Sir Francis Bacon was an English philosopher and statesman. The First Doctor and his companions watched Bacon and Queen Elizabeth I meet with William Shakespeare on the Time-Space Visualiser. They saw Bacon give Shakespeare the idea to write Hamlet. (TV: The Chase)
The Doctor claimed to have been present when Francis Bacon tried to stuff snow up a chicken. This caused Bacon to contract pneumonia and die. (PROSE: Heart of TARDIS)
When William Shakespeare stowed away aboard the Fifth Doctor's TARDIS in 1597 and died in King Richard III's place in 1485, Richard decided to go to 1597 and take up Shakespeare's life and career where it had been left off. The Doctor recommended Richard contact Francis Bacon for writing tips. (AUDIO: The Kingmaker)
Iris Wildthyme claimed that Bacon wrote the works of Shakespeare, having been told by Will that they had an arrangement. (PROSE: Minions of the Moon)
Behind the scenes
The revelation in The Kingmaker that Shakespeare was replaced by another man who then sought writing tips from Francis Bacon is a reference to the longstanding Baconian theory, which suggests Bacon wrote the plays commonly attributed to Shakespeare.