Humans have been shown to combine noisy sensory information with previous experience (priors), in qualitative and sometimes quantitative agreement with the statistically-optimal predictions of Bayesian integration. However, when the prior distribution becomes more complex than a simple Gaussian, such as skewed or bimodal, training takes much longer and performance appears suboptimal. It is unclear whether such suboptimality arises from an imprecise internal representation of the complex prior, or from additional constraints in performing probabilistic computations on complex distributions, even when accurately represented. Here we probe the sources of suboptimality in probabilistic inference using a novel estimation task in which subjects are exposed to an explicitly provided distribution, thereby removing the need to remember the prior. Subjects had to estimate the location of a target given a noisy cue and a visual representation of the prior probability density over locations, which changed on each trial. Different classes of priors were examined (Gaussian, unimodal, bimodal). Subjects' performance was in qualitative agreement with the predictions of Bayesian Decision Theory although generally suboptimal. The degree of suboptimality was modulated by statistical features of the priors but was largely independent of the class of the prior and level of noise in the cue, suggesting that suboptimality in dealing with complex statistical features, such as bimodality, may be due to a problem of acquiring the priors rather than computing with them. We performed a factorial model comparison across a large set of Bayesian observer models to identify additional sources of noise and suboptimality. Our analysis rejects several models of stochastic behavior, including probability matching and sample-averaging strategies. Instead we show that subjects' response variability was mainly driven by a combination of a noisy estimation of the parameters of the priors, and by variability in the decision process, which we represent as a noisy or stochastic posterior.