Conditional-Flow Nested Sampling for Neural Energy Landscapes

1. Integration point: wrap an existing neural energy model, diffusion latent model, VAE decoder, or Bayesian neural network inside a nested sampler. Define $\ell_\phi(x)$ as negative energy, log likelihood, or negative validation loss. Initialize $N=256$ to $1024$ live points from $p_0$ and use a conditional RealNVP, masked autoregressive flow, or neural spline flow with the threshold $\lambda$ supplied through FiLM or concatenated conditioning layers. 2. Pseudocode: sample live points $x_k\sim p_0$; identify the worst point and set $\lambda_i=\min_k\ell_\phi(x_k)$; draw $M$ candidates from $q_\theta(\cdot\mid\lambda_i)$; reject candidates below the threshold; compute exact prior and flow log densities; calculate importance weights; resample one replacement proportionally to those weights; append the live set to FIFO window $\mathcal{W}_i$; update the flow by the displayed likelihood loss; repeat. 3. Exact quantities are flow Jacobian densities, neural scores, constraints, and importance weights. Estimate only ESS, acceptance rate, and computational cost empirically. 4. First experiment: compare this method with slice or Metropolis nested sampling on a 16-dimensional eight-well neural energy landscape at equal score evaluations. The prediction is at least a $5\times$ reduction in score evaluations per accepted replacement after warm-up, with evidence error within two independent nested-sampling standard deviations. The measured volume trajectory should satisfy $\mathbb{E}[\log X_i]\approx-i/N$ within 20%; larger deviation indicates proposal failure.