Exploring handwritten digit classification: a tidy analysis of the MNIST dataset

In a recent post, I offered a definition of the distinction between data science and machine learning: that data science is focused on extracting insights, while machine learning is interested in making predictions. I use both machine learning and data science in my work: I might fit a model on Stack Overflow traffic data to determine which users are likely to be looking for a job (machine learning), but then construct summaries and visualizations that examine why the model works (data science). This is an important way to discover flaws in your model, and to combat algorithmic bias. This is one reason that data scientists are often responsible for developing machine learning components of a product. I'd like to further explore how data science and machine learning complement each other, by demonstrating how I would use data science to approach a problem of image classification.