Identifying several biased coins encountered by a hidden random walk

Suppose that attached to each site z in Z is a coin with bias theta(z), and only finitely many of these coins have non-zero bias. Allow a simple random walker to generate observations by tossing, at each move, the coin attached to its current position. Then we can determine the biases {theta(z) : z in Z}, using only the outcomes of these coin tosses and no information about the path of the random walker, up to a shift and reflection of Z. This generalizes a result of Harris and Keane.