Bayesian Inference

The prior encodes existing knowledge or belief about a parameter before observing new data. In power systems, this could represent historical failure rates of transformers or seasonal load profiles.

• Informative priors: Derived from domain expertise or historical data
• Non-informative priors: Designed to minimize influence when little is known
• Conjugate priors: Selected for mathematical convenience, ensuring the posterior belongs to the same distribution family

The prior is multiplied by the likelihood to form the posterior via Bayes' theorem.

The likelihood quantifies how probable the observed data is, given specific parameter values. It acts as the bridge between the prior and the data.

• Represents the data-generating process
• For grid state estimation, this models the probability of sensor readings given a particular voltage magnitude
• The shape of the likelihood determines how strongly the data pulls the posterior away from the prior

A sharply peaked likelihood indicates high information content in the measurements.

The posterior is the updated probability distribution after combining the prior and the likelihood. It represents the complete state of knowledge after seeing the evidence.

• Calculated as: Posterior ∝ Prior × Likelihood
• Provides a full probability density, not just a point estimate
• Enables credible intervals: "There is a 95% probability the parameter lies in this range"

In probabilistic power flow, the posterior can represent updated uncertainty about net load after observing real-time SCADA measurements.

MCMC is a family of algorithms for sampling from complex posterior distributions that cannot be computed analytically. It constructs a Markov chain whose stationary distribution equals the target posterior.

• Metropolis-Hastings: Proposes new samples and accepts or rejects based on a ratio of posterior densities
• Gibbs Sampling: Samples each parameter conditionally on all others, useful for hierarchical models
• Hamiltonian Monte Carlo (HMC): Uses gradient information to propose efficient transitions, reducing autocorrelation

MCMC is essential for high-dimensional grid state estimation where closed-form posteriors are intractable.

A conjugate prior is a prior distribution that, when combined with a specific likelihood, yields a posterior in the same parametric family. This enables closed-form updates without numerical integration.

• Beta-Binomial: Beta prior for a probability parameter with binomial data
• Normal-Normal: Normal prior for a mean with normal likelihood (known variance)
• Gamma-Poisson: Gamma prior for a rate parameter with Poisson count data

In load forecasting, a normal-normal conjugate pair allows sequential Bayesian updating of mean load estimates as new meter readings arrive.

A credible interval is the Bayesian counterpart to the frequentist confidence interval, but with a fundamentally different interpretation.

• Credible interval: "Given the observed data, there is a 95% probability the true parameter lies within this interval"
• Confidence interval: "If we repeated the experiment many times, 95% of the computed intervals would contain the true parameter"
• Highest Posterior Density (HPD): The narrowest interval containing a specified probability mass

This direct probabilistic interpretation makes credible intervals more intuitive for communicating risk to grid operators.

The initial belief about a parameter's value before observing any new data. In grid analysis, this often represents engineering knowledge about typical load behavior or historical failure rates. A strong prior anchors the inference, while a weak prior lets the data dominate.

• Conjugate Priors: Simplify math by ensuring the posterior is the same distribution family.
• Example: A Beta distribution modeling the prior probability of a transformer's insulation degradation.

The probability of observing the actual sensor data given a specific hypothesis about the grid's state. It quantifies how well a proposed model explains the measurements.

• Mechanism: Connects the abstract model to noisy real-world SCADA or PMU data.
• Example: A Gaussian likelihood representing the probability of a voltage measurement error, centered on the true but unknown voltage.

The updated, combined belief about a parameter after weighing the prior against the new evidence. This is the core output of Bayesian Inference, representing the full state of knowledge.

• Computation: Proportional to Prior × Likelihood.
• Utility: Provides not just a point estimate (like max load) but a complete probability density, enabling risk-based grid decisions.

A family of algorithms for sampling from the posterior when it's too complex to solve analytically. MCMC constructs a random walk that spends more time in high-probability regions.

• Metropolis-Hastings: A foundational MCMC method using an acceptance/rejection step.
• Gibbs Sampling: Efficiently samples from conditional distributions.
• Grid Application: Estimating complex joint distributions of line impedances and bus voltages.

An optimization-based alternative to MCMC that approximates the true posterior with a simpler, tractable distribution. VI minimizes the Kullback-Leibler (KL) Divergence between the two.

• Speed: Generally much faster than MCMC for large datasets.
• Trade-off: Provides an approximation, not exact samples.
• Use Case: Real-time distribution system state estimation where computational speed is critical.

A technique that generates predictions by averaging over multiple plausible models, weighted by their posterior probabilities. This inherently accounts for model uncertainty, not just parameter uncertainty.

• Robustness: Prevents overconfidence in a single "best" model.
• Application: Combining forecasts from different renewable generation models (e.g., physical vs. statistical) to produce a more reliable net load prediction.