Part II: Identity Thesis

Arousal: Update Rate

Introduction
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Arousal: Update Rate

Arousal measures how rapidly the system is revising its world model. The natural formalization is the KL divergence between successive belief states:

Art=KL(bt+1∣bt)=∑xbt+1(x)log⁡bt+1(x)bt(x)\arousal_t = \KL\left( \belief_{t+1} | \belief_t \right) = \sum_{\mathbf{x}} \belief_{t+1}(\mathbf{x}) \log \frac{\belief_{t+1}(\mathbf{x})}{\belief_t(\mathbf{x})}

In latent-space models, this can be approximated more directly:

Art=∣zt+1−zt∣2orI(ot;zt+1∣zt,at)\arousal_t = | \latent_{t+1} - \latent_t |^2 \quad \text{or} \quad \MI(\obs_t; \latent_{t+1} | \latent_t, \action_t)
Phenomenal Correspondence

High arousal: Large belief updates, far from any attractor, system actively navigating. Low arousal: Near a fixed point, low surprise, system at rest in a basin.