The olfactory system faces a difficult inference problem: it has to determine what odors are present based on the distributed activation of its receptor neurons. Here we derive neural implementations of two approximate inference algorithms that could be used by the brain. One is a variational algorithm (which builds on the work of Beck. et al., 2012), the other is based on sampling. Importantly, we use a more realistic prior distribution over odors than has been used in the past: we use a "spike and slab'' prior, for which most odors have zero concentration. After mapping the two algorithms onto neural dynamics, we find that both can infer correct odors in less than 100 ms, although it takes 500 ms to eliminate false positives. Thus, at the behavioral level, the two algorithms make very similar predictions. However, they make different assumptions about connectivity and neural computations, and make different predictions about neural activity. Thus, they should be distinguishable experimentally. If so, that would provide insight into the mechanisms employed by the olfactory system, and, because the two algorithms use very different coding strategies, that would also provide insight into how networks represent probabilities.