I don't have a graduate degree at all. I don't even an undergraduate degree in anything even related to CS (I double majored in Philosophy and Criminal Justice in college, and went to a pretty meager college). I don't apply for any jobs, or post a resume anywhere, but I get job offers frequently from startups to Fortune 500 companies. Not long ago, I was offered an interview w/ Apple. It's definitely possible, employers just simply want to see proof that you know what you know. Even with a degree, unless it's a very prestigious university, you'll never get a decent job without either previous experience or an excellent portfolio.

Beyond a Github, you ought to have some sort of research blog or something, showing projects, your thought process, and your statistics knowledge beyond the code, with explanations of what you're doing and why. Some employers might find your Github page, but you're way more likely to get discovered and look interesting with some sort of blog alongside the Github in my opinion. The Github profile is great to have when a technical person in the company reviews you, but many recruiters aren't engineers, and wont know really how to assess your code.

That's just one man's opinion, however. Any form of networking can help. I've never referred someone just by their github page, I want to know their personality before I'd think of hiring/interviewing/referring them personally. Many companies first will reach out to their employees for tips on who they should reach out to, which is where most of my job offers come from. People within companies have found my site/youtube channel, and I get recommendations that way for the most part. Sometimes I am found by recruiters doing searches on Google or LinkedIn.