I've a similar name - my gggg-grandmother Francesca Bajata was born in Palermo, married Onofrio Pileri and had a daughter Guiseppa in Palermo Province in about 1794.
Guiseppa had children in Ficarazzi, Palermo Province.
I don't have anything earlier on them. I think that Bajata and Baiada may be the same line, just written differently by different record keepers.
Have you anything that corresponds to this?