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By assuming a Markov-memory input process, we construct a Monte Carlo method for computing a lower bound on the feedback capacity of finite-state inter-symbol interference (ISI) channels. The transition probabilities of the Markov process are allowed to depend on previous observations of the channel outputs, where a suboptimal feedback strategy is chosen to implement the feedback. Generally, the bound is tight for low signal-to-noise ratios (SNRs). At high SNRs the tightness varies depending on the chosen channel. We provide 2 channel examples: one where the bound is not tight at high SNRs and the other where the bound is so tight that it surpasses the tightest known upper bounds on the feedforward capacity.