the death of Mr Bayoh, though the precise terms of reference have still to be confirmed. It will have a wider remit than the fatal accident inquiry that is normally held into deaths in police custody.
In his statement Mr Yousaf said: "The First Minister and I met with Mr Bayoh’s family today to express our deepest condolences and assure them of our commitment to establishing the facts surrounding this tragic incident. They are right to expect a full public examination of the circumstances of Mr Bayoh’s death and I stated my determination to put in place a process to deliver that...
"In this case, the Lord Advocate considers the remit of a FAI would not allow all the issues which require to be investigated to be addressed. FAIs can examine circumstances and factors leading up to a death, but not what follows after, and in this case the Lord Advocate has identified questions, raising issues of public interest and importance about the early stages of the post-incident management of the investigation that an FAI simply could not examine."