How is Roman religion empirical?
This is a great question. The Romans wanted to keep the peace with the gods as much as possible (pax deorum). And they determined this peace through empirical observance. Let me explain.
In the face of hardships and indicators of divine wrath, the Romans concluded that there must have been a breech in the pax deorum. The Romans had to resort to educated guesses and trial and error to determine what had gone wrong and how to repair it. In other words, the Romans would try something to appease the gods and if this failed, they would try something else. This pattern would continue, until success was obtained, which was construed as tangible proof that the pax deorum was established.
In book 5 of Livy, the Romans celebrated the first lectisternium, elected only patricians in a new election, and finally concluded that the problem was ritual imprecision in the celebration of the Latin Games. All of these points show that the Romans were empirical in their outlook.