Biological neurons exhibit dynamic excitation behavior in the form of stochastic firing, rather than stiffly giving out spikes upon reaching a fixed threshold voltage, which empowers the brain to perform probabilistic inference in the face of uncertainty. However, owing to the complexity of the stochastic firing process in biological neurons, the challenge of fabricating and applying stochastic neurons with bio-realistic dynamics to probabilistic scenarios remains to be fully addressed. In this work, a novel CuS/GeSe conductive-bridge threshold switching memristor is fabricated and singled out to realize electronic stochastic neurons, which is ascribed to the similarity between the stochastic switching behavior observed in the device and that of biological ion channels. The corresponding electric circuit of a stochastic neuron is then constructed and the probabilistic firing capacity of the neuron is utilized to implement Bayesian inference in a spiking neural network (SNN). The application prospects are demonstrated on the example of a tumor diagnosis task, where common fatal diagnostic errors of a conventional artificial neural network are successfully circumvented. Moreover, in comparison to deterministic neuron-based SNNs, the stochastic neurons enable SNNs to deliver an estimate of the uncertainty in their predictions, and the fidelity of the judgement is drastically improved by 81.2%.