Acts 16:34

And he brought them up into his house, and set food before them, and rejoiced greatly, with all his house, having believed in God.

Having believed in God ... is used as a synecdoche here for the whole bundle of things by which he had become a follower of Christ; and this use of such a figure to stand for the various things he had already done is exactly the same as the use of it earlier for the various things he was required to do. Thus, here is a perfect example of how Paul and the other New Testament writers used "faith" or "believe" as a short form for a number of related actions.

Rejoiced greatly ... The rejoicing followed his baptism, as so frequently in the New Testament; and as DeWelt was quoted earlier, "The conversions in the New Testament end, not with the testimony of the converts, but with their baptism." Scriptural conversion does not exist apart from it.