liquid_bayes

Liquid Bayes Chain - The Classics Revival

Probabilistic Control of Continuous Dynamics with Bayesian Feedback

Experimental Research Code - Functional but unoptimized, expect rough edges

What Is This?

Liquid Bayes Chain combines liquid neural networks with Bayesian inference to create a system where probabilistic confidence directly modulates continuous dynamics. The network's liquid state evolves based on Bayesian uncertainty, creating adaptive exploration-exploitation behavior.

Core Innovation: Bayesian confidence estimates control liquid time constants and dynamics, creating a feedback loop between probabilistic reasoning and continuous neural evolution.

Architecture Highlights

• Confidence-Modulated Dynamics: Bayesian uncertainty controls liquid evolution speed
• Adaptive Time Constants: Neural dynamics adjust based on confidence levels
• Probabilistic Feedback Loop: Continuous dynamics inform Bayesian updates
• Multi-Step Chain Processing: Sequential confidence-guided evolution steps
• Uncertainty Quantification: Full probabilistic output with confidence measures
• Exploration-Exploitation Balance: High confidence → stability, low confidence → exploration

Quick Start

from liquid_bayes import LiquidBayesChain

# Create liquid-Bayesian system
model = LiquidBayesChain(
    input_dim=32,
    state_dim=64,
    output_dim=10,
    num_chain_steps=4
)

# Process input with uncertainty quantification
input_signal = torch.randn(batch_size, input_dim)
output = model(input_signal, return_chain_states=True)

# Get uncertainty information
uncertainty_info = model.predict_with_uncertainty(input_signal)
print(f"Confidence: {uncertainty_info['confidence'].mean():.3f}")

Current Status

• Working: Liquid dynamics, Bayesian networks, confidence modulation, chain evolution, uncertainty quantification
• Rough Edges: No benchmarking on standard tasks, chain length optimization needed
• Still Missing: Advanced Bayesian structures, variational inference, distributed chain processing
• Performance: Good convergence on toy problems, needs validation on real tasks
• Memory Usage: Moderate, scales with chain length and state dimension
• Speed: Sequential chain processing, parallelization opportunities exist

Mathematical Foundation

The liquid dynamics evolve according to:

dx/dt = -x/τ(confidence) + W_rec·σ(x) + W_in·u + noise(1-confidence)

Bayesian confidence estimation uses:

P(belief|evidence) ∝ P(evidence|belief) × P(belief)
confidence = 1 - H(P(belief|evidence))

Where H is Shannon entropy. High confidence leads to stable dynamics (large τ), while low confidence increases exploration through noise injection and faster adaptation.

The chain processes through multiple steps:

x_{t+1} = LiquidEvolution(x_t, u, confidence_t)
confidence_{t+1} = BayesianUpdate(x_{t+1})

Research Applications

• Adaptive control systems with uncertainty
• Robotics with confidence-aware planning
• Financial modeling with risk adaptation
• Autonomous systems requiring exploration-exploitation
• Scientific computing with adaptive dynamics

Installation

pip install torch numpy scipy
# Download liquid_bayes.py from this repo

The Classics Revival Collection

Liquid Bayes Chain is part of a larger exploration of foundational algorithms enhanced with modern neural techniques:

• Evolutionary Turing Machine
• Hebbian Bloom Filter
• Hopfield Decision Graph
• Liquid Bayes Chain ← You are here
• Liquid State Space Model
• Möbius Markov Chain
• Memory Forest

Citation

@misc{liquidbayes2025,
  title={Liquid Bayes Chain: Probabilistic Control of Continuous Dynamics},
  author={Jae Parker 𓅸 1990two},
  year={2025},
  note={Part of The Classics Revival Collection}
}