中文版
贝叶斯统计的核心思想与基础知识
贝叶斯统计是以贝叶斯定理为核心，通过将先验知识和观测数据相结合，更新对参数或模型的认知的一种统计方法。它不仅强调概率的频率解释（频率统计学中概率描述事件的长期发生频率），更强调概率作为信念的不确定性量化。
1. 贝叶斯定理的基本公式
贝叶斯定理的数学表达式如下：
p ( θ ∣ D ) = p ( D ∣ θ ) p ( θ ) p ( D ) p(\theta | D) = \frac{p(D | \theta) p(\theta)}{p(D)} p(θ∣D)=p(D)p(D∣θ)p(θ)
- ( θ \theta θ )： 模型的参数（如均值、方差、回归系数等）。
- ( D D D )： 数据或观测到的证据。
- ( p ( θ ) p(\theta) p(θ) )： 参数的 先验分布，表示在观察数据之前对参数的主观信念。
- ( p ( D ∣ θ ) p(D | \theta) p(D∣θ) )： 数据的 似然函数，表示在给定参数下观测数据的概率。
- ( p ( D ) p(D) p(D) )： 边际似然 ，即数据的归一化常数，表示观测数据在所有可能参数下的加权平均概率：
p ( D ) = ∫ p ( D ∣ θ ) p ( θ ) d θ p(D) = \int p(D | \theta) p(\theta) d\theta p(D)=∫p(D∣θ)p(θ)dθ - ( p ( θ ∣ D ) p(\theta | D) p(θ∣D) )： 参数的 后验分布，表示在观察数据之后对参数的更新后的信念。
2. 核心概念解析
2.1. 先验分布 ( p ( θ ) p(\theta) p(θ) )
先验分布反映了在没有观察数据之前，对参数 ( θ \theta θ ) 的初始认知或假设。它可以基于以下方式确定：
- 主观经验： 根据已有领域知识和直觉。
- 无信息先验（Non-informative Prior）： 当没有特别的知识时，可以使用均匀分布等无偏见的分布。
- 共轭先验： 为了方便计算，可以选择与似然函数形式相同的分布作为先验（如高斯分布的均值先验也用高斯分布）。
2.2. 似然函数 ( p ( D ∣ θ ) p(D | \theta) p(D∣θ) )
似然函数是贝叶斯统计中的核心，它描述了参数 ( θ \theta θ ) 给定的条件下，数据 ( D D D ) 出现的概率。其形式通常由问题的概率模型决定。例如：
- 对于高斯分布，似然函数可以写为：
p ( D ∣ μ , σ 2 ) = ∏ i = 1 N N ( x i ∣ μ , σ 2 ) p(D | \mu, \sigma^2) = \prod_{i=1}^N \mathcal{N}(x_i | \mu, \sigma^2) p(D∣μ,σ2)=i=1∏NN(xi∣μ,σ2)
2.3. 后验分布 ( p ( θ ∣ D ) p(\theta | D) p(θ∣D) )
后验分布是贝叶斯统计的最终目标，表示在观测数据后，对参数 ( θ \theta θ ) 的更新后的信念。它通过结合先验分布和似然函数，对参数的估计进行修正。
后验分布的重要性质：
- 动态更新： 当新的数据到来时，可以用当前的后验分布作为新的先验分布，进行递归更新。
- 权衡先验和数据： 当数据量较少时，后验分布更依赖于先验分布；当数据量足够多时，后验分布逐渐被数据主导。
3. 贝叶斯统计的步骤
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建立先验分布 ( p ( θ ) p(\theta) p(θ) )：
根据问题背景，选择适当的先验分布。常用的有均匀分布、高斯分布、Beta 分布等。
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构建似然函数 ( p ( D ∣ θ ) p(D | \theta) p(D∣θ) )：
根据数据生成的概率模型确定似然函数。
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结合贝叶斯公式，计算后验分布 ( p ( θ ∣ D ) p(\theta | D) p(θ∣D) )：
利用公式：
p ( θ ∣ D ) = p ( D ∣ θ ) p ( θ ) p ( D ) p(\theta | D) = \frac{p(D | \theta) p(\theta)}{p(D)} p(θ∣D)=p(D)p(D∣θ)p(θ)其中边际似然 ( p ( D ) p(D) p(D) ) 主要用于归一化。
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推断与预测：
利用后验分布对参数进行估计（如最大后验估计 MAP）或对新数据进行预测。
4. 贝叶斯统计的核心思想：更新信念
贝叶斯统计的核心思想在于 "通过数据更新信念"：
- 先验分布： 你对某个现象的初始认知。
- 数据： 提供了关于现象的证据。
- 后验分布： 更新后的认知，即数据与先验信息的结合。
直观理解：
假设你投掷一枚硬币，初始你认为硬币是公平的（先验分布），但在连续观察 10 次结果后，9 次是正面（数据），你会调整对硬币公正性的信念（后验分布）。这种调整是贝叶斯方法的精髓。
5. 贝叶斯统计的优点
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直观性：
贝叶斯方法将概率解释为信念的强度，便于理解和操作。
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适用于小样本：
当数据较少时，先验分布提供了额外的信息来源。
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动态更新：
后验分布可以作为新的先验分布，不断更新参数的估计。
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灵活性：
贝叶斯方法能很好地处理复杂模型和不确定性，例如层次模型和缺失数据。
6. 贝叶斯统计在实际中的应用
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机器学习：
贝叶斯方法用于贝叶斯网络、贝叶斯优化等领域，解决分类、回归、超参数优化问题。
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医学统计：
利用贝叶斯方法分析实验结果、药物疗效。
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金融预测：
用于股票价格预测、不确定性分析。
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自然语言处理：
贝叶斯文本分类、主题模型（如 LDA）等。
7. 小结
贝叶斯统计的核心是通过贝叶斯定理将 先验分布 和 似然函数 结合，推导出 后验分布。它不仅提供了参数的估计，还通过概率的形式量化了不确定性。这种基于信念更新的统计思想在现代数据科学中有着广泛的应用价值。
英文版
Introduction to Bayesian Statistics
Bayesian statistics is a framework that uses Bayes' theorem to update our beliefs about parameters or models based on observed data. Unlike frequentist statistics, which interprets probability as the long-run frequency of events, Bayesian statistics views probability as a measure of uncertainty or belief about an event or parameter.
1. Bayes' Theorem
Bayes' theorem provides the mathematical foundation for Bayesian statistics:
p ( θ ∣ D ) = p ( D ∣ θ ) p ( θ ) p ( D ) p(\theta | D) = \frac{p(D | \theta) p(\theta)}{p(D)} p(θ∣D)=p(D)p(D∣θ)p(θ)
Where:
- ( θ \theta θ ): The parameter(s) of interest (e.g., mean, variance, or regression coefficients).
- ( D D D ): Observed data or evidence.
- ( p ( θ ) p(\theta) p(θ) ) : The prior distribution , representing our beliefs about ( θ \theta θ ) before observing the data.
- ( p ( D ∣ θ ) p(D | \theta) p(D∣θ) ) : The likelihood function , which gives the probability of observing ( D D D ) given ( θ \theta θ ).
- ( p ( D ) p(D) p(D) ) : The marginal likelihood , a normalization constant calculated as:
p ( D ) = ∫ p ( D ∣ θ ) p ( θ ) d θ p(D) = \int p(D | \theta) p(\theta) d\theta p(D)=∫p(D∣θ)p(θ)dθ - ( p ( θ ∣ D ) p(\theta | D) p(θ∣D) ) : The posterior distribution , representing our updated belief about ( θ \theta θ ) after observing the data.
2. Key Concepts
2.1 Prior Distribution (( p ( θ ) p(\theta) p(θ) ))
The prior distribution captures our knowledge or assumptions about ( θ \theta θ ) before seeing the data. Priors can be:
- Informative priors: Incorporate domain knowledge to reflect strong beliefs.
- Non-informative priors: Reflect minimal prior knowledge, such as uniform or flat distributions.
- Conjugate priors: Chosen to simplify calculations, where the prior and posterior share the same functional form (e.g., Gaussian prior for a Gaussian likelihood).
2.2 Likelihood (( p ( D ∣ θ ) p(D | \theta) p(D∣θ) ))
The likelihood function describes how the observed data ( D D D ) is generated given the parameter ( θ \theta θ ). For example:
- In a Gaussian model with unknown mean ( μ \mu μ ) and known variance ( σ 2 \sigma^2 σ2 ), the likelihood is:
p ( D ∣ μ ) = ∏ i = 1 N N ( x i ∣ μ , σ 2 ) p(D | \mu) = \prod_{i=1}^N \mathcal{N}(x_i | \mu, \sigma^2) p(D∣μ)=i=1∏NN(xi∣μ,σ2)
2.3 Posterior Distribution (( p ( θ ∣ D ) p(\theta | D) p(θ∣D) ))
The posterior distribution combines the prior distribution and the likelihood to represent our updated beliefs about ( \theta ) after observing the data. The posterior can be used for:
- Parameter estimation (e.g., mean, variance).
- Uncertainty quantification by providing credible intervals.
- Prediction of future observations.
2.4 Marginal Likelihood (( p ( D ) p(D) p(D) ))
Also known as the evidence, the marginal likelihood is used to normalize the posterior distribution. It can also compare models in Bayesian model selection.
3. The Bayesian Workflow
Step 1: Define the Prior
Choose a prior distribution ( p ( θ ) p(\theta) p(θ) ) based on domain knowledge or general assumptions.
Step 2: Specify the Likelihood
Define the likelihood function ( p ( D ∣ θ ) p(D | \theta) p(D∣θ) ) based on the probabilistic model of the observed data.
Step 3: Apply Bayes' Theorem
Combine the prior and likelihood to calculate the posterior distribution:
p ( θ ∣ D ) ∝ p ( D ∣ θ ) p ( θ ) p(\theta | D) \propto p(D | \theta) p(\theta) p(θ∣D)∝p(D∣θ)p(θ)
Step 4: Make Inferences
Use the posterior distribution to:
- Estimate parameters (e.g., posterior mean or mode).
- Compute credible intervals for uncertainty quantification.
- Predict future outcomes.
4. The Posterior Distribution in Bayesian Analysis
The posterior distribution ( p ( θ ∣ D ) p(\theta | D) p(θ∣D) ) represents the updated belief about the parameter ( θ \theta θ ) after observing the data ( D D D ). It balances:
- The prior (( p ( θ ) p(\theta) p(θ) )): Encodes initial beliefs or assumptions.
- The data (( p ( D ∣ θ ) p(D | \theta) p(D∣θ) )): Provides evidence through the likelihood.
Dynamic Updating
Bayesian inference allows continuous updating:
- Use the current posterior as the new prior when additional data becomes available.
- Repeat the process iteratively to refine parameter estimates.
Interpretation
Unlike frequentist point estimates, the posterior gives a probability distribution over parameters, allowing richer inferences:
- Credible intervals indicate the range where the parameter lies with high probability.
- Posterior predictive checks assess model fit to data.
5. Advantages of Bayesian Statistics
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Incorporation of Prior Knowledge
Bayesian methods explicitly allow for the inclusion of prior information, which is useful when data is scarce.
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Uncertainty Quantification
The posterior distribution quantifies uncertainty about parameters, providing more nuanced inferences.
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Flexibility
Bayesian methods handle complex hierarchical models, missing data, and non-standard distributions.
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Dynamic Learning
Bayesian inference allows continuous updating as new data becomes available.
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Probabilistic Predictions
Predictions are made with associated uncertainties, rather than single point estimates.
6. Applications of Bayesian Statistics
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Machine Learning
- Bayesian optimization for hyperparameter tuning.
- Bayesian neural networks for uncertainty-aware predictions.
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Medicine
- Estimating treatment effects and drug efficacy.
- Decision-making under uncertainty.
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Natural Language Processing
- Bayesian topic modeling (e.g., Latent Dirichlet Allocation).
- Bayesian approaches for text classification.
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Economics and Finance
- Predicting stock prices and market trends.
- Risk assessment and portfolio optimization.
7. Example: Bayesian Inference for a Gaussian Mean
Problem Setup
Suppose we observe ( N N N ) data points ( { x 1 , x 2 , . . . , x N } \{x_1, x_2, ..., x_N\} {x1,x2,...,xN} ), assumed to be drawn from a Gaussian distribution with unknown mean ( μ \mu μ ) and known variance ( σ 2 \sigma^2 σ2 ).
Prior Distribution
The prior for ( μ \mu μ ) is also Gaussian:
p ( μ ) = N ( μ ∣ μ 0 , σ 0 2 ) p(\mu) = \mathcal{N}(\mu | \mu_0, \sigma_0^2) p(μ)=N(μ∣μ0,σ02)
Likelihood
The likelihood of the data is:
p ( D ∣ μ ) = ∏ i = 1 N N ( x i ∣ μ , σ 2 ) p(D | \mu) = \prod_{i=1}^N \mathcal{N}(x_i | \mu, \sigma^2) p(D∣μ)=i=1∏NN(xi∣μ,σ2)
Posterior
Combining the prior and likelihood gives the posterior:
p ( μ ∣ D ) ∝ p ( D ∣ μ ) p ( μ ) p(\mu | D) \propto p(D | \mu) p(\mu) p(μ∣D)∝p(D∣μ)p(μ)
The posterior distribution for ( μ \mu μ ) is also Gaussian:
p ( μ ∣ D ) = N ( μ ∣ μ n , σ n 2 ) p(\mu | D) = \mathcal{N}(\mu | \mu_n, \sigma_n^2) p(μ∣D)=N(μ∣μn,σn2)
Where:
- ( μ n = σ 0 2 x ˉ + σ 2 μ 0 σ 0 2 + N σ 2 \mu_n = \frac{\sigma_0^2 \bar{x} + \sigma^2 \mu_0}{\sigma_0^2 + N\sigma^2} μn=σ02+Nσ2σ02xˉ+σ2μ0 )
- ( σ n 2 = σ 0 2 σ 2 σ 0 2 + N σ 2 \sigma_n^2 = \frac{\sigma_0^2 \sigma^2}{\sigma_0^2 + N\sigma^2} σn2=σ02+Nσ2σ02σ2 )
Here, ( μ n \mu_n μn ) is the posterior mean, a weighted average of the prior mean ( μ 0 \mu_0 μ0 ) and the sample mean ( x ˉ \bar{x} xˉ ), and ( σ n 2 \sigma_n^2 σn2 ) is the posterior variance, reflecting reduced uncertainty after observing data.
Conclusion
Bayesian statistics is a powerful paradigm that combines prior knowledge and observed data to iteratively refine our understanding of parameters. Its ability to provide probabilistic inferences and quantify uncertainty makes it an essential tool across various fields, from machine learning to medicine and finance.
后记
2024年11月28日15点28分于上海，在GPT4o大模型辅助下完成。