Linearly positive histories: Probabilities for a robust family of sequences of quantum events

Non-negative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories. The resulting linearly positive histories avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply. Thus linearly positive histories are a broad extension of decohering histories. Moreover, the resulting theory is manifestly time-reversal invariant.