Chapter 2 Conditional Distributions
[Sunday 18 May 2025 02:01:50 AM] Conditional distributions are one of the most important constructions of modern probability. However, there are multiple notions of it (which are similar, but need slightly different treatments mathematically, though most of them have the same essence). In this post, I’ll discuss some of those constructions (and book references where you can read more about them). Some of these abstractions can be really helpful if one comes across situations where naive notions of conditional probability might fail.
The prerequisites for this post will be:
- Basic notions of measure theory; measures, measurable maps etc., and related concepts.
- The standard Borel \(\sigma\)-algebra on \(\mathbb{R}\).
- The Lebesgue integral, and some related convergence theorems (monotone, dominated convergence). Also, expected values.
All these are standard notions studied in any measure theory course. For these notions, my favourite books are (Rao 2005), (Klenke 2020), (Bogachev 2007) and (Dudley 2004); I recommend the reader to go through all these books and see the minor differences in their treatments.