采样

Sampling 采样

A* sampling

CJ Maddison, 2014, NeurIPS

A pratical generic sampling algorithm that searchs for the maximum of a Gumbel process using A* search.

Monte Carlo

Monte Carlo refers to approximation algorithms relying on repeated random sampling. (stochastic simulation)

Monte Carlo for Expectation Approximation

\[\mathbb{E}_{X \sim p}[f(X)] \approx \frac1n \sum_i^n f(\bm x_i) \]

Why Monte Carlo Expectation works: Law of Large Numbers.

Monte Carlo for Integration Approximation

To calculate the integration of

\[\int_\Omega f(\bm x)d\bm x, \Omega \in \R^{d} \]

:

1. Draw \(n\) samples uniformly from \(\Omega\) at random, denoted as \(\bm x^{(1)}, \bm x^{(2)},\dots, \bm x^{(n)}\)
2. Calculate

\[V=\int_\Omega d\bm x \]

, here the shape of \(\Omega\) should be simple, or it's hard to calculate the above integration.
3. Calculate the approximation

\[V \cdot \frac1n \sum_i^n f(\bm x^{(i)}) \approx \int_\Omega f(\bm x)d\bm x \]

Why it works:

\[p_{X\sim \mathrm{Uniform}(\Omega)}(X=x) = {1\over \int_{x\in \Omega} dx} = \frac1V \begin{aligned} \int_\Omega f(x) dx &= \int_\Omega f(x) \frac{V}{V} dx \\ &= V \int_\Omega f(x) \frac{1}{V} dx \\ &= V \int_\Omega f(x) p_{X\sim \mathrm{Uniform}(\Omega)}(x) dx \\ &= V\cdot \mathbb E_{X\sim \mathrm{Uniform}(\Omega)}[f(x)] \\ &\approx V \cdot \frac1n \sum_i^n f(x^{(i)}) \end{aligned} \]

Inverse Transform Sampling

1. 求累积密度函数 \(y={\rm{cdf}}(x)\) 的逆函数 \(x={\rm{cdf}}^{-1}(y)\) ；
2. 从0~1的均匀分布中采样得到样本u，求对应的样本x。

这样看该采样： cdf值域是[0,1]，在cdf图像上，从其值域[0,1]中等可能随机选择n个点，其对应的能使cdf取得该值的x（及cdf逆函数输出值）与cdf陡峭程度有关，越笔直（陡峭）则能获得更多样本x。

Pros:

• 采样效果很好。因p(x)高峰对应的x区间，cdf在该区间上陡增。

Cons：

• 有的函数cdf逆函数难以求出。

Importance Sampling

For an objective:

\[\int_\Omega f(x) dx \]

, we introduce a random variable \(X\) following a distribution with a probability density function \(h(X)\) with \(\forall x\in A h(x)\ne 0\) , so \(\int h(x)dx=1\) , and then

\[\int_\Omega f(x)dx = \int_\Omega f(x)\frac{h(x)}{h(x)} dx = \int_\Omega \frac{f(x)}{h(x)}h(x)dx = \mathbb E_{X\sim h(X)}\left[\frac{f(X)}{h(X)}\right] \]

. Next, we approximate the expectation by Monte Carlo:

\[\int f(x) dx \approx \frac1n \sum_i^n \frac{f(x^{(i)})}{h(x^{(i)})} \]

where \(x^{(i)}\) 's are samples drawn from sample space. We call h(x) importance sampling funtion. A good importance sampling function h(x) should have the following properties:

1. \(h(x)>0\)
2. close proportional to \(|f(x)|\)
3. easy to simulate
4. easy to compute \(h(x)\)

Acceptance-Rejection Sampling (accept-reject)

(also called rejection sampling)

We can generate sample values for a target random variable \(X\) from a target distribution \(p(X)\) by instead sampling values from an introduced distribution \(q(X)\) satisfying \(\exists c>0: cq(x)> p(x),\forall x\) . The choosen distribution \(q(X)\) is called a proposal distribution.

To obtain samples as follows:

1. Obtain a sample \(x_i\) from \(q(x)\) and \(u_i\) from \(\mathrm{Uniform}(0,1)\)
2. Check \(\frac{p(x)}{cq(x)}\)

• Accept if \(u_i \le \frac{p(x)}{cq(x)}\)
• Reject otherwise.

Pros:
*

Cons:

• Maybe inefficient due to too many rejection samples

拒绝-接受采样不常用，因其proposal distribution和c不好找，容易导致拒绝率很高。

Adaptive Rejection Sampling

MCMC, Markov Chain Monte Carlo

Gibbs Sampling

Gibbs sampling is a MCMC method. 常用的高维数据采样方法。