This paper proposes a novel multi-target tracking (MTT) algorithm for scenarios with arbitrary numbers of measurements per target. We propose the variational probabilistic multi-hypothesis tracking (VPMHT) algorithm based on the variational Bayesian expectation-maximisation (VBEM) algorithm to resolve the MTT problem in the classic PMHT algorithm. With the introduction of variational inference, the proposed VPMHT handles track-loss much better than the conventional probabilistic multi-hypothesis tracking (PMHT) while preserving a similar or even better tracking accuracy. Extensive numerical simulations are conducted to demonstrate the effectiveness of the proposed algorithm.