☠️🦜 Markov Chain Monty Python Simulation 🦜☠️

This repository contains simplified, reusable, model templates for running discrete-time Markov Chain Monte Carlo (MCMC) simulations in both Python and R.

To make the code readable and self-contained, the simulation models the behavioral states of the dear, departed Mister Polly Parrot of Monty Python fame. It serves as a basic framework for tracking how an individual organism or population transitions through probabilistic life-history stages until hitting an absorbing state (Ex-Parrot).

🦜 The Transition Matrix Mathematics 🦜

The simulation tracks an organism moving through a finite state space $S$:

$$S = {\text{Resting}, \text{Pining}, \text{Stunned}, \text{Ex-Parrot}}$$

The transition probability matrix $P$, where entry $P_{ij}$ represents the probability of transitioning from state $i$ to state $j$, is defined as:

$$ P = \begin{pmatrix} 0.5 & 0.3 & 0.2 & 0.0 \\ 0.4 & 0.4 & 0.2 & 0.0 \\ 0.2 & 0.0 & 0.0 & 0.8 \\ 0.0 & 0.0 & 0.0 & 1.0 \end{pmatrix} $$

Key Concept: Absorbing States

State 4 (Ex-Parrot) is an absorbing state because $P_{4,4} = 1.0$. Once the system transitions into this state, it can never leave. In epidemiology and survival analysis, this directly mirrors irreversible multi-state disease progression or a terminal event (such as mortality, permanent recovery with full immunity, or tracking a vector moving from $S$ -> $I$ -> $R$). It represents the definitive endpoint of a biological cohort.

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🦜 Model Validation & Diagnostics 🦜

Because raw mathematical simulations can hide sampling bugs or structural biases, both the Python and R implementations include interactive diagnostics to validate the model's health before you interpret the biological results.

Convergence Diagnostics

Both scripts are explicitly segmented into Execution Cells (# %%). Running the simulation and plotting the diagnostic cells side-by-side reveals two metrics:

1. The Trace Plot (Sampling Timeline): Tracks the path of the simulation step-by-step.
   • Expected Behavior: A healthy run will show rapid, chaotic "fuzzy caterpillar" noise jumping dynamically between the transient biological states (Resting, Pining, Stunned) before flatlining permanently at the bottom once the parrot joins the choir invisible.
   • Failure Modes: A sharp, clean diagonal cliff or bolt at the very start indicates a Burn-in Failure (the simulation started in an unnaturally rare state and took too long to find the true baseline). A slow, lazy wave indicates Low Effective Sample Size (ESS), meaning the model is too "sticky" and isn't moving freely between states.
2. The Marginal Posterior Distribution (State Frequency): A categorical histogram showing the final, long-term proportion of time spent in each state. This verifies the empirical probability of an organism occupying a specific health bracket (e.g., the exact percentage of the timeline spent "Pining for the Fjords" versus "Resting").

Directory Requirements

To run the diagnostics locally, ensure your folders contain the standard environment configurations:

• /r/: Requires ggplot2 and patchwork installed in your R library.
• /python/: Requires running pip install -r requirements.txt to load numpy, pandas, matplotlib, and seaborn.

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