Consider Private Information Retrieval (PIR), where a client wants to retrieve one file out of K files that are replicated in N different servers and the client selection must remain private when up to T servers may collude. Additionally, suppose that the client has noisy side information about each of the K files, and the side information about a specific file is obtained by passing this file through one of D possible discrete memoryless test channels, where D≤K. While the statistics of the test channels are known by the client and by all the servers, the specific mapping between the files and the test channels is unknown to the servers. We study this problem when the client wants to preserve the privacy of its desired file selection and the mapping For this problem setup, we derive the optimal download rate. Our problem setup generalizes PIR with private noiseless side information and PIR with private side information under storage constraints.