It depends. Skim through H&H to see. I find it more intuitive and modern, but it also covers way more territory. At some point, the material will be hard because it's very advanced mathematics.
However, by then perhaps you have already adjusted. There's also a solution manual. Furthermore, many difficult proofs are in the appendix. So it's more of a calculus book if you want to ignore the analysis part.
I am hoping to get some experience with probalistic modelling, maximum likelihood etc. I work in bioinformatics and spent most of my time on algorithms development but probability/stats/ML are becoming the norm now. I find it hard to follow papers and develop new methods.
The skills needed will vary a lot. Hence my concern about studying H&H. It's a good idea, as real analysis is the foundation. But it will take too much of your time to get to something useful. Probably you should try to learn more applied material in parallel and let both threads merge in the future.
For maximum likelihood, you need to learn convex optimization right after real analysis. The canonical reference is [1], but there's also a very simple and pragmatic linear algebra textbook by the same author that also covers some of the optimization basics [2]. This might be a good entry point, certainly easier than Spivak or H&H. There's also [3,4], which you probably know about. These are great and emphasize the modeling part. Maximum likelihood (via EM) is in the appendix, and you don't really need to know a lot of math to get going.
If you prefer a Bayesian or a variational point of view, modeling is the really important part. MCMC and message passing algorithms tend to be reused. For high level modeling of study results (e.g. differential expression on complicated designs), Gelman's Stan books [5] are a delight to learn from. If you need to roll your own custom inference, you should learn about graphical data structures such as factor graphs [6,7]. Here, knowhow from H&H is also required.
Thanks so much, it's nice to get suggestions from someone in the same area!
These are the most helpful and practical suggestions I have encountered. You've hit the nail on the head with the exact problem I have been having working through books like Spivak and Axler. It always felt like I wasn't learning anything practical towards my work and that anything useful was a long ways away. I do enjoy the books and the material and the suggestion to pursue them in parallel is something I wish I thought about.
However, by then perhaps you have already adjusted. There's also a solution manual. Furthermore, many difficult proofs are in the appendix. So it's more of a calculus book if you want to ignore the analysis part.