We propose a generic construction for fully secure decentralized multiauthority predicate encryption. In such multiauthority predicate encryption scheme, ciphertexts are associated with one or more predicates from various authorities and only if a user has a set of decryption keys that evaluates all predicates to true, the user is able to recover the message. In our decentralized system, anyone can create a new authority and issue decryption keys for their own predicates. We introduce the concept of a multi-authority admissible pair encoding scheme and, based on these encodings, we give a generic conversion algorithm that allows us to easily combine various predicate encryption schemes into a multi-authority predicate encryption variant. The resulting encryption schemes are proven fully secure under standard subgroup decision assumptions in the random oracle model. Finally, by instantiating several concrete multi-authority admissible pair encoding schemes and applying our conversion algorithm, we are able to create a variety of novel multi-authority predicate encryption schemes.