Explore the basic principles and implementation process of nested sampling algorithms

The nested sampling algorithm is an efficient Bayesian statistical inference algorithm used to calculate the integral or summation under complex probability distributions. It works by decomposing the parameter space into multiple hypercubes of equal volume, and gradually and iteratively "pushing out" one of the smallest volume hypercubes, and then filling the hypercube with random samples to better estimate the integral value of the probability distribution. . Through continuous iteration, the nested sampling algorithm can obtain high-precision integral values ​​and boundaries of parameter space, which can be applied to statistical problems such as model comparison, parameter estimation, and model selection. The core idea of ​​this algorithm is to transform complex integration problems into a series of simple integration problems, and approach the real integral value by gradually reducing the volume of the parameter space. Each iteration step obtains samples from the parameter space through random sampling and performs weighted calculations according to the probability density function of the samples to obtain an estimate of the integral value. The advantage of the nested sampling algorithm is that it can handle various complex probability distributions and performs well in terms of computational efficiency and accuracy.

The nested sampling algorithm was originally proposed by Skilling in 2004. It is widely used in data analysis and model comparison in astronomy, statistics, physics, biology and other fields. Below we will introduce the basic idea and implementation process of the nested sampling algorithm through a simple example.

Suppose we have a probability density function p(x) of a normal distribution, and we want to calculate its integral value over the entire real number interval, that is, solve for ∫p(x)dx. According to the properties of the normal distribution, we know that the integral value of p(x) is 1. In order to verify this property, we can use the nested sampling algorithm for calculation. The basic idea of ​​this algorithm is to approximate the integral value by randomly sampling on a normal distribution and performing a weighted summation of the sampling points. By repeatedly performing the process of sampling and weighted summation, we can obtain an integral value close enough to 1 to verify the properties of the normal distribution.

First, we decompose the parameter space [-∞, ∞] into multiple hypercubes V_i with equal volumes. The volume of each hypercube is ΔV = 1/N, where N is the number of hypercubes. We use x_i to represent a random sample in the i-th hypercube, and then calculate the value of p(x_i). To ensure that each hypercube can be filled, we need to randomly sample some samples from one hypercube and fill these samples into other hypercubes. This way, each hypercube is filled and we get a more accurate estimate of the probability density function.

Then, we need to select a hypercube V_{\text{min}} whose probability density function value is the smallest. In order to implement this process, we need to remove the sample with the smallest probability density function value in V_{\text{min}}, that is, remove all samples with the smallest probability density function value in x_i from V_{\text{min}} . During this process, we need to record the volume and minimum probability density function value of V_{\text{min}} and use them as reference values ​​for the next iteration.

Repeat the above process until all hypercubes are "pushed out", at which point we have a complete probability density function estimate and an approximation of the integral value. The specific implementation process is as follows:

import numpy as np

def log_likelihood(x):
"""定义概率密度函数"""
return -0.5 * x ** 2

def nested_sampling(N, log_likelihood):
"""嵌套采样算法实现"""
log_X = -np.inf
logL = [log_likelihood(np.random.randn()) for i in range(N)]
for i in range(N):
# 找到最小的概率密度函数值的样本
idx = np.argmin(logL)
logL[idx] = np.inf
# 计算当前的体积和概率密度函数值
log_X_new = logL[idx] - np.log(N - i)
logL_new = log_likelihood(np.random.randn())
# 更新 X 和 logL
log_X = np.logaddexp(log_X,log_X_new)
logL[idx] = logL_new
# 返回结果
return log_X, log_X - np.log(N)

Among them, N represents the number of hypercubes, log_likelihood is the logarithmic value of the probability density function, log_X is the approximate value of the logarithmic integral value, and logL is the minimum probability density function in each hypercube. The logarithmic value of the value, np.logaddexp is the logarithmic addition function, used to avoid numerical underflow or overflow.

In the above code, we first define a probability density function log_likelihood of a normal distribution, and then implement the nested sampling algorithm through the nested_sampling function. In this function, we first initialize the value of log_X to negative infinity, then iterate through the loop N times to find the sample with the smallest probability density function value, calculate the current volume and probability density function value, update the values ​​of log_X and logL, and Return the final result.

It should be noted that we did not directly calculate the integral value in the above code, but calculated its logarithmic value log_X. This is because in actual calculations, the probability density function Values ​​are often very small and may cause numeric underflow or overflow. Therefore, we usually compute integrals using logarithmic values, which avoids numerical problems and allows for better handling of products and integrals of probability density functions.

The nested sampling algorithm is a very effective statistical inference algorithm that can be used to calculate the integral or summation under complex probability distributions. Its main idea is to decompose the parameter space into multiple hypercubes of equal volume, and then continuously iterate by randomly sampling and "pushing out" the hypercube to obtain high-precision integral values ​​and boundaries of the parameter space. Nested sampling algorithms are widely used in data analysis and model comparison in astronomy, statistics, physics, biology and other fields.

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