Tia's father is not Marquis Rozan, because he was too old and ill to bed Miraila. Her father was likely one of the marquis' grandsons, whom Miraila likely seduced. Although the marquis' daughter suspected this, she never took action and accepted Tia's presence to avoid the shame and exposure of a cheating scandal to the family reputation.
The emperor got cross at Miraila getting pregnant by another man, so she decided to take over the Rozan inheritance to give herself more social security/leverage. She struck during the marquis' funeral by poisoning the dinner served to the present family members and blaming the Hanson servant family for it. The murder was sloppily planned and implemented; the only reason Miraila fully escaped sentencing was due to the emperor's intervention, who ultimately still loved Miraila too much to see her prosecuted. He pulled strings and bribed parties to get her fully acquitted.