Alphabeta MathPOMDPs and belief-state planning
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Combinatorics
Machine Learning Algorithms
Measure Theory
Probability
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- limsup, liminf, and Subsequential Limits
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Lemma
A finite initial segment of the naturals has its index as cardinality
Definition
Finite partially observable Markov decision process
Definition
Belief states and the finite Bayes filter
Algorithm
Exhaustive finite-horizon POMDP alpha-vector backup
Lemma
The finite POMDP Bayes filter is a probability kernel
Theorem
The belief state is a sufficient statistic for finite POMDP control
Theorem
Finite-horizon POMDP values are piecewise-linear and convex
Sources
- Bob Dumas and John E. McCarthy, Transition to Higher Mathematics, §1.2: Sets
- Kaelbling, Littman, and Cassandra, Planning and Acting in Partially Observable Stochastic Domains, Brown Technical Report CS-96-08
- Russ Tedrake, Planning Under Uncertainty: Partially-observable Markov decision processes, MIT Underactuated Robotics course notes
- Pascal Poupart, CS 886 Lecture 13: POMDPs, University of Waterloo