How does the brain represent information and process it? By controlling and analyzing the input to and output from the brain -- the sensory stimulus and the behavior -- we have made great advances in modeling cognitive computation which is often well described by simple mathematical models such as a network of noisy leaky integrators and comparators. However, despite extensive efforts, how the brain implements those computations as a biophysical system is still largely unknown. One of the main obstacles is the subsampling problem: there are many (hundreds to hundreds of millions of) neurons across brain areas involved in transforming sensory information to producing the behavior, however, our experimental technology has been limited to observing neural signals from a small fraction of neurons in a handful of areas at the same time. This vast subsampling has limited our ability to infer the physical implementation of cognitive computation in the brain. We had to heavily rely on theoretical principles, intuition, and imagination as to how they might be implemented by a complex network of spiking neurons. Fortunately, recent advances in neural recording technology is allowing us access to several orders of magnitude more neurons at a high temporal precision. This is opening up new opportunities to directly infer the neural implementations: how external and internal information is represented in the population, and how it is transformed over time and across areas. Even for the simplest cognitive processes, such a bottom-up approach has not been successful so far in uncovering the underlying neural dynamics. In this presentation, I lay a principled approach that can tackle the subsampling problem by exploiting the low-dimensional structure of tasks and neural variability. This new approach to studying neural codes and computation is possible through advances in statistical and machine learning techniques aimed at extracting interpretable, i.e., scientifically useful, mathematical models. We have developed interpretable probabilistic models and Bayesian inference algorithms suitable for reverse engineering neural computation from large-scale population data.