I received a reply from Staffordshire Record Office
"It is difficult to be sure, where the census only uses initials, but I doubt this PB is the same person because she is from Hanley which is in Stoke on Trent in North Staffordshire, while your Prudence is from the very south of the county. Unfortunately we only have a very partial collection of surviving records from Stafford County Asylum. We have no patient registers for this period, and a gap in the surviving female case books until 1872. There is a full national index of asylum patients on the Ancestry website, created from returns sent to central government, but there is no matching entry for Prudence which confirms PB must be someone else. If she had been there, she still would have had a death certificate in her proper name.
As she has apparently vanished, she may have left the marriage for whatever reason (or her husband did, depending on whether he was living in the same place or moved away), and she may have passed as being married to someone else, so doesn’t show up under her own surname. Did you check in case she had reverted to using her maiden name? Or were there any other children that she could have lived with on a later census? (though in that case of course she would be under her married name). Did her husband stay in the same area, and when did he die? These are probably all questions you have already considered, in which case I don’t have any other suggestions to help, I’m afraid."
