Package 'bnns'

Title: Bayesian Neural Network with 'Stan'
Description: Offers a flexible formula-based interface for building and training Bayesian Neural Networks powered by 'Stan'. The package supports modeling complex relationships while providing rigorous uncertainty quantification via posterior distributions. With features like user chosen priors, clear predictions, and support for regression, binary, and multi-class classification, it is well-suited for applications in clinical trials, finance, and other fields requiring robust Bayesian inference and decision-making. References: Neal(1996) <doi:10.1007/978-1-4612-0745-0>.
Authors: Swarnendu Chatterjee [aut, cre, cph]
Maintainer: Swarnendu Chatterjee <[email protected]>
License: MIT + file LICENSE
Version: 0.1.2.9000
Built: 2025-01-14 06:10:42 UTC
Source: https://github.com/swarnendu-stat/bnns

Help Index

Generic Function for Fitting Bayesian Neural Network Models

Description

This is a generic function for fitting Bayesian Neural Network (BNN) models. It dispatches to methods based on the class of the input data.

Usage

bnns(
  formula,
  data,
  L = 1,
  nodes = rep(2, L),
  act_fn = rep(2, L),
  out_act_fn = 1,
  iter = 1000,
  warmup = 200,
  thin = 1,
  chains = 2,
  cores = 2,
  seed = 123,
  prior_weights = NULL,
  prior_bias = NULL,
  prior_sigma = NULL,
  verbose = FALSE,
  refresh = max(iter/10, 1),
  normalize = TRUE,
  ...
)

Arguments

formula

A symbolic description of the model to be fitted. The formula should specify the response variable and predictors (e.g., y ~ x1 + x2). y must be continuous for regression (out_act_fn = 1), numeric 0/1 for binary classification (out_act_fn = 2), and factor with at least 3 levels for multi-classification (out_act_fn = 3).
data

A data frame containing the variables in the model.
L

An integer specifying the number of hidden layers in the neural network. Default is 1.
nodes

An integer or vector specifying the number of nodes in each hidden layer. If a single value is provided, it is applied to all layers. Default is 16.
act_fn

An integer or vector specifying the activation function(s) for the hidden layers. Options are:

  • 1 for tanh

  • 2 for sigmoid (default)

  • 3 for softplus

  • 4 for ReLU

  • 5 for linear

out_act_fn

An integer specifying the activation function for the output layer. Options are:

  • 1 for linear (default)

  • 2 for sigmoid

  • 3 for softmax

iter

An integer specifying the total number of iterations for the Stan sampler. Default is 1e3.
warmup

An integer specifying the number of warmup iterations for the Stan sampler. Default is 2e2.
thin

An integer specifying the thinning interval for Stan samples. Default is 1.
chains

An integer specifying the number of Markov chains. Default is 2.
cores

An integer specifying the number of CPU cores to use for parallel sampling. Default is 2.
seed

An integer specifying the random seed for reproducibility. Default is 123.
prior_weights

A list specifying the prior distribution for the weights in the neural network. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "normal", "uniform", and "cauchy".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "normal": Provide mean (mean of the distribution) and sd (standard deviation).

    • For "uniform": Provide alpha (lower bound) and beta (upper bound).

    • For "cauchy": Provide mu (location parameter) and sigma (scale parameter).

If prior_weights is NULL, the default prior is a normal(0, 1) distribution. For example:

  • list(dist = "normal", params = list(mean = 0, sd = 1))

  • list(dist = "uniform", params = list(alpha = -1, beta = 1))

  • list(dist = "cauchy", params = list(mu = 0, sigma = 2.5))

prior_bias

A list specifying the prior distribution for the biases in the neural network. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "normal", "uniform", and "cauchy".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "normal": Provide mean (mean of the distribution) and sd (standard deviation).

    • For "uniform": Provide alpha (lower bound) and beta (upper bound).

    • For "cauchy": Provide mu (location parameter) and sigma (scale parameter).

If prior_bias is NULL, the default prior is a normal(0, 1) distribution. For example:

  • list(dist = "normal", params = list(mean = 0, sd = 1))

  • list(dist = "uniform", params = list(alpha = -1, beta = 1))

  • list(dist = "cauchy", params = list(mu = 0, sigma = 2.5))

prior_sigma

A list specifying the prior distribution for the sigma parameter in regression models (out_act_fn = 1). This allows for setting priors on the standard deviation of the residuals. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "half-normal" and "inverse-gamma".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "half-normal": Provide sd (standard deviation of the half-normal distribution).

    • For "inverse-gamma": Provide shape (shape parameter) and scale (scale parameter).

If prior_sigma is NULL, the default prior is a half-normal(0, 1) distribution. For example:

  • list(dist = "half_normal", params = list(mean = 0, sd = 1))

  • list(dist = "inv_gamma", params = list(alpha = 1, beta = 1))

verbose

TRUE or FALSE: flag indicating whether to print intermediate output from Stan on the console, which might be helpful for model debugging.
refresh

refresh (integer) can be used to control how often the progress of the sampling is reported (i.e. show the progress every refresh iterations). By default, refresh = max(iter/10, 1). The progress indicator is turned off if refresh <= 0.
normalize

Logical. If TRUE (default), the input predictors are normalized to have zero mean and unit variance before training. Normalization ensures stable and efficient Bayesian sampling by standardizing the input scale, which is particularly beneficial for neural network training. If FALSE, no normalization is applied, and it is assumed that the input data is already pre-processed appropriately.
...

Currently not in use.

Details

The function serves as a generic interface to different methods of fitting Bayesian Neural Networks. The specific method dispatched depends on the class of the input arguments, allowing for flexibility in the types of inputs supported.

Value

The result of the method dispatched by the class of the input data. Typically, this would be an object of class "bnns" containing the fitted model and associated information.

References

  1. Bishop, C.M., 1995. Neural networks for pattern recognition. Oxford university press.

  2. Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M.A., Guo, J., Li, P. and Riddell, A., 2017. Stan: A probabilistic programming language. Journal of statistical software, 76.

  3. Neal, R.M., 2012. Bayesian learning for neural networks (Vol. 118). Springer Science & Business Media.

See Also

bnns.default

Examples

# Example usage with formula interface:
data <- data.frame(x1 = runif(10), x2 = runif(10), y = rnorm(10))
model <- bnns(y ~ -1 + x1 + x2,
  data = data, L = 1, nodes = 2, act_fn = 1,
  iter = 1e1, warmup = 5, chains = 1
)

# See the documentation for bnns.default for more details on the default implementation.

Bayesian Neural Network Model Using Formula(default) Interface

Description

Fits a Bayesian Neural Network (BNN) model using a formula interface. The function parses the formula and data to create the input feature matrix and target vector, then fits the model using bnns.default.

Usage

## Default S3 method:
bnns(
  formula,
  data,
  L = 1,
  nodes = rep(2, L),
  act_fn = rep(2, L),
  out_act_fn = 1,
  iter = 1000,
  warmup = 200,
  thin = 1,
  chains = 2,
  cores = 2,
  seed = 123,
  prior_weights = NULL,
  prior_bias = NULL,
  prior_sigma = NULL,
  verbose = FALSE,
  refresh = max(iter/10, 1),
  normalize = TRUE,
  ...
)

Arguments

formula

A symbolic description of the model to be fitted. The formula should specify the response variable and predictors (e.g., y ~ x1 + x2).
data

A data frame containing the variables in the model.
L

An integer specifying the number of hidden layers in the neural network. Default is 1.
nodes

An integer or vector specifying the number of nodes in each hidden layer. If a single value is provided, it is applied to all layers. Default is 16.
act_fn

An integer or vector specifying the activation function(s) for the hidden layers. Options are:

  • 1 for tanh

  • 2 for sigmoid (default)

  • 3 for softplus

  • 4 for ReLU

  • 5 for linear

out_act_fn

An integer specifying the activation function for the output layer. Options are:

  • 1 for linear (default)

  • 2 for sigmoid

  • 3 for softmax

iter

An integer specifying the total number of iterations for the Stan sampler. Default is 1e3.
warmup

An integer specifying the number of warmup iterations for the Stan sampler. Default is 2e2.
thin

An integer specifying the thinning interval for Stan samples. Default is 1.
chains

An integer specifying the number of Markov chains. Default is 2.
cores

An integer specifying the number of CPU cores to use for parallel sampling. Default is 2.
seed

An integer specifying the random seed for reproducibility. Default is 123.
prior_weights

A list specifying the prior distribution for the weights in the neural network. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "normal", "uniform", and "cauchy".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "normal": Provide mean (mean of the distribution) and sd (standard deviation).

    • For "uniform": Provide alpha (lower bound) and beta (upper bound).

    • For "cauchy": Provide mu (location parameter) and sigma (scale parameter).

If prior_weights is NULL, the default prior is a normal(0, 1) distribution. For example:

  • list(dist = "normal", params = list(mean = 0, sd = 1))

  • list(dist = "uniform", params = list(alpha = -1, beta = 1))

  • list(dist = "cauchy", params = list(mu = 0, sigma = 2.5))

prior_bias

A list specifying the prior distribution for the biases in the neural network. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "normal", "uniform", and "cauchy".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "normal": Provide mean (mean of the distribution) and sd (standard deviation).

    • For "uniform": Provide alpha (lower bound) and beta (upper bound).

    • For "cauchy": Provide mu (location parameter) and sigma (scale parameter).

If prior_bias is NULL, the default prior is a normal(0, 1) distribution. For example:

  • list(dist = "normal", params = list(mean = 0, sd = 1))

  • list(dist = "uniform", params = list(alpha = -1, beta = 1))

  • list(dist = "cauchy", params = list(mu = 0, sigma = 2.5))

prior_sigma

A list specifying the prior distribution for the sigma parameter in regression models (out_act_fn = 1). This allows for setting priors on the standard deviation of the residuals. The list must include two components:

  • dist: A character string specifying the distribution type. Supported values are "half-normal" and "inverse-gamma".

  • params: A named list specifying the parameters for the chosen distribution:

    • For "half-normal": Provide sd (standard deviation of the half-normal distribution).

    • For "inverse-gamma": Provide shape (shape parameter) and scale (scale parameter).

If prior_sigma is NULL, the default prior is a half-normal(0, 1) distribution. For example:

  • list(dist = "half_normal", params = list(mean = 0, sd = 1))

  • list(dist = "inv_gamma", params = list(alpha = 1, beta = 1))

verbose

TRUE or FALSE: flag indicating whether to print intermediate output from Stan on the console, which might be helpful for model debugging.
refresh

refresh (integer) can be used to control how often the progress of the sampling is reported (i.e. show the progress every refresh iterations). By default, refresh = max(iter/10, 1). The progress indicator is turned off if refresh <= 0.
normalize

Logical. If TRUE (default), the input predictors are normalized to have zero mean and unit variance before training. Normalization ensures stable and efficient Bayesian sampling by standardizing the input scale, which is particularly beneficial for neural network training. If FALSE, no normalization is applied, and it is assumed that the input data is already pre-processed appropriately.
...

Currently not in use.

Details

The function uses the provided formula and data to generate the design matrix for the predictors and the response vector. It then calls helper function bnns_train to fit the Bayesian Neural Network model.

Value

An object of class "bnns" containing the fitted model and associated information, including:

  • fit: The fitted Stan model object.

  • data: A list containing the processed training data.

  • call: The matched function call.

  • formula: The formula used for the model.

Examples

# Example usage:
data <- data.frame(x1 = runif(10), x2 = runif(10), y = rnorm(10))
model <- bnns(y ~ -1 + x1 + x2,
  data = data, L = 1, nodes = 2, act_fn = 3,
  iter = 1e1, warmup = 5, chains = 1
)

Measure Performance for Binary Classification Models

Description

Evaluates the performance of a binary classification model using a confusion matrix and accuracy.

Usage

measure_bin(obs, pred, cut = 0.5)

Arguments

obs

A numeric or integer vector of observed binary class labels (0 or 1).
pred

A numeric vector of predicted probabilities for the positive class.
cut

A numeric threshold (between 0 and 1) to classify predictions into binary labels.

Value

A list containing:

conf_mat

A confusion matrix comparing observed and predicted class labels.

accuracy

The proportion of correct predictions.

ROC

ROC generated using pROC::roc

AUC

Area under the ROC curve.

Examples

obs <- c(1, 0, 1, 1, 0)
pred <- c(0.9, 0.4, 0.8, 0.7, 0.3)
cut <- 0.5
measure_bin(obs, pred, cut)
# Returns: list(conf_mat = <confusion matrix>, accuracy = 1, ROC = <ROC>, AUC = 1)

Measure Performance for Multi-Class Classification Models

Description

Evaluates the performance of a multi-class classification model using log loss and multiclass AUC.

Usage

measure_cat(obs, pred)

Arguments

obs

A factor vector of observed class labels. Each level represents a unique class.
pred

A numeric matrix of predicted probabilities, where each row corresponds to an observation, and each column corresponds to a class. The number of columns must match the number of levels in obs.

Details

The log loss is calculated as:

−1N∑i=1N∑c=1Cyiclog⁡(pic)-\frac{1}{N} \sum_{i=1}^N \sum_{c=1}^C y_{ic} \log(p_{ic})

where yicy_{ic} is 1 if observation ii belongs to class cc, and picp_{ic} is the predicted probability for that class.

The AUC is computed using the pROC::multiclass.roc function, which provides an overall measure of model performance for multiclass classification.

Value

A list containing:

log_loss

The negative log-likelihood averaged across observations.

ROC

ROC generated using pROC::roc

AUC

The multiclass Area Under the Curve (AUC) as computed by pROC::multiclass.roc.

Examples

library(pROC)
obs <- factor(c("A", "B", "C"), levels = LETTERS[1:3])
pred <- matrix(
  c(
    0.8, 0.1, 0.1,
    0.2, 0.6, 0.2,
    0.7, 0.2, 0.1
  ),
  nrow = 3, byrow = TRUE
)
measure_cat(obs, pred)
# Returns: list(log_loss = 1.012185, ROC = <ROC>, AUC = 0.75)

Measure Performance for Continuous Response Models

Description

Evaluates the performance of a continuous response model using RMSE and MAE.

Usage

measure_cont(obs, pred)

Arguments

obs

A numeric vector of observed (true) values.
pred

A numeric vector of predicted values.

Value

A list containing:

rmse

Root Mean Squared Error.

mae

Mean Absolute Error.

Examples

obs <- c(3.2, 4.1, 5.6)
pred <- c(3.0, 4.3, 5.5)
measure_cont(obs, pred)
# Returns: list(rmse = 0.1732051, mae = 0.1666667)

Predict Method for "bnns" Objects

Description

Generates predictions from a fitted Bayesian Neural Network (BNN) model.

Usage

## S3 method for class 'bnns'
predict(object, newdata = NULL, ...)

Arguments

object

An object of class "bnns", typically the result of a call to bnns.default.
newdata

A matrix or data frame of new input data for which predictions are required. If NULL, predictions are made on the training data used to fit the model.
...

Additional arguments (currently not used).

Details

This function uses the posterior distribution from the Stan model in the bnns object to compute predictions for the provided input data.

Value

A matrix/array of predicted values(regression)/probabilities(classification) where first dimension corresponds to the rows of newdata or the training data if newdata is NULL. Second dimension corresponds to the number of posterior samples. In case of out_act_fn = 3, the third dimension corresponds to the class.

See Also

bnns, print.bnns

Examples

# Example usage:
data <- data.frame(x1 = runif(10), x2 = runif(10), y = rnorm(10))
model <- bnns(y ~ -1 + x1 + x2,
  data = data, L = 1, nodes = 2, act_fn = 2,
  iter = 1e1, warmup = 5, chains = 1
)
new_data <- data.frame(x1 = runif(5), x2 = runif(5))
predictions <- predict(model, newdata = new_data)
print(predictions)

Print Method for "bnns" Objects

Description

Displays a summary of a fitted Bayesian Neural Network (BNN) model, including the function call and the Stan fit details.

Usage

## S3 method for class 'bnns'
print(x, ...)

Arguments

x

An object of class "bnns", typically the result of a call to bnns.default.
...

Additional arguments (currently not used).

Value

The function is called for its side effects and does not return a value. It prints the following:

  • The function call used to generate the "bnns" object.

  • A summary of the Stan fit object stored in x$fit.

See Also

bnns, summary.bnns

Examples

# Example usage:
data <- data.frame(x1 = runif(10), x2 = runif(10), y = rnorm(10))
model <- bnns(y ~ -1 + x1 + x2,
  data = data, L = 1, nodes = 2, act_fn = 2,
  iter = 1e1, warmup = 5, chains = 1
)
print(model)

relu transformation

Description

relu transformation

Usage

relu(x)

Arguments

x

A numeric vector or matrix on which relu transformation is going to be applied.

Value

A numeric vector or matrix after relu transformation.

Examples

relu(matrix(1:4, , nrow = 2))

sigmoid transformation

Description

sigmoid transformation

Usage

sigmoid(x)

Arguments

x

A numeric vector or matrix on which sigmoid transformation is going to be applied.

Value

A numeric vector or matrix after sigmoid transformation.

Examples

sigmoid(matrix(1:4, nrow = 2))

Apply Softmax Function to a 3D Array

Description

This function applies the softmax transformation along the third dimension of a 3D array. The softmax function converts raw scores into probabilities such that they sum to 1 for each slice along the third dimension.

Usage

softmax_3d(x)

Arguments

x

A 3D array. The input array on which the softmax function will be applied.

Details

The softmax transformation is computed as:

softmax(xijk)=exp⁡(xijk)∑lexp⁡(xijl)\text{softmax}(x_{ijk}) = \frac{\exp(x_{ijk})}{\sum_{l} \exp(x_{ijl})}

This is applied for each pair of indices ⁠(i, j)⁠ across the third dimension (k).

The function processes the input array slice-by-slice for the first two dimensions ⁠(i, j)⁠, normalizing the values along the third dimension (k) for each slice.

Value

A 3D array of the same dimensions as x, where the values along the third dimension are transformed using the softmax function.

Examples

# Example: Apply softmax to a 3D array
x <- array(runif(24), dim = c(2, 3, 4)) # Random 3D array (2x3x4)
softmax_result <- softmax_3d(x)

softplus transformation

Description

softplus transformation

Usage

softplus(x)

Arguments

x

A numeric vector or matrix on which softplus transformation is going to be applied.

Value

A numeric vector or matrix after softplus transformation.

Examples

softplus(matrix(1:4, nrow = 2))

Summary of a Bayesian Neural Network (BNN) Model

Description

Provides a comprehensive summary of a fitted Bayesian Neural Network (BNN) model, including details about the model call, data, network architecture, posterior distributions, and model fitting information.

Usage

## S3 method for class 'bnns'
summary(object, ...)

Arguments

object

An object of class bnns, representing a fitted Bayesian Neural Network model.
...

Additional arguments (currently unused).

Details

The function prints the following information:

  • Call: The original function call used to fit the model.

  • Data Summary: Number of observations and features in the training data.

  • Network Architecture: Structure of the BNN including the number of hidden layers, nodes per layer, and activation functions.

  • Posterior Summary: Summarized posterior distributions of key parameters (e.g., weights, biases, and noise parameter).

  • Model Fit Information: Bayesian sampling details, including the number of iterations, warmup period, thinning, and chains.

  • Notes: Remarks and warnings, such as checks for convergence diagnostics.

Value

A list (returned invisibly) containing the following elements:

  • "Number of observations": The number of observations in the training data.

  • "Number of features": The number of features in the training data.

  • "Number of hidden layers": The number of hidden layers in the neural network.

  • "Nodes per layer": A comma-separated string representing the number of nodes in each hidden layer.

  • "Activation functions": A comma-separated string representing the activation functions used in each hidden layer.

  • "Output activation function": The activation function used in the output layer.

  • "Stanfit Summary": A summary of the Stan model, including key parameter posterior distributions.

  • "Iterations": The total number of iterations used for sampling in the Bayesian model.

  • "Warmup": The number of iterations used as warmup in the Bayesian model.

  • "Thinning": The thinning interval used in the Bayesian model.

  • "Chains": The number of Markov chains used in the Bayesian model.

  • "Performance": Predictive performance metrics, which vary based on the output activation function.

The function also prints the summary to the console.

See Also

bnns, print.bnns

Examples

# Fit a Bayesian Neural Network
data <- data.frame(x1 = runif(10), x2 = runif(10), y = rnorm(10))
model <- bnns(y ~ -1 + x1 + x2,
  data = data, L = 1, nodes = 2, act_fn = 2,
  iter = 1e1, warmup = 5, chains = 1
)

# Get a summary of the model
summary(model)