We prove a powerful inequality which provides very tight gaussian tail bounds “” for probabilities on product state spaces . Talagrand’s Inequality has found lots of applications in probability and combinatorial optimization and, if one can apply it, it generally outperforms inequalities like Azzuma-Hoeffding.
- Laplacian; Adjoints; Harmonic fn; Green’s fn; Forward Eqn; Backward Eqn.
- Markov Chains and Martingales; Green’s Functions and occupancy; Potential functions; time-reversal and adjoints.
We show that relative entropy decreases for continuous time Markov chains.
We consider a system consisting of interacting objects. As we let the number of objects increase, we can characterize the limiting behaviour of the system.
In the Cross Entropy Method, we wish to estimate the likelihood
Here is a random variable whose distribution is known and belongs to a parametrized family of densities . Further is often a solution to an optimization problem.
Sanov’s asks how likely is it that the empirical distribution some IIDRV’s is far from the distribution. And shows that the relative entropy determines the likelihood of being far.
Entropy and Relative Entropy occur sufficiently often in these notes to justify a (somewhat) self-contained section. We cover the discrete case which is the most intuitive.