Bayesian statistics, a probability-based approach to statistical analysis, is increasingly used in the field of data analysis in Python. Different from the traditional frequentist school, Bayesian statistics makes full use of prior information and constantly updates the estimation of parameters according to new data. This article will introduce in detail the advanced technical points of Bayesian statistics in Python data analysis, including Bayesian inference, probability programming and Markov chain Monte Carlo, etc.
1. Bayesian inference
Bayesian inference is one of the core methods of Bayesian statistics. It uses Bayesian formula to calculate the posterior probability and obtain more accurate estimates by updating the prior probability. In Python, Bayesian inference analysis can be performed using the PyMC3 library.
1.1 Prior distribution
A key part of Bayesian inference is the prior distribution, which represents initial beliefs about unknown parameters. In PyMC3, we can use various probability distributions (such as normal distribution, uniform distribution, etc.) to establish prior distributions.
import pymc3 as pm
with pm.Model() as model:
# 定义先验分布
mu = pm.Normal('mu', mu=0, sd=1)
sigma = pm.HalfNormal('sigma', sd=1)
1.2 Posterior sampling
Posterior sampling is the core step of Bayesian inference, which obtains the posterior probability distribution of parameters through sampling methods. In PyMC3, methods such as MCMC (Markov Chain Monte Carlo) and variational inference can be used for posterior sampling.
with model:
# 执行马尔科夫链蒙特卡洛采样
trace = pm.sample(5000, tune=1000)
1.3 Posterior Analysis
Posterior analysis is the process of analyzing and interpreting the posterior sampling results. PyMC3 provides a wealth of tools and functions for posterior analysis.
# 查看参数的后验概率分布直方图
pm.plot_posterior(trace)
# 汇总参数的统计指标
pm.summary(trace)
# 计算参数的HPD置信区间
pm.stats.hpd(trace['mu'])
2. Probabilistic programming
Probabilistic programming is a programming paradigm based on probabilistic models that unifies the definition of models and the process of inference into a single framework. In Python, libraries such as PyMC3 and Edward can be used for probabilistic programming to achieve flexible definition and inference of models.
2.1 PyMC3 Probability Model
PyMC3 provides an intuitive and flexible way to define probabilistic models by using Python syntax and conventions to describe random variables and their relationships.
import pymc3 as pm
with pm.Model() as model:
# 定义随机变量
x = pm.Normal('x', mu=0, sd=1)
y = pm.Normal('y', mu=x, sd=1, observed=data)
2.2 Edward Probabilistic Programming
Edward is another popular probabilistic programming toolkit that can define probabilistic models using a high-level API and provides various inference algorithms.
import tensorflow as tf
import edward as ed
# 定义随机变量
x = ed.Normal(loc=0, scale=1)
y = ed.Normal(loc=tf.gather(x, indices), scale=1, observed=data)
3. Markov Chain Monte Carlo
Markov Chain Monte Carlo (MCMC) is a parameter estimation method commonly used in Bayesian statistics, which samples through the Markov chain and converges to the target distribution under certain conditions. In Python, MCMC sampling can be performed using libraries such as PyMC3 and Stan.
3.1 MCMC sampling in PyMC3
PyMC3 provides sample()functions to perform MCMC sampling, supporting multiple sampling algorithms (such as NUTS, Metropolis-Hastings, etc.) and parameter tuning options.
with model:
# 使用NUTS算法执行MCMC采样
trace = pm.sample(5000, tune=1000, nuts_kwargs={'target_accept': 0.9})
3.2 Stan's MCMC Sampling
Stan is another popular probabilistic programming language and library that provides powerful MCMC sampling and model inference capabilities.
import stan
# 编写Stan模型代码
stan_code = """
data {
int<lower=0> N;
vector[N] y;
}
parameters {
real mu;
real<lower=0> sigma;
}
model {
y ~ normal(mu, sigma);
}
"""
# 编译并拟合模型
stan_model = stan.build(stan_code, data=data)
fit = stan_model.sample(num_chains=4, num_samples=5000)
in conclusion
Through the introduction of this article, you understand the advanced technical points of Bayesian statistics in Python data analysis, including the concept and application of Bayesian inference, the principle and implementation of probabilistic programming, and Markov Chain Monte Carlo (MCMC) ) and its usage in Python. These advanced technical points can help you more fully understand and apply the role of Bayesian statistics in data analysis.
Bayesian inference is a statistical inference method that calculates the posterior probability distribution of parameters by combining prior knowledge and observation data. In Bayesian inference, we treat parameters as random variables and use the Bayesian formula to compute posterior probabilities from prior probabilities and likelihood functions. An important step in Bayesian inference is posterior sampling, which approximates the posterior probability distribution by generating samples that conform to the posterior distribution. Commonly used posterior sampling methods include Markov chain Monte Carlo (MCMC) and variational inference, etc.
Probabilistic programming is a programming paradigm that unifies probabilistic models and inference procedures into a single framework. It allows us to use the Python language to describe the structure and parameter relationships of probabilistic models, and use inference algorithms for model inference and parameter estimation. PyMC3 and Edward are two commonly used probabilistic programming libraries that provide high-level APIs to define probabilistic models and support a variety of inference algorithms.
Markov Chain Monte Carlo (MCMC) is a Markov chain-based sampling method for generating samples from complex distributions. The core idea of MCMC is to perform a series of iterations on the current state through the transition matrix of the Markov chain, so that the final state converges to the target distribution. In Python, libraries such as PyMC3 and Stan provide convenient interfaces to perform MCMC sampling and support a variety of sampling algorithms and parameter tuning options.
Bayesian statistics has a wide range of applications in Python data analysis. Advanced techniques such as Bayesian inference, probabilistic programming, and Markov chain Monte Carlo can more accurately estimate parameters, perform model selection, and perform predictive analysis. In practical applications, it is very important to select appropriate tools and methods for analysis and modeling according to the needs of specific problems and the characteristics of data.