Lord Bracadale, who has presided over some of the most high-profile criminal trials in recent Scottish history, will chair the public inquiry as it examines the circumstances leading up to Mr Bayoh's death, as well as the events following it.
Mr Bayoh, 32, died in 2015 after being restrained by officers responding to a call in Kirkcaldy, Fife.
It is understood Mr Yousaf will meet Lord Bracadale and the Bayoh family in the coming weeks to discuss terms of reference for the inquiry, which will determine its scope and parameters.
Lord Bracadale said: "I welcome the invitation to chair the public inquiry into the circumstances surrounding the death of Sheku Bayoh.
"I know that Mr Bayoh's family and other interested parties, as well as the general public, will expect a careful and thorough examination of the facts.
"I very much hope that my inquiry will provide an open and transparent means of exploring the issues."
Mr Yousaf said the inquiry will help to identify lessons and improvements for the future.
"I am delighted that Lord Bracadale has agreed to chair this public inquiry," he said.
"Lord Bracadale brings a wealth of knowledge and experience to this important task of providing a clear understanding of the circumstances surrounding Mr Bayoh's death.
"The public inquiry and its recommendations will identify lessons and improvements for the future to help prevent deaths in similar circumstances and build trust and confidence in policing."