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April 18, 2023 1
MARKOV CHAIN MONTE CARLO:A Workhorse for Modern Scientific Computation
Xiao-Li MengDepartment of Statistics
Harvard University
2April 18, 2023
Introduction
3April 18, 2023
Applications of Monte Carlo
Physics
Chemistry
Astronomy
Biology
Environment
Engineering
Traffic
Sociology
Education
Psychology
Arts
Linguistics
History
Medical Science
Economics
Finance
Management
Policy
Military
Government
Business
…
4April 18, 2023
Monte Carlo 之应用A Chinese version of the previous slide
物理
化学
天文
生物
环境
工程
交通
社会
教育
心理
人文
语言
历史
医学
经济
金融
管理
政策
军事
政府
商务…
5April 18, 2023
Monte Carlo Integration
Suppose we want to compute
where f(x) is a probability density. If
we have samples x1,…,xn » f(x), we can estimate I by
6April 18, 2023
Monte Carlo Optimization
We want to maximize p(x) Simulate from
f(x) / p(x).
As ! 1, the simulated
draws will be more and
more concentrated around
the maximizer of p(x)
-5 0 5
0.0
0.1
0.2
0.3
0.4
x
dens
ity
-5 0 5
0.00
0.05
0.10
0.15
x
dens
ity
-5 0 50.0
e+00
8.0
e-09
x
dens
ity
=1
=2
=20
7April 18, 2023
Simulating from a Distribution
What does it mean?Suppose a random variable (随机变量 ) X can only take two values:
Simulating from the distribution of X means that we want a collection of 0’s and 1’s:
such that about 25% of them are 0’s and about 75%of them are 1’s, when n, the simulation size is large.
The {xi, i = 1,…,n} don’t have to be independent
8April 18, 2023
Simulating from a Complex Distribution
Continuous variable X, described by a density function f(x)
Complex: the form of f(x) the dimension of x
9April 18, 2023
Markov Chain Monte Carlo
where {U(t), t=1,2,…} are identically and independently distributed.
Under regularity conditions,
So We can treat {x(t), t= N0, …, N} as an approximate sample from f(x), the stationary/limiting distribution.
10April 18, 2023
Gibbs Sampler
Target density: We know how to simulate form the conditional
distributions
For the previous example,
N(,2)Normal Distribution
“Bell Curve”
11April 18, 2023
12April 18, 2023
Statistical Inference Point Estimator:
Variance Estimator:
Interval Estimator:
13April 18, 2023
Gibbs Sampler (k steps)
Select an initial value (x1(0), x2
(0) ,…, xk(0)).
For t = 0,1,2, …, N Step 1: Draw x1
(t+1) from f(x1|x2(t), x3
(t),…, xk(t))
Step 2: Draw x2(t+1) from f(x2|x1
(t+1), x3(t),…, xk
(t))
…
Step K:Draw xk(t+1) from f(xk|x1
(t+1), x2(t+1),…, xk-1
(t+1))
Output {(x1(t), x2
(t),…, xk(t) ): t= 1,2,…,N}
Discard the first N0 draws
Use {(x1(t), x2
(t),…, xk(t) ): t= N0+1,2,…,N} as (approximate) samples from
f(x1, x2,…, xk).
14April 18, 2023
Data Augmentation
We want to simulate from
But this is just the marginal distribution of
So once we have simulations:
{(x(t), y(t): t= 1,2,…,N)},
we also obtain draws:
{x(t): t= 1,2,…,N)}
0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
de
nsi
ty
15April 18, 2023
A More Complicated Example
16April 18, 2023
Metropolis-Hastings algorithm
Simulate from an approximate distribution q(z1|z2), then
Step 0: Select z(0);
Now for t = 1,2,…,N, repeat
Step 1: draw z1 from q(z1|z2=z(t))
Step 2: Calculate
Step 3: set
Discard the first N0 draws
Accept
reject
17April 18, 2023
M-H Algorithm: An Intuitive Explanation
Assume , then
18April 18, 2023
M-H: A Terrible Implementation
We choose q(z|z2)=q(z)=(x-4)(y-4)
Step 1: draw x » N(4,1), y » N(4,1);
Dnote z1=(x,y)
Step 2: Calculate
Step 3: draw u » U[0,1]
Let
19April 18, 2023
Why is it so bad?
20April 18, 2023
M-H: A Better Implementation
Starting from some arbitrary (x(0),y(0))
Step 1: draw x » N(x(t),1), y » N(y(t),1)“random walk”
Step 2: dnote z1=(x,y), calculate
Step 3: draw u » U[0,1]Let
21April 18, 2023
Much Improved!
22April 18, 2023
Further Discussion
How large should N0 and N be?
Not an easy problem! Key difficulty:
multiple modes in unknown area
We would like to know all (major) modes, as well as their surrounding mass.
Not just the global mode
We need “automatic, Hill-climbing” algorithms.
) The Expectation/Maximization (EM) Algorithm, which can be viewed as a deterministic version of Gibbs Sampler.
23April 18, 2023
Drive/Drink Safely,
Don’t become a Statistic;
Go to Graduate School,
Become a Statistician!