BOCD-NIG: Normal-Inverse-Gamma Model

Overview

The BOCD-NIG model is a conjugate Bayesian approach for detecting changepoints in univariate data with unknown mean and variance. It uses the Normal-Inverse-Gamma (NIG) distribution as the conjugate prior for efficient sequential inference.

When to Use BOCD-NIG

Best suited for:

  • Univariate continuous data streams
  • Data with unknown mean and variance
  • Fast real-time processing requirements
  • Applications requiring simple interpretability
  • When data is approximately normally distributed

Advantages:

  • Computationally efficient (conjugate updates)
  • Closed-form posterior updates
  • Low memory footprint
  • Fast execution

Limitations:

  • Assumes univariate normal data
  • Less robust to outliers and multimodal distributions
  • May underperform on non-Gaussian data

Parameters

Initialization 

from pybocd import BOCDNIG

model = BOCDNIG(
    m_0=0.0,           # Prior mean
    kappa_0=1.0,       # Prior precision (inverse variance scale)
    alpha_0=1.0,       # Prior shape for inverse-gamma distribution
    beta_0=1.0,        # Prior rate for inverse-gamma distribution
    l=200.0,           # Expected run length (transition probability)
    threshold=1e-4     # Pruning threshold for negligible weights
)

Parameter Descriptions

Parameter Description
m_0 Prior mean of the normal distribution
kappa_0 Prior precision scaling factor (higher = stronger prior belief)
alpha_0 Shape parameter of the inverse-gamma distribution for variance
beta_0 Rate parameter of the inverse-gamma distribution for variance
l Expected run length; probability of changepoint = 1/l at each time step
threshold Minimum weight threshold for maintaining run-length hypotheses

Parameter Tuning Guidelines

  • m_0: Set based on prior knowledge of the data’s mean. If unknown, use 0 or the first few observations.
  • kappa_0: Higher values indicate stronger confidence in m_0. Use smaller values for more flexibility.
  • alpha_0 and beta_0:44 Set based on prior knowledge of variance. Inverse-gamma distribution’s mean is beta_0 / (alpha_0 - 1) for alpha_0 > 1. Use small values for weak priors.
  • l: Choose based on expected frequency of changepoints. Smaller values make the model more sensitive to changes.
  • threshold: Adjust to balance computational efficiency and accuracy. Lower values retain more hypotheses but increase computation.

Usage Example 

import numpy as np
from pybocd import BOCDNIG

# Generate synthetic data with a changepoint
np.random.seed(42)
data = np.concatenate(
    [
        np.random.normal(0, 1, 100),  # Segment 1: mean=0, std=1
        np.random.normal(3, 1, 100),  # Segment 2: mean=3, std=1
    ]
)

# Initialize model
model = BOCDNIG(m_0=0.0, kappa_0=1.0, alpha_0=1.0, beta_0=1.0, l=200.0, threshold=1e-4)

# Process data
for t, x in enumerate(data):
    model.add_data(x)

    # Get the most probable run length (MAP estimate)
    run_length_dist = model.run_length_dist
    run_length = model.run_length
    max_idx = np.argmax(run_length_dist)
    max_path = run_length[max_idx]
    print(f"Time {t}: MAP = {max_path}")

Tuning the Model

Expected Run Length (l)

A higher l means changepoints are less likely. Choose based on domain knowledge:

  • l = 50: Expect a changepoint every ~50 observations
  • l = 200: Expect a changepoint every ~200 observations
  • l = 1000: Expect a changepoint every ~1000 observations

Pruning Threshold

The threshold parameter removes run-length hypotheses with negligible posterior weight, reducing computation:

model = BOCDNIG(..., threshold=1e-2)  # More aggressive pruning (faster)
model = BOCDNIG(..., threshold=1e-6)  # Less pruning (more accurate)

Prior Parameters

For weak priors (less informative):

BOCDNIG(m_0=0.0, kappa_0=0.1, alpha_0=0.5, beta_0=0.5)

For strong priors (more informative):

BOCDNIG(m_0=5.0, kappa_0=10.0, alpha_0=10.0, beta_0=10.0)

References

For theoretical details, see the original BOCD paper and the pybocd documentation.