Overview and various implementations of Bayesian estimation

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Overview of Bayesian Estimation Techniques

Bayesian inference is a method of statistical inference based on a probabilistic framework and is a machine learning technique for dealing with uncertainty. The objective of Bayesian inference is to combine data and prior knowledge (prior distribution) to estimate the probability distribution of unknown parameters. The following is an overview of the Bayesian estimation procedure.

Setting the prior distribution: In Bayesian estimation, the prior distribution of the unknown parameters is first set. The prior is the probability distribution of the parameters before the data are available, and is set based on existing knowledge and assumptions.
Observing the Data: The next step in Bayesian estimation is to observe the data. The data obtained here are used to estimate the parameters.
Calculating the likelihood: The likelihood is the conditional probability of the parameters given the data, and Bayesian estimation involves the step of modeling the relationship between the data and the parameters in order to calculate the likelihood.
Calculating the posterior distribution of the parameter: Using Bayes’ theorem, the prior distribution and the likelihood are combined to calculate the posterior distribution of the parameter. The posterior distribution is the probability distribution of the parameters after accounting for the data.
Make inferences and predictions: Use the posterior distribution of the parameters to make various inferences and predictions. This can include calculating means and confidence intervals for parameters or making predictions for new data.

The use of Bayesian inference allows for data analysis and decision making while accounting for uncertainty, and also has the advantage of being flexible to successive learning and addition of information.

Algorithms used for Bayesian estimation

Various algorithms and methods exist for Bayesian estimation. Some representative algorithms are described below.

Markov Chain Monte Carlo (MCMC): MCMC is one of the main methods in Bayesian estimation and is used to sample the posterior distribution of parameters. Typical MCMC algorithms include the Metropolis-Hastings algorithm and Gibbs sampling. These algorithms can efficiently explore the parameter space and generate samples from the posterior distribution. For more information on MCMC, see “Markov Chain Monte Carlo (MCMC) Methods and Bayesian Estimation.
Variational Bayesian (VB): VB is a variational inference method for approximating the posterior distribution; the principle of VB is to estimate the posterior distribution by choosing from a predefined set of approximate distributions. Variational Bayesian methods have advantages such as computational efficiency and the possibility of obtaining analytical approximate solutions. For more information on variational Bayesian methods, see “About Variational Bayesian Learning.
Particle Filter (PF): PF is a Bayesian estimation technique for nonlinear state-space models that uses particles in state space to approximate the posterior distribution, resampling particles over a series of time steps and using weighted particles to represent the posterior distribution. For more information on particle filters, see “Geospatial Information in Motion – Simulation and Data Assimilation.
Bayesian Optimization: Bayesian optimization is a Bayesian estimation technique for the optimization problem of the objective function. Bayesian optimization combines prior knowledge (prior distribution) and observed data to estimate the optimal solution of the objective function and is used for experimental evaluation and computationally expensive optimization problems such as active learning and hyperparameter optimization. For Bayesian optimization, see “Nonparametric Bayesian and Gaussian Processes.

Libraries and platforms used for Bayesian estimation

The following libraries and platforms are commonly used to implement Bayesian inference

PyMC3: PyMC3 provides a powerful library for building Bayesian statistical models in Python. pyMC3 supports MCMC sampling and variational inference, and provides a flexible and easy-to-use API. and supports a wide range of Bayesian modeling problems.
Stan: Stan provides a platform for Bayesian statistical modeling and high-performance MCMC sampling. Stan offers interfaces in C++, Python, R, etc., and uses a dedicated language for describing stochastic models (the Stan language), with a focus on efficient estimation and It focuses on model scalability and supports large, complex models.
Edward: Edward will be a library of Bayesian inference that runs on TensorFlow, providing a variety of inference methods, such as variational inference and stochastic gradient methods (SGHMC, SGNHT, etc.), leveraging TensorFlow’s capabilities, It is also possible to build complex models such as Bayesian neural networks.
Pyro: Pyro is a library for Bayesian inference based on PyTorch that provides a framework for probabilistic programming and supports inference methods such as variational inference and importance sampling Pyro is flexible and extensible, combining probabilistic models with neural networks It is suitable for building Bayesian-like models.
Anglican: Anglican is a Clojure-based probabilistic programming platform that combines the functional language programming capabilities of Clojure with probabilistic programming. Anglican allows you to build and infer models for Bayesian inference.

For examples of applications of Bayesian estimation

Bayesian inference has been widely applied in a variety of domains, and several applications of Bayesian inference are discussed below.

Machine Learning and Pattern Recognition: Bayesian inference is often used in the domains of machine learning and pattern recognition. For example, Bayesian linear regression and Bayesian neural networks using Bayesian estimation can make predictions and classifications that account for parameter uncertainty.
Medical research: Bayesian inference is also used for data analysis and evaluation in medical research. This is specifically used to analyze the results of clinical trials to estimate treatment effects and to use medical data to predict disease progression and assess risk.
FINANCE AND RISK MANAGEMENT: Bayesian estimation is also applied to financial market modeling and risk management. This is specifically the use of Bayesian estimation in stock price forecasting and portfolio optimization, as well as in risk-informed decision making.
Economics and Social Sciences: Bayesian estimation is also widely used in economics and social sciences. Specifically, it has been applied to elucidate and forecast various economic and social phenomena, such as demand forecasting, market modeling, and modeling of social phenomena.
Natural Sciences and Engineering: Bayesian inference is also used in the natural sciences and engineering, including physics, chemistry, environmental sciences, and engineering. It is specifically applied to the analysis of experimental data, signal processing, filtering of sensor data, and estimation of model parameters.

Because of its ability to handle uncertainty, Bayesian estimation is a powerful tool in estimation and prediction based on real data, and is used in a variety of domains. The following sections describe each of these applications in detail.

On Machine Learning and Pattern Recognition Using Bayesian Inference

<Overview>.

Bayesian inference is widely used in the domains of machine learning and pattern recognition. Below we describe some specific applications of machine learning and pattern recognition using Bayesian estimation.

Bayesian Linear Regression: Bayesian linear regression is a method of estimating and predicting parameters in a linear regression model using Bayesian estimation. Bayesian linear regression assumes a normal distribution as a prior distribution and estimates the posterior distribution of the parameters based on observed data. This allows for forecasting and estimation that takes into account parameter uncertainty. For more information on Bayesian linear regression, please refer to the “Machine Learning Professional Series: Bayesian Deep Learning Reading Note.
Bayesian Neural Networks: Bayesian neural networks are a combination of neural networks and Bayesian estimation. While parameters are estimated at points in ordinary neural networks, Bayesian neural networks estimate the posterior distribution of parameters, which enables prediction and calculation of confidence intervals that reflect the uncertainty of the model. For more information on Bayesian neural networks, please refer to “Machine Learning Professional Series: Bayesian Deep Learning Reading Memo” etc.
Bayesian Non-negative Matrix Factorization: Bayesian non-negative matrix factorization is a method used for dimensionality reduction and feature extraction of matrix data. By decomposing the data matrix into non-negative matrix factors, the structure and patterns of the data can be extracted and combined with Bayesian estimation to estimate the factor matrix and adaptively select dimensions for the model. For more information on stochastic matrix factorization, see Clustering Techniques for Asymmetric Relational Data – Stochastic Block Models and Infinite Relational Models.
Gaussian Processes: Gaussian processes are nonlinear regression and classification methods based on Bayesian inference. In Gaussian processes, data are modeled by a Gaussian process to estimate the posterior distribution. Gaussian processes are flexible models that allow for forecasting and confidence interval calculations that account for model uncertainty. For more information on Gaussian processes, see “Nonparametric Bayesian and Gaussian Processes.

Thus, Bayesian inference can be applied to machine learning and pattern recognition to model uncertainty and improve the reliability of predictions. One of the advantages of Bayesian estimation is that it can incorporate prior knowledge and be effective for small data problems. Below we discuss a few examples of implementations of Bayesian linear regression.

<Bayesian linear regression implementation in python>

A common Python implementation of Bayesian linear regression is to use the PyMC3 library. A simple example of Bayesian linear regression implementation using PyMC3 is described below.

import pymc3 as pm
import numpy as np

# Training Data
X_train = np.array([1, 2, 3, 4, 5])
y_train = np.array([2, 4, 6, 8, 10])

# Model Building
with pm.Model() as model:
    # prior distribution
    intercept = pm.Normal('intercept', mu=0, sd=10)
    slope = pm.Normal('slope', mu=0, sd=10)
    sigma = pm.HalfNormal('sigma', sd=1)
    
    # linear model
    y_pred = intercept + slope * X_train
    
    # likelihood function
    likelihood = pm.Normal('y', mu=y_pred, sd=sigma, observed=y_train)
    
    # sampling
    trace = pm.sample(2000, tune=1000, cores=1)  # MCMC Sampling

# Confirmation of results
pm.summary(trace)
pm.traceplot(trace)

In this example, a Bayesian linear regression model is constructed using the training data X_train and y_train, with intercept (intercept), slope (slope), and standard deviation of observed error (sigma) as parameters of the model as a prior distribution. The linear model predictions y_pred are calculated using the intercept and slope, and the observed data y_train is incorporated into the model as a normally distributed likelihood function.

Sampling is done using the pm.sample() method, taking 2000 samples, with the first 1000 being ignored as burn-in (adjustment period). The results of the sampling are stored in a trace object. Finally, pm.summary() and pm.traceplot() are used to display summary statistics and trace plots of the sampling results.

Medical Research Using Bayesian Estimation

<Overview>

Bayesian inference is used in various areas of medical research. One specific example is the estimation of the effect of drug administration, as described in “Statistical Science Incorporating Individual and Regional Differences: Examples in the Medical Field. Other practical examples are described below.

Analysis of clinical trials: Bayesian inference can be useful in the analysis of data from clinical trials. This includes, for example, the use of Bayesian estimation when analyzing data from randomized controlled trials to evaluate the effect of a new treatment. Bayesian estimation can be used to estimate probability distributions of outcomes and confidence intervals for treatment effects.
Dose Finding: Bayesian estimation is also useful in assessing the efficacy and safety of drugs. In particular, dose-finding sometimes uses Bayesian estimation to find the optimal dose for a patient. Bayesian estimation can be applied to model the relationship between dose and response, to estimate the optimal dose, and to assess the risk of side effects.
Survival Analysis: Survival analysis is an important area of medical research. Bayesian estimation is used to estimate hazard ratios and survival curves in survival analysis. Bayesian estimation can be used to estimate model parameters for survival analysis and confidence intervals for survival probabilities.
Prediction of personalized medicine: Bayesian estimation has also been applied in the study of personalized medicine. In personalized medicine, the goal is to determine the optimal treatment strategy based on patient characteristics and pathophysiology. Bayesian estimation enables us to model the relationship between patient characteristics and treatment effects, and to make individualized predictions and decisions.

As described above, Bayesian estimation has become a useful method to account for uncertainty in medical data and to perform statistical estimation as a basis for forecasting and decision making. The following section discusses some of these and discusses the time-varying grandiosity of the analysis of clinical trials.

<Example of a clinical trial analysis implementation in python>

To implement the analysis of clinical trials by Bayesian estimation in Python, several libraries are required. In the following example, we use the libraries NumPy, SciPy, and PyMC3. First, install these libraries.

$ pip install numpy scipy pymc3

Next, we will analyze the clinical trial using Bayesian estimation in Python in the following manner.

import numpy as np
import pymc3 as pm

# Data Preparation
# Number of subjects and successes in Group 1
n1 = 50
k1 = 35

# Number of subjects and successes in Group 2
n2 = 50
k2 = 40

# Setting the prior distribution
alpha = 1  # Hyperparameters of the prior distribution for Group 1
beta = 1   # Hyperparameters of the prior distribution for Group 2

# Model Definition
with pm.Model() as model:
    # Setting the prior distribution
    theta1 = pm.Beta('theta1', alpha, beta)
    theta2 = pm.Beta('theta2', alpha, beta)
    
    # Likelihood settings
    y1 = pm.Binomial('y1', n=n1, p=theta1, observed=k1)
    y2 = pm.Binomial('y2', n=n2, p=theta2, observed=k2)
    
    # Performing Estimation
    trace = pm.sample(1000, tune=1000, cores=1)  # sampling
    
# Output of estimation results
pm.summary(trace, hdi_prob=0.95)

In this example, the success rates (θ1 and θ2) for the two groups are estimated, using the Beta distribution as the prior distribution and the Binomial distribution as the likelihood. The estimation results are output using the pm.summary function to display the Bayesian confidence interval (95% HDI).

Financing and Risk Management Using Bayesian Estimation

<Overview>

Bayesian estimation has a wide range of applications in the areas of finanace and risk management. The following are some examples of finanace and risk management applications of Bayesian estimation.

Portfolio Optimization: Portfolio optimization is the problem of constructing an investment portfolio by combining multiple assets. By applying Bayesian estimation to this problem, the expected rate of return and risk of each asset can be estimated to determine the optimal portfolio proportions. It also makes it possible to incorporate prior knowledge and predictive models as well as historical data.
Valuation and Pricing: Bayesian estimation can also be applied to option pricing and the valuation of financial instruments. Specifically, Bayesian estimation can be used in the Black-Scholes model and its derivatives to estimate volatility and the probability distribution of prices. For more information on the Black-Scholes model, see “Intuitive Methods in Economic Mathematics: A Note on Probability and Statistics.
Risk Management: Bayesian estimation is also used as a risk management technique. For example, Bayesian estimation may be used to estimate risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR), and it can be used to detect outliers, estimate the probability of events, and conduct stress tests, It can also be used to address a wide range of aspects of risk management.
Valuation of derivatives: Bayesian estimation can also be useful in the valuation of derivatives. For example, Bayesian estimation can be used to estimate hidden variables of options (e.g., rate of change in volatility), which can help in option valuation and hedging strategy decisions.

Bayesian estimation is a powerful technique for accounting for information uncertainty and making risk assessment and decision making more reliable in the areas of finanace and risk management. The implementation of some of these for risk management is discussed below.

<Implementation in python of risk management using Bayesian estimation>

An example implementation of risk management using Bayesian estimation is shown below.

import numpy as np
import pymc3 as pm

# Data Preparation
data = np.array([0, 1, 0, 0, 1, 1, 0, 1, 1, 1])  # 0: イベントなし, 1: イベントあり

# Setting the prior distribution
alpha = 1  # Hyperparameters of prior distribution
beta = 1

# Model Definition
with pm.Model() as model:
    # Setting the prior distribution
    theta = pm.Beta('theta', alpha, beta)
    
    # Likelihood settings
    y = pm.Bernoulli('y', p=theta, observed=data)
    
    # Performing Estimation
    trace = pm.sample(1000, tune=1000, cores=1)  # sampling

# Output of estimation results
pm.summary(trace, hdi_prob=0.95)

In this example, the probability of occurrence of risk (occurrence of an event) is estimated Bayesian, with the data being a sequence of 0s and 1s, where 0 indicates that the event did not occur and 1 indicates that the event did occur. In the model, θ is the probability of occurrence of the risk, the beta distribution is used as the prior distribution, and the Bernoulli distribution is set as the likelihood.

The estimation results are output using the pm.summary function to display the Bayesian confidence interval (95% HDI), and Bayesian estimation allows estimating the posterior distribution of the probability of occurrence of the risk.

Economics and social science using Bayesian estimation

<Overview>

Bayesian inference has a wide range of applications in economics and social science research. The following are examples of applications of Bayesian estimation in economics and the social sciences.

Economic forecasting: Bayesian inference can be useful in constructing economic forecasting models. Specifically, Bayesian estimation is used to estimate the probability distribution of economic variables and forecast intervals when constructing models to forecast economic indicators such as economic growth rate, inflation rate, and unemployment rate.
Marketing Analysis: Bayesian inference is also applied in marketing analysis. Specifically, Bayesian inference can be used to model customer purchasing behavior and demand forecasting.
Social Network Analysis: Bayesian inference can also be useful in social network analysis. Specifically, Bayesian inference can be used to model relationships and connections between people as a network, estimate the propagation of information and influence on the network, estimate the network structure and characteristics of individual nodes, and contribute to the understanding and prediction of social networks.
Educational Evaluation: Bayesian inference is also applied in educational evaluation. This is specifically the case when building models to evaluate the effects of learning and educational policies, such as using Bayesian inference to evaluate and make predictions that take into account the characteristics of individual learners and schools.

In these ways, Bayesian estimation is a powerful method for estimation and forecasting in economics and social science research, taking into account highly uncertain data and complex relationships. The following is a python implementation of social network estimation using Bayesian estimation from the above.

<Implementation in python of social network estimation using Bayesian estimation>

There are several approaches to estimating social networks using Bayesian estimation. Here, we present an example implementation of social network estimation using a Dirichlet process, a type of nonparametric Bayesian model.

import numpy as np
import pymc3 as pm
import networkx as nx
import matplotlib.pyplot as plt

# Data Preparation
adjacency_matrix = np.array([[0, 1, 1, 0],
                             [1, 0, 1, 1],
                             [1, 1, 0, 1],
                             [0, 1, 1, 0]])

n = adjacency_matrix.shape[0]  # Number of nodes

# Model Definition
with pm.Model() as model:
    # Parameters of the Dirichlet process
    alpha = 1  # Hyperparameters of prior distribution
    
    # Generation of stochastic matrices by Dirichlet processes
    p_matrix = pm.Dirichlet('p_matrix', np.full((n, n), alpha), shape=(n, n))
    
    # Generation of links by Bernoulli distribution
    adjacency_matrix_pred = pm.Bernoulli('adjacency_matrix_pred', p=p_matrix, observed=adjacency_matrix)
    
    # Performing Estimation
    trace = pm.sample(1000, tune=1000, cores=1)  # sampling

# Output of estimation results
pm.summary(trace, hdi_prob=0.95)

# Visualization of estimated social networks
plt.figure(figsize=(6, 6))
g = nx.from_numpy_matrix(np.mean(trace['p_matrix'], axis=0))
pos = nx.circular_layout(g)
nx.draw_networkx(g, pos=pos, with_labels=True, node_color='skyblue', edge_color='gray', width=1, alpha=0.8)
plt.axis('off')
plt.show()

In this example, the structure of the social network is estimated from a given adjacency matrix, where the adjacency matrix consists of zeros and ones, representing the presence or absence of links between nodes. The model uses a Dirichlet process to generate the probability matrix and a Bernoulli distribution to model the presence of links. The estimated results are output using the pm.summary function to display the Bayesian confidence interval (95% HDI). In addition, a graph is drawn using NetworkX and Matplotlib to visualize the estimated social network.

On natural science and engineering using Bayesian inference

<Overview>

Bayesian inference has applications in various fields of natural science and engineering. The following are some examples of natural science and engineering applications of Bayesian inference.

Physics: Bayesian inference is widely used in the field of physics. This includes, specifically, the use of Bayesian estimation to estimate physical parameters and model selection in the analysis of experimental data in particle physics, and the use of Bayesian estimation to estimate parameters of cosmological models in the analysis of observational data in astronomy.
Chemistry: In the field of chemistry, Bayesian estimation is used to estimate reaction rate constants and to estimate the structure of molecules. Bayesian estimation can be used to combine experimental data and prior knowledge to obtain more reliable estimation results.
Biology: Bayesian inference is also an important method in the study of biology. In the fields of genetics and evolutionary biology, Bayesian inference is used to estimate gene evolution and phylogenetic relationships among species, and it may also be used in modeling biological systems and analyzing biological responses.
Engineering: Bayesian inference has a variety of applications in engineering. This is specifically the case in signal and image processing, where Bayesian estimation is used for noise reduction and signal restoration, and also in parameter estimation for control systems and machine learning models.

The following is a python implementation of molecular structure estimation using Bayesian estimation.

<Implementation in python of molecular structure estimation using Bayesian estimation>

Molecular structure estimation using Bayesian inference is performed by estimating the posterior distribution of parameters (e.g., coordinates of atoms) that represent the structure of a molecule. An example implementation is shown below.

import numpy as np
import pymc3 as pm
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Data Preparation
observed_data = np.array([[0.0, 0.0, 0.0],
                          [1.0, 0.0, 0.0],
                          [0.0, 1.0, 0.0]])

n_atoms = observed_data.shape[0]  # atomic number
n_dimensions = observed_data.shape[1]  # rank

# Setting the prior distribution
prior_mean = np.zeros((n_atoms, n_dimensions))
prior_cov = np.eye(n_atoms)

# Model Definition
with pm.Model() as model:
    # Set parameter pre-distribution
    params = pm.MvNormal('params', mu=prior_mean, cov=prior_cov, shape=(n_atoms, n_dimensions))
    
    # Likelihood settings
    likelihood = pm.MvNormal('likelihood', mu=params, observed=observed_data, shape=(n_atoms, n_dimensions))
    
    # Performing Estimation
    trace = pm.sample(1000, tune=1000, cores=1)  # sampling

# Output of estimation results
pm.summary(trace, hdi_prob=0.95)

# Visualization of the inferred structure of the molecule
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
for i in range(n_atoms):
    ax.scatter(trace['params'][:, i, 0], trace['params'][:, i, 1], trace['params'][:, i, 2], marker='o', alpha=0.2)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
plt.show()

In this example, the molecular structure is estimated from the given observed data (coordinates of atoms), variables (params) representing the coordinates of atoms are introduced as parameters in the model, and a multivariate normal distribution is used as the prior distribution. The multivariate normal distribution is also set as the likelihood, and the relationship between the observed data and the parameters is modeled. The estimation results are output using the pm.summary function to display the Bayesian confidence interval (95% HDI), and the coordinates of the atoms are plotted using Matplotlib’s 3D plot to visualize the estimated molecular structure.

Reference Books and Reference Information

For more detailed information on Bayesian inference, please refer to “Probabilistic Generative Models” “Bayesian Inference and Machine Learning with Graphical Models” and “Nonparametric Bayesian and Gaussian Processes.

A good reference book on Bayesian estimation is “The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of C“

“Think Bayes: Bayesian Statistics in Python“

“Bayesian Modeling and Computation in Python“

“Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ, 2nd Edition“