{
 "artifacts": null,
 "category": "sampling",
 "description": "Use a conditional normalizing flow to replace inner-loop MCMC when sampling states or parameters under progressively tighter neural energy or likelihood constraints. The flow is trained online from recent live sets, and proposals are corrected by importance weighting and resampling, so flow bias does not directly corrupt the nested estimate.",
 "formulas_latex": [
  "$$\\pi_i(x)=\\frac{p_0(x)\\,\\mathbf{1}[\\ell_\\phi(x)\u003e\\lambda_i]}{X_i},\\qquad X_i=\\int p_0(x)\\mathbf{1}[\\ell_\\phi(x)\u003e\\lambda_i]dx.$$",
  "$$w_j=\\frac{p_0(x_j)\\mathbf{1}[\\ell_\\phi(x_j)\u003e\\lambda_i]}{q_\\theta(x_j\\mid\\lambda_i)},\\qquad \\mathrm{ESS}=\\frac{(\\sum_j w_j)^2}{\\sum_j w_j^2}.$$",
  "$$X_i=t_iX_{i-1},\\qquad t_i\\sim\\mathrm{Beta}(N,1),\\qquad \\mathbb{E}[\\log t_i]=-\\frac{1}{N}.$$",
  "$$\\theta\\leftarrow\\theta-\\alpha\\nabla_\\theta\\left[-\\frac{1}{|\\mathcal{W}_i|}\\sum_{x\\in\\mathcal{W}_i}\\log q_\\theta(x\\mid\\lambda(x))\\right],$$"
 ],
 "id": 3133,
 "implementation": "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.",
 "math_summary": "Let $x\\in\\mathbb{R}^d$ be a neural-network state, latent code, parameter vector, or generated sample, and let $\\ell_\\phi(x)$ be a score such as negative energy or log likelihood. At nested level $i$, the constrained prior is $\\pi_i(x)=p_0(x)\\mathbf{1}[\\ell_\\phi(x)\u003e\\lambda_i]/X_i$, where $p_0$ is the base prior, $\\lambda_i$ is the current threshold, and $X_i$ is the remaining prior volume. A conditional invertible flow $q_\\theta(x\\mid\\lambda_i)$ proposes candidates. Their unnormalized importance weights are $w_j=p_0(x_j)\\mathbf{1}[\\ell_\\phi(x_j)\u003e\\lambda_i]/q_\\theta(x_j\\mid\\lambda_i)$. Weighted resampling approximates the constrained target even when the flow is imperfect. With $N$ live points, ideal nested-volume shrinkage satisfies $X_i=t_iX_{i-1}$ with $t_i\\sim\\mathrm{Beta}(N,1)$ and expected log shrinkage $-1/N$. The flow is trained by maximum likelihood on a sliding window of recent live sets.",
 "math_tags": [
  "probability",
  "statistics",
  "optimization",
  "information-theory"
 ],
 "ml_areas": [
  "diffusion-sampling",
  "vae",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.03193",
  "arxiv_url": "https://arxiv.org/abs/2609.03193",
  "summary_what_math_gives_to_ml": "The paper offers a constructive replacement for expensive constrained MCMC in nested sampling: a single normalizing flow conditioned on the current likelihood or energy bound is trained from a sliding window of recent live points, then used for parallel importance-weighted proposals and rejection resampling. Its transferable asset is self-consistent amortized sampling of changing constrained ensembles without requiring a pre-existing training set or structured prior. For neural networks, the strongest applications are accelerating exploration of Bayesian neural-network posteriors, neural energy landscapes, diffusion latents, or world-model state distributions while monitoring flow efficiency as a quantitative diagnostic of multimodality and target drift.",
  "title": "Generative Nested Sampling of Atomistic Thermodynamic Landscapes",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 8
 },
 "solves": [
  "speedup",
  "sample-efficiency",
  "stability"
 ],
 "title": "Conditional-Flow Nested Sampling for Neural Energy Landscapes",
 "url": "https://synthcore.org/idea/3133/conditional-flow-nested-sampling-for-neural-energy-landscape",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}