BayesianArd / Regressor Layer

Bayesian Automatic Relevance Determination (ARD) Regression. Fits the weights of a regression model, using an ARD prior. The weights of the regression model are assumed to be in Gaussian distributions. Also estimate the parameters lambda (precisions of the distributions of the weights) and alpha (precision of the distribution of the noise). The estimation is done by an iterative procedure (Evidence Maximization).

Mathematical formulation: $p (y ∣ X, w, α) = N (y ∣ Xw, α^{- 1})$ $p (w ∣ λ) = j = 1 \prod D N (w_{j} ∣0, λ_{j}^{- 1})$ where:

y is the target vector
X is the feature matrix
w are the weights
α is observation noise precision
λ are feature-specific precisions

Key characteristics:

Automatic feature selection
Probabilistic predictions
Uncertainty estimation
Sparse solutions

Advantages:

Feature relevance learning
Robust to overfitting
Automatic complexity control
Uncertainty quantification

Common applications:

Feature selection
Scientific modeling
Robust regression
Uncertainty-aware prediction
High-dimensional data

Outputs:

Predicted Table: Results with predictions
Validation Results: Cross-validation metrics
Test Metric: Hold-out performance
Feature Importances: Learned relevances

Table

Predicted Table

Validation Results

Test Metric

Feature Importances

SelectFeatures

[column, ...]

Feature column selection for Bayesian ARD:

Requirements:

Data properties:
- Numeric values
- No missing data
- Finite numbers
- Standardized features
ARD considerations:
- Feature relevance
- Prior knowledge
- Correlation structure
- Sparsity patterns
Preprocessing needs:
- Scaling crucial
- Outlier handling
- Feature engineering
- Quality checks
Bayesian aspects:
- Prior compatibility
- Uncertainty structure
- Model assumptions
- Information content

Note: If empty, uses all numeric columns except target

SelectTarget

column

Target column specification for Bayesian ARD:

Requirements:

Data type:
- Numeric continuous
- No missing values
- Finite numbers
- Real-valued
Statistical properties:
- Gaussian-like preferred
- Noise structure
- Scale considerations
- Outlier patterns
Bayesian modeling:
- Prior compatibility
- Uncertainty levels
- Noise estimation
- Model assumptions
Quality aspects:
- Distribution check
- Range verification
- Transformation needs
- Uncertainty assessment

Note: Must be a single numeric column

Params

oneof

DefaultParams

Default configuration for Bayesian ARD Regression:

Core settings:

Model structure:
- Intercept included
- 300 iterations
- 1e-3 tolerance
Hyperparameters:
- α prior: Gamma(1e-6, 1e-6)
- λ prior: Gamma(1e-6, 1e-6)
- λ threshold: 1e+4
Computation:
- No score monitoring
- Automatic pruning
- Variational updates

Best suited for:

Initial modeling
Feature selection
Uncertainty analysis
Robust predictions

CustomParams

Customizable parameters for Bayesian ARD Regression:

Parameter categories:

Optimization:
- Iteration control
- Convergence criteria
- Computation monitoring
Prior specification:
- Noise precision (α)
- Feature precisions (λ)
- Hyperparameters
Model structure:
- Feature pruning
- Intercept handling
- Relevance determination

Trade-offs:

Computation vs precision
Sparsity vs accuracy
Prior knowledge vs data

MaxIter

u32

300

Maximum iterations for variational optimization:

Convergence process:

Update weights (w)
Update hyperparameters (α, λ)
Check convergence criteria

Typical ranges:

Simple problems: 100-200
Standard: 300
Complex: 500-1000

Impact factors:

Data dimensionality
Prior specifications
Convergence needs
Precision requirements

Tol

f64

0.001

Convergence tolerance threshold:

Stopping criterion: $∣∣ w_{t + 1} - w_{t} ∣∣ < t o l$

Typical values:

Strict: 1e-4 to 1e-5
Standard: 1e-3
Relaxed: 1e-2

Trade-offs:

Precision vs speed
Stability vs convergence
Resource usage

Alpha1

f64

0.000001

Shape parameter for the Gamma distribution prior over the alpha parameter.

Gamma distribution shape: $p (α) = G amma (α_{1}, α_{2})$

Effects:

Controls noise model
Influences uncertainty
Affects model flexibility

Selection guide:

Small: Weak prior (1e-6)
Large: Strong prior (>1)
Based on noise knowledge

Alpha2

f64

0.000001

Inverse scale parameter (rate parameter) for the Gamma distribution prior over the alpha parameter.

Inverse scale parameter in: $p (α) = G amma (α_{1}, α_{2})$

Impact:

Prior variance control
Uncertainty scaling
Model robustness

Typical values:

Uninformative: 1e-6
Informative: Based on data
Match with alpha_1

Lambda1

f64

0.000001

Shape parameter for feature precision priors:

Prior structure: $p (λ_{j}) = G amma (λ_{1}, λ_{2})$

Effects:

Feature relevance prior
Sparsity control
Selection strength

Usage:

Small: Flexible selection
Large: Strong sparsity
Based on feature beliefs

Lambda2

f64

0.000001

Rate parameter for feature precision priors:

Inverse scale in: $p (λ_{j}) = G amma (λ_{1}, λ_{2})$

Controls:

Prior variance
Feature shrinkage
Selection diversity

Selection:

Weak: 1e-6 (default)
Strong: Based on priors
Match with lambda_1

ComputeScore

bool

false

Objective function monitoring: whether to compute the objective function at each step of the model.

When True:

Tracks convergence
Monitors likelihood
Validates updates
Enables diagnostics

Trade-offs:

Additional computation
Better monitoring
Memory usage
Debugging capability

ThresholdLambda

f64

10000

Feature pruning threshold:

Pruning rule: $R e m o v e w_{j} i f λ_{j} > t h res h o l d_l amb d a$

Effects:

Controls sparsity
Improves efficiency
Reduces complexity
Feature selection

Selection guide:

Lower: More features
Higher: Stronger pruning
Default: 1e+4

FitIntercept

bool

true

Intercept calculation control:

Model forms: With intercept: $y = Xw + b + ϵ$ Without intercept: $y = Xw + ϵ$

Effects when True:

Centers predictions
Improves fit
Not subject to ARD
Standard modeling

Effects when False:

Origin-constrained
Full ARD control
Theory-driven
Domain-specific needs

GridSearch

Hyperparameter optimization for Bayesian ARD Regression:

Search space organization:

Optimization parameters:
- Convergence control
- Iteration limits
- Monitoring options
Prior specifications:
- Noise precision (α)
- Feature precisions (λ)
- Pruning thresholds
Model structure:
- Intercept options
- Feature selection
- Complexity control

Best practices:

Log-scale for precisions
Multiple initializations
Prior sensitivity
Convergence verification

MaxIter

[u32, ...]

300

Iteration limit search space:

Search patterns:

Basic range:
- [100, 300, 500]
- Standard problems
Extended search:
- [300, 500, 1000]
- Complex problems
Thorough analysis:
- [200, 400, 600, 800]
- Convergence study

Tol

[f64, ...]

0.001

Tolerance threshold search:

Search ranges:

High precision:
- [1e-5, 1e-4, 1e-3]
- Accurate solutions
Standard range:
- [1e-4, 1e-3, 1e-2]
- Balanced approach
Quick convergence:
- [1e-3, 1e-2]
- Faster solutions

Alpha1

[f64, ...]

0.000001

Noise precision shape parameter search:

Search spaces:

Uninformative:
- [1e-6, 1e-4, 1e-2]
- Weak priors
Informative:
- [0.1, 1.0, 10.0]
- Strong priors
Mixed approach:
- [1e-6, 1e-3, 1.0]
- Range exploration

Alpha2

[f64, ...]

0.000001

Noise precision rate parameter search:

Search strategies:

Match alpha_1:
- Similar values
- Balanced prior
Scale exploration:
- [1e-6, 1e-4, 1e-2]
- Various scales
Prior sensitivity:
- Multiple combinations
- Impact analysis

Lambda1

[f64, ...]

0.000001

Feature precision shape parameter search:

Search patterns:

Weak selection:
- [1e-6, 1e-4, 1e-2]
- Flexible ARD
Strong selection:
- [0.1, 1.0, 10.0]
- Aggressive ARD
Comprehensive:
- Mixed strengths
- Selection study

Lambda2

[f64, ...]

0.000001

Feature precision rate parameter search:

Search spaces:

Match lambda_1:
- Balanced priors
- Standard approach
Scale variation:
- [1e-6, 1e-4, 1e-2]
- Different scales
Selection impact:
- Various combinations
- Feature sensitivity

ComputeScore

[bool, ...]

false

Score computation evaluation:

Options:

No monitoring: [false]
- Faster computation
- Standard training
With monitoring: [true]
- Convergence tracking
- Diagnostic information
Compare both: [true, false]
- Performance impact
- Resource trade-off

ThresholdLambda

[f64, ...]

10000

Feature pruning threshold search:

Search ranges:

Conservative:
- [1e+3, 1e+4, 1e+5]
- Gradual pruning
Aggressive:
- [1e+4, 1e+6, 1e+8]
- Strong sparsity
Comprehensive:
- Multiple scales
- Pruning analysis

FitIntercept

[bool, ...]

true

Intercept inclusion search:

Options:

Standard: [true]
- Default modeling
- Better fit
Zero-intercept: [false]
- Theory-driven
- Full ARD
Compare: [true, false]
- Model evaluation
- Impact analysis

RefitScore

enum

R2score

Regression model evaluation metrics:

Purpose:

Model performance evaluation
Error measurement
Quality assessment
Model comparison

Selection criteria:

Error distribution
Scale sensitivity
Domain requirements
Business objectives

Default ~

Model's native scoring method:

Typically R² score
Model-specific implementation
Standard evaluation
Quick assessment

R2score ~

Coefficient of determination (R²):

Formula: $R^{2} = 1 - \frac{\sum ( y _{i} - y ^ _{i} ) ^{2}}{\sum ( y _{i} - y ˉ ) ^{2}}$

Properties:

Range: (-∞, 1]
1: Perfect prediction
0: Constant model
Negative: Worse than mean

Best for:

General performance
Variance explanation
Model comparison
Standard reporting

ExplainedVariance ~

Explained variance score:

Formula: $1 - \frac{Va r ( y - y ^ )}{Va r ( y )}$

Properties:

Range: (-∞, 1]
Accounts for bias
Variance focus
Similar to R²

Best for:

Variance analysis
Bias assessment
Model stability

MaxError ~

Maximum absolute error:

Formula: $max (∣ y_{i} - \overset{y}{^}_{i} ∣)$

Properties:

Worst case error
Original scale
Sensitive to outliers
Upper error bound

Best for:

Critical applications
Error bounds
Safety margins
Risk assessment

NegMeanAbsoluteError ~

Negative mean absolute error:

Formula: $- \frac{1}{n} \sum ∣ y_{i} - \overset{y}{^}_{i} ∣$

Properties:

Linear error scale
Robust to outliers
Original units
Negated for optimization

Best for:

Robust evaluation
Interpretable errors
Outlier presence

NegMeanSquaredError ~

Negative mean squared error:

Formula: $- \frac{1}{n} \sum (y_{i} - \overset{y}{^}_{i})^{2}$

Properties:

Squared error scale
Outlier sensitive
Squared units
Negated for optimization

Best for:

Standard optimization
Large error penalty
Statistical analysis

NegRootMeanSquaredError ~

Negative root mean squared error:

Formula: $- \frac{1}{n} \sum (y_{i} - \overset{y}{^}_{i})^{2}$

Properties:

Original scale
Outlier sensitive
Interpretable units
Negated for optimization

Best for:

Standard reporting
Interpretable errors
Model comparison

NegMeanSquaredLogError ~

Negative mean squared logarithmic error:

Formula: $- \frac{1}{n} \sum (lo g (1 + y_{i}) - lo g (1 + \overset{y}{^}_{i}))^{2}$

Properties:

Relative error scale
For positive values
Sensitive to ratios
Negated for optimization

Best for:

Exponential growth
Relative differences
Positive predictions

NegMedianAbsoluteError ~

Negative median absolute error:

Formula: $- m e d ian (∣ y_{i} - \overset{y}{^}_{i} ∣)$

Properties:

Highly robust
Original scale
Outlier resistant
Negated for optimization

Best for:

Robust evaluation
Heavy-tailed errors
Outlier presence

NegMeanPoissonDeviance ~

Negative Poisson deviance:

Formula: $- 2 \sum (y_{i} lo g (\frac{y _{i}}{y ^ _{i}}) - (y_{i} - \overset{y}{^}_{i}))$

Properties:

For count data
Non-negative values
Poisson assumption
Negated for optimization

Best for:

Count prediction
Event frequency
Rate modeling

NegMeanGammaDeviance ~

Negative Gamma deviance:

Formula: $- 2 \sum (lo g (\frac{y _{i}}{y ^ _{i}}) + \frac{y _{i} - y ^ _{i}}{y ^ _{i}})$

Properties:

For positive continuous data
Constant CV assumption
Relative errors
Negated for optimization

Best for:

Positive continuous data
Multiplicative errors
Financial modeling

NegMeanAbsolutePercentageError ~

Negative mean absolute percentage error:

Formula: $- \frac{100}{n} \sum ∣ \frac{y _{i} - y ^ _{i}}{y _{i}} ∣$

Properties:

Percentage scale
Scale independent
For non-zero targets
Negated for optimization

Best for:

Relative performance
Scale-free comparison
Business metrics

D2AbsoluteErrorScore ~

D² score with absolute error:

Formula: $1 - \frac{\sum ∣ y _{i} - y ^ _{i} ∣}{\sum ∣ y _{i} - y ˉ ∣}$

Properties:

Range: (-∞, 1]
Robust version of R²
Linear error scale
Outlier resistant

Best for:

Robust evaluation
Non-normal errors
Alternative to R²

D2PinballScore ~

D² score with pinball loss:

Properties:

Quantile focus
Asymmetric errors
Risk assessment
Distribution modeling

Best for:

Quantile regression
Risk analysis
Asymmetric costs
Distribution tails

D2TweedieScore ~

D² score with Tweedie deviance:

Properties:

Compound Poisson-Gamma
Flexible dispersion
Mixed distributions
Insurance modeling

Best for:

Insurance claims
Mixed continuous-discrete
Compound distributions
Specialized modeling

Split

oneof

DefaultSplit

Standard train-test split configuration optimized for general classification tasks.

Configuration:

Test size: 20% (0.2)
Random seed: 98
Shuffling: Enabled
Stratification: Based on target distribution

Advantages:

Preserves class distribution
Provides reliable validation
Suitable for most datasets

Best for:

Medium to large datasets
Independent observations
Initial model evaluation

Splitting uses the ShuffleSplit strategy or StratifiedShuffleSplit strategy depending on the field stratified. Note: If shuffle is false then stratified must be false.

CustomSplit

Configurable train-test split parameters for specialized requirements. Allows fine-tuning of data division strategy for specific use cases or constraints.

Use cases:

Time series data
Grouped observations
Specific train/test ratios
Custom validation schemes

RandomState

u64

Random seed for reproducible splits. Ensures:

Consistent train/test sets
Reproducible experiments
Comparable model evaluations

Same seed guarantees identical splits across runs.

Shuffle

bool

true

Data shuffling before splitting. Effects:

true: Randomizes order, better for i.i.d. data
false: Maintains order, important for time series

When to disable:

Time dependent data
Sequential patterns
Grouped observations

TrainSize

f64

0.8

Proportion of data for training. Considerations:

Larger (e.g., 0.8-0.9): Better model learning
Smaller (e.g., 0.5-0.7): Better validation

Common splits:

0.8: Standard (80/20 split)
0.7: More validation emphasis
0.9: More training emphasis

Stratified

bool

false

Maintain class distribution in splits. Important when:

Classes are imbalanced
Small classes present
Representative splits needed

Requirements:

Classification tasks only
Cannot use with shuffle=false
Sufficient samples per class

Cv

oneof

DefaultCv

Standard cross-validation configuration using stratified 3-fold splitting.

Configuration:

Folds: 3
Method: StratifiedKFold
Stratification: Preserves class proportions

Advantages:

Balanced evaluation
Reasonable computation time
Good for medium-sized datasets

Limitations:

May be insufficient for small datasets
Higher variance than larger fold counts
May miss some data patterns

CustomCv

Configurable stratified k-fold cross-validation for specific validation requirements.

Features:

Adjustable fold count with NFolds determining the number of splits.
Stratified sampling
Preserved class distributions

Use cases:

Small datasets (more folds)
Large datasets (fewer folds)
Detailed model evaluation
Robust performance estimation

NFolds

u32

Number of cross-validation folds. Guidelines:

3-5: Large datasets, faster training
5-10: Standard choice, good balance
10+: Small datasets, thorough evaluation

Trade-offs:

More folds: Better evaluation, slower training
Fewer folds: Faster training, higher variance

Must be at least 2.

KfoldCv

K-fold cross-validation without stratification. Divides data into k consecutive folds for iterative validation.

Process:

Splits data into k equal parts
Each fold serves as validation once
Remaining k-1 folds form training set

Use cases:

Regression problems
Large, balanced datasets
When stratification unnecessary
Continuous target variables

Limitations:

May not preserve class distributions
Less suitable for imbalanced data
Can create biased splits with ordered data

NSplits

u32

Number of folds for cross-validation. Selection guide: Recommended values:

5: Standard choice (default)
3: Large datasets/quick evaluation
10: Thorough evaluation/smaller datasets

Trade-offs:

Higher values: More thorough, computationally expensive
Lower values: Faster, potentially higher variance

Must be at least 2 for valid cross-validation.

RandomState

u64

Random seed for fold generation when shuffling. Important for:

Reproducible results
Consistent fold assignments
Benchmark comparisons
Debugging and validation

Set specific value for reproducibility across runs.

Shuffle

bool

true

Whether to shuffle data before splitting into folds. Effects:

true: Randomized fold composition (recommended)
false: Sequential splitting

Enable when:

Data may have ordering
Better fold independence needed

Disable for:

Time series data
Ordered observations

StratifiedKfoldCv

Stratified K-fold cross-validation maintaining class proportions across folds.

Key features:

Preserves class distribution in each fold
Handles imbalanced datasets
Ensures representative splits

Best for:

Classification problems
Imbalanced class distributions
When class proportions matter

Requirements:

Classification tasks only
Sufficient samples per class
Categorical target variable

NSplits

u32

Number of stratified folds. Guidelines: Typical values:

5: Standard for most cases
3: Quick evaluation/large datasets
10: Detailed evaluation/smaller datasets

Considerations:

Must allow sufficient samples per class per fold
Balance between stability and computation time
Consider smallest class size when choosing

RandomState

u64

Seed for reproducible stratified splits. Ensures:

Consistent fold assignments
Reproducible results
Comparable experiments
Systematic validation

Fixed seed guarantees identical stratified splits.

Shuffle

bool

false

Data shuffling before stratified splitting. Impact:

true: Randomizes while maintaining stratification
false: Maintains data order within strata

Use cases:

true: Independent observations
false: Grouped or sequential data

Class proportions maintained regardless of setting.

ShuffleSplitCv

Random permutation cross-validator with independent sampling.

Characteristics:

Random sampling for each split
Independent train/test sets
More flexible than K-fold
Can have overlapping test sets

Advantages:

Control over test size
Fresh splits each iteration
Good for large datasets

Limitations:

Some samples might never be tested
Others might be tested multiple times
No guarantee of complete coverage

NSplits

u32

Number of random splits to perform. Consider: Common values:

5: Standard evaluation
10: More thorough assessment
3: Quick estimates

Trade-offs:

More splits: Better estimation, longer runtime
Fewer splits: Faster, less stable estimates

Balance between computation and stability.

RandomState

u64

Random seed for reproducible shuffling. Controls:

Split randomization
Sample selection
Result reproducibility

Important for:

Debugging
Comparative studies
Result verification

TestSize

f64

0.2

Proportion of samples for test set. Guidelines: Common ratios:

0.2: Standard (80/20 split)
0.25: More validation emphasis
0.1: More training data

Considerations:

Dataset size
Model complexity
Validation requirements

It must be between 0.0 and 1.0.

StratifiedShuffleSplitCv

Stratified random permutation cross-validator combining shuffle-split with stratification.

Features:

Maintains class proportions
Random sampling within strata
Independent splits
Flexible test size

Ideal for:

Imbalanced datasets
Large-scale problems
When class distributions matter
Flexible validation schemes

NSplits

u32

Number of stratified random splits. Guidelines: Recommended values:

5: Standard evaluation
10: Detailed analysis
3: Quick assessment

Consider:

Sample size per class
Computational resources
Stability requirements

RandomState

u64

Seed for reproducible stratified sampling. Ensures:

Consistent class proportions
Reproducible splits
Comparable experiments

Critical for:

Benchmarking
Research studies
Quality assurance

TestSize

f64

0.2

Fraction of samples for stratified test set. Best practices: Common splits:

0.2: Balanced evaluation
0.3: More thorough testing
0.15: Preserve training size

Consider:

Minority class size
Overall dataset size
Validation objectives

It must be between 0.0 and 1.0.

TimeSeriesSplitCv

Time Series cross-validator. Provides train/test indices to split time series data samples that are observed at fixed time intervals, in train/test sets. It is a variation of k-fold which returns first k folds as train set and the k + 1th fold as test set. Note that unlike standard cross-validation methods, successive training sets are supersets of those that come before them. Also, it adds all surplus data to the first training partition, which is always used to train the model. Key features:

Maintains temporal dependence
Expanding window approach
Forward-chaining splits
No future data leakage

Use cases:

Sequential data
Financial forecasting
Temporal predictions
Time-dependent patterns

Note: Training sets are supersets of previous iterations.

NSplits

u32

Number of temporal splits. Considerations: Typical values:

5: Standard forward chaining
3: Limited historical data
10: Long time series

Impact:

Affects training window growth
Determines validation points
Influences computational load

MaxTrainSize

u64

Maximum size of training set. Should be strictly less than the number of samples. Applications:

0: Use all available past data
>0: Rolling window of fixed size

Use cases:

Limit historical relevance
Control computational cost
Handle concept drift
Memory constraints

TestSize

u64

Number of samples in each test set. When 0:

Auto-calculated as n_samples/(n_splits+1)
Ensures equal-sized test sets

Considerations:

Forecast horizon
Validation requirements
Available future data

Gap

u64

Number of samples to exclude from the end of each train set before the test set.Gap between train and test sets. Uses:

Avoid data leakage
Model forecast lag
Buffer periods

Common scenarios:

0: Continuous prediction
>0: Forward gap for realistic evaluation
Match business forecasting needs