Implementeert Underdamped Langevin Dynamics

[hmd101]

Task 1: Core Underdamped Implementation (BAOAB Splitting)

File: src/psyphy/inference/mcmc/underdamped.py

Requirements:

• Implement BAOAB splitting Leimkuhler & Matthews 2013
• Define ULDState NamedTuple
• Support configurable friction γ and temperature T
• Optional: BAB splitting as simpler alternative
• No Metropolis correction by default (biased but faster)

Splitting schemes:

1. BAOAB (default, recommended):

   • B: half-step deterministic force
   • A: half-step position
   • O: exact Ornstein-Uhlenbeck noise
   • A: half-step position
   • B: half-step deterministic force
   • More accurate --> Exact Ornstein-Uhlenbeck sampling (O step)

2. BAB (optional, simpler):

   • B: half-step force + approximate noise
   • A: full-step position
   • B: half-step force
   • Simpler but less accurate

Implementation note:

• Could add as splitting='bab' option
• could add Metropolis-adjusted = False option

Test cases:

• 1D Gaussian posterior: verify mean and variance recovery
• 2D correlated Gaussian: check mixing across correlation

Open questions:

• do we need multiple chain support?