Overview of Black-Box Variational Inference (BBVI)
Black-Box Variational Inference (BBVI) is a type of variational inference method for approximating the posterior distribution of complex probabilistic models in probabilistic programming and Bayesian statistical modeling, where the posterior distribution is difficult to solve analytically and approximate Variational inference is a method that uses approximate methods to perform inference when the posterior distribution is difficult to solve analytically.
BBVI is called “Black-Box” because the probability model to be inferred is treated as a black box and can be applied without depending on the internal structure of the model itself or the form of the likelihood function. BBVI is a method that enables inference even without knowing the internal structure of the model.
The main idea of BBVI is to introduce a variational distribution of the posterior distribution and use this variational distribution to approximate the true posterior distribution. The variational distribution is designed to concentrate on regions of high likelihood for a given set of parameters and is trained to minimize the appropriate distance from the true posterior distribution.
The BBVI procedure is outlined as follows:
1. Selection of variate distribution: A variate distribution is selected to approximate. Typically, the variate distribution is chosen to be sufficiently flexible with respect to the posterior distribution, but still computationally feasible.
2. Maximize the Evidence Lower BOund (ELBO): Maximize the lower bound of evidence, which is the objective function of variational inference, using the Kullback-Leibler (KL) divergence as described in “Overview of Kullback-Leibler Variational Inference and Various Algorithms and Implementations“. Leibler Divergence), which is also described in “Overview of Kullback-Leibler Variance Estimation and Various Algorithms and Implementations”, and approximates the posterior distribution by maximizing it.
3. Optimization using the gradient method: To maximize the ELBO, the parameters of the variate distribution are updated using the gradient method or its variants as described in “Overview of the Gradient Method, Algorithms and Examples of Implementation“. At this point, it is necessary to evaluate the likelihood function of the model, and BBVI is characterized by the fact that it is as black box as possible.
BBVI is useful when prior knowledge of the model is limited or when dealing with complex stochastic models, and is particularly applicable to large data sets and high-dimensional parameter spaces.
Algorithms used in Black-Box Variational Inference (BBVI)
Black-Box Variational Inference (BBVI) requires solving optimization problems that maximize the Evidence Lower Bound (ELBO) as part of variational inference. This section describes optimization algorithms and methods commonly used in BBVI. 1.
1. Gradient Ascent:
In BBVI, the gradient method is commonly used to maximize the ELBO, which is the lower bound of the KL divergence between the variational distribution and the true posterior distribution, and maximizing this lower bound brings us closer to the true posterior distribution. The gradient of the parameters is obtained by computing the derivative with respect to the ELBO parameters, which is then used to update the parameters. For more information, see “Overview of the Gradient Method with Algorithms and Examples of Implementations.
2. Stochastic Gradient Ascent (SGA):
When the data is large, it is computationally expensive to calculate the ELBO gradient over the entire data set. Stochastic Gradient Ascent estimates and updates the gradient using a randomly selected subset (mini-batch) of data, which allows the model to be trained efficiently on large data sets. For details, see “Overview of Stochastic Gradient Descent (SGD), its algorithms and examples of implementation“.
3. Natural Gradient Descent:
The usual gradient method assumes that the parameter space is a Euclidean space. However, the parameters of a probability distribution are usually constrained, and the natural gradient method updates the gradient by considering the appropriate metric in the parameter space (the inverse of the Fisher information matrix described in “Overview of the Fisher Information Matrix and Related Algorithms and Examples of Implementations“). For details, please refer to “Overview of the Natural Gradient Method, Algorithms, and Examples of Implementations.
4. Black-Box Optimization:
In BBVI, the model and likelihood function are sometimes considered black boxes, so black-box optimization methods are sometimes applied to variational inference.
Application of Black-Box Variational Inference (BBVI)
Black-Box Variational Inference (BBVI) has been applied to a variety of Bayesian modeling problems. Specific applications are described below.
1. stochastic programming:
BBVI is widely used in probabilistic programming as described in “Probabilistic Programming with Clojure. Probabilistic programming is a method of describing a probabilistic model and performing inference based on that model, and BBVI is useful when the model is a black box.
2. deep generative models:
BBVI is used in Generative Models in Deep Learning. For example, the Variational Autoencoder (VAE) described in “Variational Autoencoder (VAE) Overview, Algorithm and Example Implementation
3. Large Data Sets:
BBVI is also an effective method for large data sets.” Stochastic Gradient Descent (SGD): An Overview, Algorithm, and Implementation Examples“, BBVI can be used to train models while reducing computational cost by computing gradients in mini-batches using the stochastic gradient method.
4 Bayesian Neural Networks:
BBVI is also used in Bayesian neural networks (BNN), which incorporate neural networks into Bayesian models and are useful for inferring uncertainty in the parameters of the network. For details, please refer to “Overview of Bayesian Deep Learning and Examples of Applications and Implementations.
5. Statistical Modeling:
BBVI has also been applied to various statistical modeling problems. These include, for example, the hierarchical Bayesian models described in “Individuality and Parameter Estimation (Interpreting Hierarchical Bayesian Models)” and models with time dependence.
6. black box optimization:
BBVI has also been applied as a black box optimization method, and is used for optimization problems in situations where the objective function is not known and is assumed to be differentiable.
As these examples show, BBVI can be a widely applied method to deal with the complexity of various Bayesian modeling and the large size of the data.
Example Implementation of Black-Box Variational Inference (BBVI)
Examples of BBVI implementations vary by programming language and library. Here we show a simple example implementation of BBVI using Python and NumPy. Note that in actual applications, more advanced probabilistic programming frameworks and libraries (Stan, PyMC3, Edward, TensorFlow Probability, etc.) will generally be used.
The following example implements BBVI, which approximates a one-dimensional normal distribution.
import numpy as np
import scipy.stats as stats
def normal_density(x, mean, std):
"""
Probability density function of normal distribution
"""
return np.exp(-(x - mean)**2 / (2 * std**2)) / np.sqrt(2 * np.pi * std**2)
def sample_from_q(params, num_samples=1):
"""
Variational distribution that generates a sample from the parameters q
"""
return np.random.normal(params[0], np.exp(params[1]), num_samples)
def bbvi(target_log_density, q_density, q_params, num_samples=100, num_iterations=1000, learning_rate=0.01):
"""
Black box variational reasoning (BBVI) implementation
"""
for _ in range(num_iterations):
# Sampling parameters
samples = sample_from_q(q_params, num_samples)
# Calculate the expected value of the gradient of the sample
grad_expected_log_density = np.mean(target_log_density(samples) * (samples - q_params[0]) / np.exp(q_params[1]))
# Parameter Update
q_params[0] += learning_rate * grad_expected_log_density
q_params[1] += learning_rate * 0.5 * (np.mean(samples**2) / np.exp(q_params[1]) - 1)
return q_params
# Probability density function of the object to approximate the normal distribution
def target_log_density(x):
return np.log(normal_density(x, 5, 2))
# Initial parameters of the variational distribution q
q_params = [0.0, 0.0]
# BBVI Execution
q_params = bbvi(target_log_density, normal_density, q_params)
# Display Results
print("Parameters of the true distribution: mean=5, std=2")
print("Variational distribution q learning results: mean={}, std={}".format(q_params[0], np.exp(q_params[1])))In this example, the normal_density function defines the probability density function of the normal distribution, the sample_from_q function generates samples from the variational distribution q, and the bbvi function implements BBVI.
Black-Box Variational Inference (BBVI) Challenges and Countermeasures
Black-Box Variational Inference (BBVI) is a powerful variational inference method, but it also faces several challenges. The main challenges of BBVI and their countermeasures are described below.
1. convergence to a locally optimal solution:
Challenge: BBVI may converge to a locally optimal solution, and the destination of convergence depends on the shape of the variational distribution and the initial values of the hyperparameters.
Solution: Starting from multiple initial values and searching for different local optimal solutions may find a better solution. It is also important to adjust the hyperparameters of the variational optimizer (e.g., learning rate).
2. sample size and computational cost:
Challenge: Increasing sample size increases computational cost due to the use of Monte Carlo methods. Also, sampling becomes more difficult in high dimensional parameter spaces.
Solution: Introducing faster sampling methods, more efficient computational methods, or using mini-batch gradient methods can be effective. Also, adjusting the sample size to the characteristics of the model can be considered.
3. appropriate choice of variational distribution:
Challenge: It can be difficult to select an appropriate variate distribution, and the shape of the variate distribution can affect the fit to the true posterior distribution.
Solution: It is important to select variational distributions based on domain knowledge and experience. The use of flexible variational distribution families may also be considered.
4. high-dimensional parameter space:
Challenge: High dimensional parameter space can lead to high computational cost and slow convergence.
Solution: Efficient sampling and variational inference methods need to be devised, especially in high-dimensional spaces, and dimensionality reduction of the model and the application of partial variational inference can be considered.
5. application to non-Gaussian distributions:
Challenge: BBVI generally assumes a Gaussian distribution for variational distributions, and it is sometimes difficult to apply BBVI to non-Gaussian distributions.
Solution: It is necessary to devise ways to use variational distributions other than Gaussian, or to improve inference methods in response to changes in variational distributions.
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.
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Reference books for learning the basics and theory
1. “Pattern Recognition and Machine Learning” by Christopher M. Bishop
Summary: The basics of variational Bayes are well explained; although BBVI itself is not listed, it is very important as prerequisite knowledge.
In particular: Chapter 10, “Approximate Inference”
2. “Machine Learning: A Probabilistic Perspective” by Kevin P. Murphy
Summary: Very good book as a base for understanding BBVI.
In particular: chapter 21, “Variational Inference”
3. “Bayesian Reasoning and Machine Learning” by David Barber
Abstract: It is free and can be read online. The book presents a wealth of algorithms for variational methods.
Professional literature and papers directly related to BBVI
4. “Black Box Variational Inference” (Ranganath et al., 2014)
Abstract: The original BBVI paper. A method that uses a score function estimator (REINFORCE) to enable gradient estimation independent of the likelihood function.
Key points: the innovation of using a score function for gradient estimation is described.
5. “Auto-Encoding Variational Bayes” (Kingma and Welling, 2013)
Summary: Different approach from BBVI, but introduces the reparameterization trick and is often used in conjunction with BBVI.
Very important in terms of applications (foundational paper for VAE).
6. “Variational Inference: A Review for Statisticians” (Blei et al., 2017)
Abstract: Comprehensive review of variational inference; very clear on BBVI’s position.
Resources to learn about implementation and applications.
7. “Probabilistic Programming and Bayesian Methods for Hackers” by Cameron Davidson-Pilon
Full-fledged tutorial book available on GitHub, with plenty of examples of implementations in PyMC and elsewhere.
8. “Deep Learning” by Ian Goodfellow, Yoshua Bengio, and Aaron Courville
Chapter 19 deals with variational methods, VAE, and the reparameterization trick, which is useful for learning about techniques to combine with BBVI.
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