## Student Probability Seminar, Fall 2017

# Monday 2-3 PM, 891 Evans

This is the UC Berkeley Student Probability Seminar, run by students in both the Mathematics and Statistics Departments at UC Berkeley. We meet on Mondays in 891 Evans from 2-3pm.

This semester we will be reading out of the book * Markov Chains and Mixing Times (2nd edition) * by Peres and Levine.
You can find the book for free here. We will alternate between
talks out of this book and talks on whatever topic is of interest.

**Previous Seminars**

DATE |
SPEAKER |
TITLE (click for abstract below) |

September 11 | Organizational Meeting | Setting Topics for Semester to Come |

September 18 | Nick Bhattacharya | MCMC and Mixing of Markov Chains |

September 25 | No speaker | |

October 2 | Satyaki Mukherjee | Couplings for Bounding Mixing Times |

October 9 | Milind Hegde | Lower Bounds on Mixing Times |

October 16 | TBD | |

October 23 | TBD |

### Title and Abstracts

**Nick Bhattacharya**

### MCMC and Mixing of Markov Chains

We review some basic language about mixing times and stationarity. We also discuss two methods for sampling a distribution using a Markov Chain (known as MCMC) - Glauber Dynamics (also known as Gibbs sampling and the Metropolis chain.

This will be a talk directly out of chapters 3-5 of MCMT.

**Satyaki Mukherjee**

### Couplings for Bounding Mixing Times

One technique for bounding mixing times is to pick a good coupling of our given chain with another and follow them until some random time. We focus on two techniques - a special case of what's called a Grand Coupling and selecting Strong Stationary Times. These methods are applied to examples from card shuffling and random walks on graphs.

This talk is from chapters 5 and 6 of MCMT.

**Milind Hegde**

### Lower Bounds on Mixing Times

Lower bounds on mixing times require a very different technique from upper bounds discussed last week. We outline three methods for finding lower bounds, which require increasingly more information about transition structure of our chain. We provide some examples from card shuffling and random walks - in particular, we show the upper bound of last week's card shuffling example was tight.

This talk is from chapter 7 of MCMT.