  We study the problem of simultaneous predictive inference on multiple
outcomes missing at random. We consider a suite of possible simultaneous
coverage properties, conditionally on the missingness pattern and on the --
possibly discretized/binned -- feature values. For data with discrete feature
distributions, we develop a procedure which attains feature- and
missingness-conditional coverage; and further improve it via pooling its
results after partitioning the unobserved outcomes. To handle general
continuous feature distributions, we introduce methods based on discretized
feature values. To mitigate the issue that feature-discretized data may fail to
remain missing at random, we propose propensity score
$\epsilon$-discretization. This approach is inspired by the balancing property
of the propensity score, namely that the missing data mechanism is independent
of the outcome conditional on the propensity [Rosenbaum and Rubin (1983)]. We
show that the resulting pro-CP method achieves propensity score discretized
feature- and missingness-conditional coverage, when the propensity score is
known exactly or is estimated sufficiently accurately. Furthermore, we consider
a stronger inferential target, the squared-coverage guarantee, which penalizes
the spread of the coverage proportion. We propose methods -- termed pro-CP2 --
to achieve it with similar conditional properties as we have shown for usual
coverage. A key novel technical contribution in our results is that propensity
score discretization leads to a notion of approximate balancing, which we
formalize and characterize precisely. In extensive empirical experiments on
simulated data and on a job search intervention dataset, we illustrate that our
procedures provide informative prediction sets with valid conditional coverage.
