Analytical Derivation Roadmap

This roadmap collects planned analytical derivations that extend the current handwritten notes on Poisson changepoint models, likelihood optimization, and Bayesian approximations.

Logical Extensions

These derivations extend the changepoint model structure while staying close to the current Poisson-emission framework.

• Multi-trial changepoints with shared or hierarchical emissions — Extend the single-timeseries likelihood so repeated trials share changepoint structure, emission parameters, or hierarchical priors.
• Multivariate changepoints with non-covariate Poisson emissions — Replace scalar counts with vectors of counts while keeping the emission model non-covariate and state-specific.
• Multi-trial, multivariate changepoints — Combine repeated trials and multivariate emissions into one likelihood, making explicit which parameters are shared across trials and dimensions.
• Dirichlet-distributed changepoints — Model normalized changepoint spacings with a Dirichlet distribution so changepoints remain ordered while allowing flexible transition timing.
• Dirichlet process distributed changepoints — Move from a fixed number of changepoints to a nonparametric construction that can adapt the number of latent states or transitions.

Pragmatic Extensions

These updates target count data workflows where the basic Poisson changepoint model is useful but too restrictive for real experimental data.

Negative Binomial Emissions

Swap the Poisson probability mass function for a Negative Binomial PMF with an additional per-state dispersion or clustering parameter. This is useful for biological and physical count data where variance often exceeds the mean; the dispersion parameter can absorb overdispersion that a pure Poisson model might otherwise explain by adding spurious changepoints.

Autoregressive or History-Dependent Rates

Modify the instantaneous log-rate so it includes previous observations, such as a term proportional to (y_{t-1}). This lets the model separate true state transitions from short-timescale dynamics like refractory periods, feedback echoes, or adaptation within a state.

Covariate-Modulated Transitions

Express changepoint locations or transition boundaries as functions of external, time-varying design variables. This connects latent state changes to observable events such as stimuli, behavior, or drug delivery, moving the model from unsupervised segmentation toward controlled prediction.

State-Specific Transition Smoothness

Give each sigmoid boundary its own steepness parameter instead of using one global transition smoothness. This allows one transition to be abrupt while another is gradual, avoiding a single compromise smoothness across all state boundaries.

Advanced Frameworks and Approximations

These derivations broaden the mathematical toolkit beyond direct gradient-based maximum likelihood estimation.

Discrete Hidden Markov Models

Replace continuous deterministic sigmoid gates with a discrete stochastic latent state governed by a Markov transition matrix. The central derivation is the Forward-Backward algorithm, which uses dynamic programming to compute exact marginal state probabilities over time.

Expectation-Maximization and Baum-Welch

Replace direct optimization with an iterative expected complete-data likelihood procedure. The E-step computes posterior state probabilities, while the M-step solves emission-rate updates by weighting observations by posterior state responsibilities.

Mean-Field Variational Inference

Replace point estimates and local Laplace approximations with a factorized proxy posterior. The main derivation is the evidence lower bound, followed by coordinate updates that fit the approximate posterior by minimizing divergence from the intractable target posterior.

Polya-Gamma Data Augmentation

Introduce auxiliary Polya-Gamma variables to transform non-conjugate likelihood terms into conditionally Gaussian forms. This can make Gibbs sampling or closed-form Variational Bayes updates available by adding latent variables that simplify the algebra.