- Concept: Bayesian inference updates the probability of a model parameter's value based on observed time series data.
- Process: It combines prior knowledge about the parameter (prior probability) with the likelihood of observing the data given the parameter values to produce an updated probability (posterior probability).
- Bayes' Theorem: The process is governed by the formula \( P(\text{Parameter}|\text{Time Series}) = \frac{P(\text{Time Series}|\text{Parameter}) \times P(\text{Parameter})}{P(\text{Time Series})} \).
- Application: This approach is particularly valuable in time series analysis for incorporating prior knowledge and handling complex, dynamic data.
LANL was my first postdoc after finishing graduate school, and USC was my final postdoc before getting my green card. At USC, I spearheaded a project to develop a Bayesian inference framework for model parameter estimation from time series data. The challenge was to create a model that could accurately predict and analyze complex data patterns. By employing Bayesian methods, I was able to construct a framework that not only provided precise estimations but also accounted for uncertainties in the data.
Bayesian Inference in Time Series Analysis
Markov Chain Monte Carlo (MCMC) in Bayesian Time Series Analysis
- Purpose: MCMC is used to approximate the posterior distribution of a parameter when it's too complex to calculate directly in the context of time series data.
- Method: It generates a series of samples through a Markov chain process, where the distribution of these samples converges to the posterior distribution of the parameter.
- Algorithms: Includes techniques like Metropolis-Hastings and Gibbs sampling, tailored to explore the parameter space efficiently.
- Utility: The samples from MCMC provide a way to estimate and understand the posterior distribution, allowing for predictions and uncertainty quantification in time series models.