This paper is dedicated to lossless data compression with probability estimation using neural networks. First, we propose a probability estimation architecture based on a chain of neural predictors, so that each unit of the chain is defined as a neural network with the minimum possible number of weights, which is sufficient for efficient compression of data generated by Markov sources of a given order. We show that this architecture allows us to minimize the overall number of weights participating in the probability estimation process depending on the statistical properties of the input data. Second, in order to improve compression efficiency, we introduce an information inheritance mechanism, where the probability estimate obtained by a low-order unit is used at the next higher-order unit. Experimental results show that the proposed lossless data compressor equipped with the chained probability estimation architecture provides compression ratios close to the state-of-the-art PAC compressor. At the same time, it outperforms PAC by a factor of 1.2 to 6.3 in encoding throughput and by a factor of 2.8 to 12.3 in decoding throughput on a consumer GPU.