Free Energy Minimization

An algorithm is derived for inferring a binary vector s given noisy observations of Asmodulo2, where A is a binary matrix. The binary vector is replaced by a vector of probabilities, optimized by free energy minimization. Experiments on the inference of the state of a linear feedback shift register indicate that this algorithm supersedes Meier and Staaelbach's polynomial algorithm. (As + n) mod 2 = z (1) where A is a binary matrix. Our task is to infer s given z and A, and given assumptions about the statistical properties of s and n. This problem arises in the decoding of a noisy signal transmitted using a linear code A, and in the inference of the sequence of a linear feedback shift register (LFSR) from noisy observations 1, 2]. I assume that the prior probability distribution of s and n is separable thus: P(s; n) = Q n P(s n) Q m P(n m). The log probability of z as a function of s can be written in terms of the noise free vector t(s) = As mod2: log P(zjs; A) = X m t m (s) g m + const: (2)