  Symbolic data analysis (SDA) aggregates large individual-level datasets into
a small number of distributional summaries, such as random rectangles or random
histograms. 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 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 block-Poisson estimator, and a much faster, but approximate,
method based on 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.
