Lectures

Lecture 1 (1/5)

• Key questions: distributions, sampling, features of a system
• Random Sampling
• Ising Model
• The main idea is that sampling allows integration, scales better than grids, and solves impossible sums.

Lecture 2 (1/7)

Example 1:

Let $x(t)$ be a particle with the following ODE

$$\begin{array}{r}\frac{dx}{dt}=-\frac{du}{dx}\end{array}$$

Let $u(x)=\frac{1}{2}{x}^2$. Then

$$\begin{array}{rl}u(x)& =\frac{1}{2}{x}^2\\ \frac{du}{dx}& =x\end{array}$$

such that

$$\frac{dx}{dt}=-x$$

Upon solving this ODE, we get

$$x(t)=x(0)\cdot e^{-t}$$

So, as $t\to \infty ,x\to 0$.

Example 2:

We could have a double well

$$u(x)=\frac{1}{4}(x^2-1{)}^2$$

In the SDE case, given

$$dx=-u^{\prime }(x)dt+b(t)d{B}_t$$

In order, the terms are:

• $dx$ is the change $x$
• $u^{\prime }(x)$ is the deterministic force
• $b(t)d{B}_t$ is some “random” force. In particular, $d{B}_t$ is the Brownian increment, or “white noise”.

So when we calculate the position of the particle via $x(t)$, the $u^{\prime }(t)$ term tells us how we move via the force, with some randomness via $b(t)d{B}_t$. This leads to

• new frequency in a system

what does frequency mean?

• new types of resonance/transport effects.

Lecture 3 (1/9)

Reference for this section is Rubinstein Ch.2

• Inversion
• Transformation of Random Variables

Lecture 4 (1/12)

Reference: Rubinstein Ch. 2, 2.4, 2.3.4

• Box-Muller Method
• Acceptance-Rejection

Lecture 5 (1/14)

• Acceptance-Rejection
• [[Acceptance-Rejection#theorem-uniformity-on-region-a|Theorem (Uniformity on Region $A$)]]
• [[Acceptance-Rejection#theorem-from-uniformity-on-a-to-target-fx|Theorem (From Uniformity on $A$ to Target $f(x)$)]]
• [[Acceptance-Rejection#theorem-a-r-generates-z-sim-f|Theorem (A-R Generates $Zsimf$)]]

Lecture 6 (1/16)

• Efficiency
• Modified A-R
• Example 1 (Sampling Standard Normal via A-R)
• Discrete Case

Lecture 7 (1/21)

• Monte-Carlo Integration
• Variance Reduction

Lecture 8 (1/23)

• Importance Sampling
• Example 1

Lecture 9 (1/26)

• Review of previous $2$ lectures.
• Markov Chains. For the purposes of this class we only consider Discrete Time/State Homogeneous Markov Chains
• Stochastic Process

Lecture 10 (1/28)

• Markov Chains

Lecture 11 (1/31)

• Long-Run Markov Chains
• Invariant Distribution

Lecture 12 (2/2)

• Long-Run Markov Chains
• Metropolis Algorithm

Lecture 13 (2/4)

• Example 1 (1D Ising Model)
• Theorem (Metropolis Satisfies Detailed Balance)

Lecture 14 (2/6)

• Review on Theorem (Metropolis Satisfies Detailed Balance)
• Gibbs Sampler

Lecture 15 (2/9)

• Review for midterm

Lecture 16 (2/13)

• Review on Metropolis Algorithm
• Gibbs Sampler

Lecture 17 (2/19)

• Remarks. $\leftarrow$ important!

Lecture 18 (2/20)

• Stochastic Process
• Stochastic Differential Equations
• Brownian Motion and SDE Formulation

Lecture 19 (2/23)

• Definition (Brownian Motion, Wiener Process)
• Properties of Brownian Motion
• Example 1

Lecture 20 (2/25)

• Simulating Brownian Motion
• Ornstein-Uhlenbeck Process (OU)

Lecture 21 (2/27)

• Moments of the OU Process
• Ito Calculus

Lecture 22 (3/2)

• Ito’s Lemma
• ODE Case
• SDE Case
• Example 1 (Discrete-Time Step with Ito’s Lemma)

Lecture 23 (3/4)

• Example 2 (Simple Ito Lemma)
• Backwards Ito SDE
• Stratanovich SDE
• Remarks (Ito v.s Ito Backwards)

Lecture 24 (3/6)

• Example 3 (Apply OU on Ito SDE)
• Example 4 (Geometric BM)
• Example 5 (Geometric BM with Log Expectation)
• Example 6 (Arithmetic BM with Drift)
• Applying Units
• 2D SDEs

Lecture 25 (3/9)

• Numerics and Convergence

Lecture 26 (3/11)

• Kolmogorov Forward Equation

Lecture 27 (3/13)

• Finals review