Overview of Bayesian Deep Learning and Examples of Applications and Implementations

Bayesian Deep Learning

Bayesian deep learning refers to an attempt to incorporate the principles of Bayesian statistics into deep learning. In ordinary deep learning, model parameters are treated as non-probabilistic values and optimization algorithms are used to find the optimal parameters, but Bayesian deep learning treats these parameters themselves as probability distributions and aims to train and infer models by taking uncertainty into account. This is called “Bayesian deep learning”. For more information on the application of uncertainty to machine learning, please refer to “Uncertainty and Machine Learning Techniques” and “Overview of Statistical Learning Theory (Non-Equationary Explanation).

Bayesian deep learning approaches are particularly useful in cases of data uncertainty and limited data, where a Bayesian approach allows models to not only adapt to the data, but also to make predictions and decisions based on the degree of uncertainty and relationships among parameters.

Bayesian deep learning can be realized using methods such as probabilistic programming and stochastic gradient methods, which allow prior and posterior distributions to be defined and combined with data to estimate parameter uncertainties.

However, Bayesian deep learning is computationally very complex and can be computationally expensive compared to conventional optimization algorithms. However, it is considered a worthwhile approach when sufficient computational resources are available or when one wishes to make reliable inferences from limited data.

Algorithms used in Bayesian deep learning

The algorithms used in Bayesian deep learning are based on methods such as probabilistic programming and stochastic gradient methods. Below we describe several algorithms related to Bayesian deep learning.

• Bayesian Neural Networks (BNN): In ordinary neural networks, which are also described in “Implementing Neural Networks and Error Back Propagation Using Clojure” weights and biases are treated as fixed values, but BNNs model these parameters as probability distributions. By using probability distributions, BNNs can model uncertainty based on data, and training of BNNs is done by estimating the posterior distribution by combining the prior distribution and the likelihood function. See also “Overview of Bayesian Neural Networks and Examples of Algorithms and Implementations” for more information on BNNs.
• Stochastic Gradient Method (SGMCMC): The stochastic gradient method is a combination of the stochastic optimization algorithm described in “Overview of Gradient Methods and Examples of Algorithms and Implementations” and the Markov Chain Monte Carlo (MCMC) method described in “Overview of Markov Chain Monte Carlo Methods and Implementations. This method makes it possible to estimate the posterior distribution of a Bayesian deep learning model. Typical methods include SGLD, described in “Stochastic Gradient Langevin Dynamics (SGLD): Overview, Algorithm, and Examples of Implementation,” and SGHMC, described in “Overview, Algorithm, and Examples of Implementation of Stochastic Gradient Hamiltonian Monte Carlo (SGHMC).
• Probabilistic Programming: “Probabilistic programming, also described in “Probabilistic Programming with Clojure (Probabilistic Programming)” uses a high-level programming language for describing probabilistic models to define models and estimate the posterior distribution of the models. The method is then used to estimate the posterior distribution of the model.” Libraries such as Stan and Pyro are used for probabilistic programming, as described in “Analysis of Markov Chain Monte Carlo (MCMC) Models with Stan in R and Clojure.
• Variational Bayesian Methods: Variational Bayesian methods, also described in “Overview and Various Implementations of Variational Bayesian Learning” aim to find an approximate distribution that is close to the true posterior distribution. Algorithms based on variational inference optimize the model aiming to minimize the distance between the variational distribution and the true posterior distribution. Typical methods include Variational Autoencoders (VAE) as described in “Overview and Examples of Conditional Generative Models.

These algorithms are used in various aspects of Bayesian deep learning to help achieve learning and inference that account for data uncertainty and model complexity. However, Bayesian deep learning can be challenging in terms of computational issues and resource requirements, so it is important to select and tune the appropriate algorithm.

Application Examples of Bayesian Deep Learning

The following are examples of applications using Bayesian deep learning.

• Prediction with uncertainty: Bayesian deep learning can be useful for assessing uncertainty in the output of a predictive model. For example, in areas such as medical diagnostics and automated driving, it is important to understand how confident the model is making predictions, and Bayesian deep learning can indicate uncertainty by estimating the prediction distribution.
• Learning models with limited data: When data is limited, regular deep learning models are at increased risk of overlearning. Bayesian deep learning uses prior distributions to control model uncertainty and can perform good inference with limited data while suppressing overlearning.
• Anomaly Detection: Bayesian deep learning may be used to detect deviations from normal data patterns. In anomaly detection, the model learns the distribution of normal data and evaluates how much new data deviates from that distribution.
• Applications in uncertain domains: In modeling physical phenomena, for example in robotics and control systems, the parameters of the model may be uncertain. Bayesian deep learning can be used to estimate model parameter uncertainty and improve control to adapt to real-world conditions.
• Signal Processing: When processing noisy sensor data or signals, Bayesian deep learning can help to isolate signals and remove noise by taking into account data uncertainty.

Example implementation of Bayesian deep learning in python

An example implementation of Bayesian deep learning is shown using a probabilistic programming library called Pyro, which is built on top of PyTorch, a flexible framework for building Bayesian models and performing inference.

The following example uses Pyro to build a Bayesian neural network (BNN) to demonstrate the basic ideas of Bayesian deep learning.

import torch
import pyro
import pyro.distributions as dist
from pyro.nn import PyroModule, PyroSample
from pyro.infer import SVI, Trace_ELBO

# Generate data
torch.manual_seed(42)
num_samples = 100
x = torch.linspace(-5, 5, num_samples).reshape(-1, 1)
y = 3 * x + 2 + 0.5 * torch.randn(num_samples, 1)

# Define Bayesian neural network
class BayesianRegression(PyroModule):
    def __init__(self, input_dim, output_dim):
        super().__init__()
        self.linear = PyroModule[torch.nn.Linear](input_dim, output_dim)
        self.linear.weight = PyroSample(
            dist.Normal(0, 1).expand([output_dim, input_dim]).to_event(2)
        )
        self.linear.bias = PyroSample(dist.Normal(0, 10).expand([output_dim]).to_event(1))

    def forward(self, x, y=None):
        sigma = pyro.sample("sigma", dist.Uniform(0, 10))
        mean = self.linear(x).squeeze(-1)
        with pyro.plate("data", x.shape[0]):
            obs = pyro.sample("obs", dist.Normal(mean, sigma), obs=y)
        return mean

# Build a model
input_dim = 1
output_dim = 1
model = BayesianRegression(input_dim, output_dim)

# Set up SVI to make inferences
guide = pyro.infer.autoguide.AutoDiagonalNormal(model)
optimizer = torch.optim.Adam(guide.parameters(), lr=0.01)
svi = SVI(model, guide, optimizer, loss=Trace_ELBO())

# Perform inference
num_steps = 1000
for step in range(num_steps):
    loss = svi.step(x, y)
    if step % 100 == 0:
        print(f"Step {step}/{num_steps}, Loss = {loss:.2f}")

# Display of inference results
posterior = guide.get_posterior()
samples = posterior.sample(torch.Size([1000]))
predicted_mean = model(x)
confidence_interval = predicted_mean.percentile([2.5, 97.5], dim=0)

import matplotlib.pyplot as plt
plt.figure(figsize=(10, 6))
plt.scatter(x, y, label="True Data")
plt.plot(x, confidence_interval[0], color='r', label='Lower 95% CI')
plt.plot(x, confidence_interval[1], color='g', label='Upper 95% CI')
plt.plot(x, predicted_mean.mean(dim=0), color='b', label='Predicted Mean')
plt.legend()
plt.show()

In this example, Pyro is used to build a Bayesian neural network to display the predictive distribution for the data generated.

Reference Information and Reference Books

For more information on Bayesian deep learning, see “Machine Learning Professional Series: Bayesian Deep Learning Reading Notes. For more information on Bayesian inference and other probabilistic generative models, see “On Probabilistic Generative Models” “Bayesian Inference and Machine Learning with Graphical Models” and “Nonparametric Bayesian and Gaussian Processes. see “Deep Learning.

A good reference book on Bayesian deep learning is “Bayesian Deep Learning”

“Enhancing Deep Learning with Bayesian Inference: Create more powerful, robust deep learning systems with Bayesian deep learning in Python”

“Probabilistic Deep Learning: With Python, Keras and TensorFlow Probability”等を参照のこと。