Symbolic data analysis (SDA) aggregates large individual-level datasets into a small number of distributional summaries, such as random rectangles or random histograms. The inference is carried out using these summaries in place of the original dataset, resulting in computational gains at the loss of some information. In likelihood-based SDA, the likelihood function is characterised by an integral with a large exponent, which limits the method's utility as for typical models the integral is unavailable in closed form. In addition, the likelihood function is known to produce biased parameter estimates in some circumstances. Our article develops a Bayesian framework for SDA methods in these settings that resolves the issues resulting from integral intractability and biased parameter estimation using pseudo-marginal Markov chain Monte Carlo methods. We develop an exact but computationally expensive method based on path sampling and the Poisson estimator, and a much faster, but approximate, method based on a Taylor expansion. Through simulation and real-data examples we demonstrate the performance of the developed methods, showing large reductions in computation time compared to the full-data analysis, with only a small loss of information.

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