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A common challenge in modeling inertial confinement fusion (ICF) experiments with computer simulations is that many of the simulation inputs are unknown and cannot be directly measured. Often, parameters that are measured in the experiment are used to infer the unknown inputs by solving the inverse problem: finding the set of simulation inputs that result in outputs consistent with the experimental observations. In ICF, this process is often referred to as a "post‐shot analysis." Post‐shot analyses are challenging as the inverse problem is often highly degenerate, the input parameter space is vast, and simulations are computationally expensive. In this work, deep neural network models equipped with model uncertainty estimates are used to train inverse models, which map directly from output to input space, to find the distribution of post‐shot simulations that are consistent with experimental observations. The inverse model approach is compared to Markov chain Monte Carlo (MCMC) sampling of the forward model, which maps from input to output space, for parameter inference tasks of varying complexity. The inverse models perform best when searching vast parameter spaces for post‐shot simulations that are consistent with a large number of observables, where MCMC sampling can be prohibitively expensive. We demonstrate how augmenting inverse models with autoencoders enable the inclusion of several dozen observables in the inverse mapping, reducing the degeneracy of the model and improving the accuracy of the post‐shot analysis. [ABSTRACT FROM AUTHOR]