I have a graduate-level background in pure mathematics (Measure Theory, Functional Analysis, Operator Algebra, etc.) I also have a job that requires some knowledge of probability theory (from basic principles to machine learning techniques).
My question: Can someone provide some canonical reading and reference materials that:
- Self-contained introduction to Probability theory
- Don't shy away from measure theoretic methodologies and proofs
- Provide a heavy emphasis on applied techniques.
Basically, I want a book that will teach me applied probability theory geared towards pure mathematicians. Something starting with the basic axioms of probability theory and introducing applied concepts with mathematical rigor.
As per the comments, I'll elaborate on what I need. I am doing basic-to-advanced data mining. Logistic Regression, Decision Trees, basic Stats and Probability (variance, standard deviation, likelihood, probability, likelihood, etc.), Supervised and Unsupervised machine learning (mainly clustering (K-Means, Hierarchal, SVM)).
With the above in mind, I want a book that will start at the beginning. Defining probability measures, but then also showing how those result in basic summation probabilities (which I know, intuitively, happen by integration over discrete sets). From there, it could go into: Markov Chains, Bayesian.... all the while discussing the foundational reasoning behind the theory, introducing the concepts with rigorous mathematics, but then showing how these methods are applied in the real world (specifically to data mining).
- Does such a book or reference exist?
PS - I realize this is similar in scope to this question. However, I'm looking for Probability theory and not statistics (as similar as the two fields are).