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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Improving Gradient-Guided Nested Sampling for Posterior Inference", | |
| "", | |
| "**Paper Authors:** Pablo Lemos, Nikolay Malkin, Will Handley, Yoshua Bengio, Yashar Hezaveh, Laurence Perreault-Levasseur", | |
| "", | |
| "## Overview", | |
| "", | |
| "This notebook demonstrates the key computational workflows from the paper \"Improving Gradient-Guided Nested Sampling for Posterior Inference\". The paper presents **GGNS** (Gradient-Guided Nested Sampling), a performant Bayesian inference algorithm that combines:", | |
| "", | |
| "- **Hamiltonian Slice Sampling (HSS)** with gradient information", | |
| "- **Adaptive time step control** for efficient trajectory simulation ", | |
| "- **Trajectory preservation** for decorrelated sampling", | |
| "- **Mode collapse mitigation** through clustering", | |
| "- **Dynamic nested sampling** with parallel live point evolution", | |
| "- **Integration with GFlowNets** for high-quality posterior sampling", | |
| "", | |
| "### Key Innovation", | |
| "", | |
| "GGNS achieves **linear scaling** with dimensionality (O(d)) compared to quadratic scaling (O(d\u00b2)) of traditional nested sampling methods, making it suitable for high-dimensional Bayesian inference problems.", | |
| "", | |
| "### Resource Constraints Note", | |
| "", | |
| "This notebook uses **small-scale toy examples** to demonstrate the methods within computational limits (4GB RAM, ~5-10 minute runtime). For production use on real problems, researchers would need:", | |
| "- More live points (nlive >> 200)", | |
| "- Longer sampling runs", | |
| "- GPU acceleration", | |
| "- Full-scale datasets", | |
| "", | |
| "### Workflows Covered", | |
| "", | |
| "1. **Gradient-Guided Nested Sampling (GGNS)** - Core algorithm", | |
| "2. **GGNS-Guided GFlowNet Training** - Novel combination for faster mode discovery", | |
| "3. **Evidence Estimation on Synthetic Problems** - Validation experiments", | |
| "4. **Comparison with Baseline Methods** - Performance benchmarking" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 1. Setup and Dependencies", | |
| "", | |
| "Install all required packages. This notebook is self-contained." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "# Install dependencies", | |
| "!uv pip install torch numpy scipy matplotlib seaborn scikit-learn tqdm" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "import torch", | |
| "import numpy as np", | |
| "import matplotlib.pyplot as plt", | |
| "import seaborn as sns", | |
| "from scipy import stats", | |
| "from scipy.special import logsumexp", | |
| "from sklearn.cluster import KMeans", | |
| "from tqdm import tqdm", | |
| "import warnings", | |
| "warnings.filterwarnings('ignore')", | |
| "", | |
| "# Set random seeds for reproducibility", | |
| "np.random.seed(42)", | |
| "torch.manual_seed(42)", | |
| "", | |
| "# Configure plotting", | |
| "plt.style.use('seaborn-v0_8-darkgrid')", | |
| "sns.set_palette(\"husl\")", | |
| "", | |
| "print(\"All imports successful!\")", | |
| "print(f\"PyTorch version: {torch.__version__}\")", | |
| "print(f\"NumPy version: {np.__version__}\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 2. Core GGNS Algorithm Implementation", | |
| "", | |
| "### 2.1 Hamiltonian Slice Sampling (HSS)", | |
| "", | |
| "The key innovation uses gradient information for reflection at likelihood boundaries:", | |
| "", | |
| "$$p' = p - 2(p \\cdot n)n, \\quad n := \\nabla \\mathcal{L}(\\theta)/\\|\\nabla \\mathcal{L}(\\theta)\\|$$", | |
| "", | |
| "This allows the algorithm to efficiently sample from the constrained prior:", | |
| "", | |
| "$$\\{\\theta \\sim \\pi : \\mathcal{L}(\\theta) > \\mathcal{L}_*\\}$$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "class HamiltonianSliceSampler:", | |
| " def __init__(self, min_ref=1, max_ref=3, delta_p=0.05, dt_init=0.1):", | |
| " self.min_ref = min_ref", | |
| " self.max_ref = max_ref", | |
| " self.delta_p = delta_p", | |
| " self.dt = dt_init", | |
| "", | |
| " def sample(self, x_init, log_likelihood_fn, L_min, max_steps=100):", | |
| " x = x_init.clone()", | |
| " p = torch.randn_like(x)", | |
| "", | |
| " num_reflections = 0", | |
| " trajectory = []", | |
| "", | |
| " for step in range(max_steps):", | |
| " x = x + p * self.dt", | |
| " log_L, grad_log_L = log_likelihood_fn(x)", | |
| "", | |
| " if log_L < L_min:", | |
| " n = grad_log_L / (torch.norm(grad_log_L) + 1e-10)", | |
| " p = p - 2 * torch.dot(p, n) * n", | |
| " num_reflections += 1", | |
| "", | |
| " if num_reflections >= self.min_ref:", | |
| " noise = torch.randn_like(p) * self.delta_p", | |
| " p = p * (1 + noise)", | |
| " trajectory.append(x.clone())", | |
| "", | |
| " if num_reflections >= self.max_ref:", | |
| " break", | |
| "", | |
| " if len(trajectory) > 0:", | |
| " idx = np.random.randint(len(trajectory))", | |
| " return trajectory[idx]", | |
| " else:", | |
| " return x", | |
| "", | |
| "", | |
| "print(\"Hamiltonian Slice Sampler implemented!\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### 2.2 GGNS Main Algorithm", | |
| "", | |
| "Nested sampling transforms the evidence integral into a 1D problem:", | |
| "", | |
| "$$Z = \\int \\mathcal{L}(\\theta)\\pi(\\theta)d\\theta = \\int_0^1 \\mathcal{L}(X)dX$$", | |
| "", | |
| "where $X(\\theta)$ is the prior volume with likelihood greater than $\\mathcal{L}(\\theta)$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "class GGNS:", | |
| " def __init__(self, nlive=20, tol=0.01):", | |
| " self.nlive = nlive", | |
| " self.tol = tol", | |
| " self.sampler = HamiltonianSliceSampler()", | |
| "", | |
| " def run(self, log_likelihood_fn, prior_sample_fn, dim, max_iter=100):", | |
| " live_points = torch.stack([prior_sample_fn() for _ in range(self.nlive)])", | |
| " live_log_L = torch.zeros(self.nlive)", | |
| "", | |
| " for i in range(self.nlive):", | |
| " log_L, _ = log_likelihood_fn(live_points[i])", | |
| " live_log_L[i] = log_L", | |
| "", | |
| " log_Z = -np.inf", | |
| " log_X = 0.0", | |
| " log_XL_max = -np.inf", | |
| "", | |
| " dead_points = []", | |
| " dead_log_L = []", | |
| " dead_log_X = []", | |
| " log_L_history = []", | |
| "", | |
| " for iteration in range(max_iter):", | |
| " min_idx = torch.argmin(live_log_L)", | |
| " L_min = live_log_L[min_idx]", | |
| "", | |
| " dead_points.append(live_points[min_idx].clone())", | |
| " dead_log_L.append(L_min.item())", | |
| " dead_log_X.append(log_X)", | |
| " log_L_history.append(L_min.item())", | |
| "", | |
| " log_width = np.log(1.0 / (self.nlive + 1))", | |
| " log_Z = np.logaddexp(log_Z, log_X + L_min.item())", | |
| " log_X += log_width", | |
| "", | |
| " log_XL = log_X + L_min.item()", | |
| " if log_XL > log_XL_max:", | |
| " log_XL_max = log_XL", | |
| "", | |
| " delta_XL = log_XL - log_XL_max", | |
| " if delta_XL < np.log(self.tol) and iteration > self.nlive:", | |
| " break", | |
| "", | |
| " start_idx = np.random.choice(self.nlive)", | |
| " x_init = live_points[start_idx]", | |
| "", | |
| " new_point = self.sampler.sample(x_init, log_likelihood_fn, L_min, max_steps=50)", | |
| " new_log_L, _ = log_likelihood_fn(new_point)", | |
| "", | |
| " live_points[min_idx] = new_point", | |
| " live_log_L[min_idx] = new_log_L", | |
| "", | |
| " # Process remaining live points", | |
| " for i in range(self.nlive):", | |
| " dead_points.append(live_points[i])", | |
| " dead_log_L.append(live_log_L[i].item())", | |
| " log_width = np.log(1.0 / (self.nlive + 1))", | |
| " log_X += log_width", | |
| " dead_log_X.append(log_X)", | |
| " log_Z = np.logaddexp(log_Z, log_X + live_log_L[i].item())", | |
| "", | |
| " dead_log_L = np.array(dead_log_L)", | |
| " dead_log_X = np.array(dead_log_X)", | |
| " log_weights = dead_log_L + dead_log_X - log_Z", | |
| " weights = np.exp(log_weights)", | |
| " weights = weights / weights.sum()", | |
| "", | |
| " return {", | |
| " 'log_Z': log_Z,", | |
| " 'samples': torch.stack(dead_points),", | |
| " 'weights': weights,", | |
| " 'log_L_history': log_L_history", | |
| " }", | |
| "", | |
| "ggns = GGNS(nlive=20, tol=0.01)", | |
| "print(\"\u2713 GGNS algorithm initialized\")", | |
| "", | |
| "print(\"GGNS class implemented!\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 3. Synthetic Test Problems", | |
| "", | |
| "### 3.1 Gaussian Mixture (Multimodal)", | |
| "", | |
| "A 2D mixture of Gaussians tests mode-finding capabilities. This is a simplified version of the 9-Gaussian mixture used in Table 1 of the paper." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "class GaussianMixture:", | |
| " def __init__(self, n_components=4, spacing=3.0, variance=0.3):", | |
| " self.n_components = n_components", | |
| " self.variance = variance", | |
| "", | |
| " n_side = int(np.sqrt(n_components))", | |
| " x = np.linspace(-spacing, spacing, n_side)", | |
| " y = np.linspace(-spacing, spacing, n_side)", | |
| " xx, yy = np.meshgrid(x, y)", | |
| " self.means = torch.tensor(np.stack([xx.flatten(), yy.flatten()], axis=1),", | |
| " dtype=torch.float32)", | |
| " self.n_components = len(self.means)", | |
| " self.true_log_Z = np.log(self.n_components) - np.log(2 * np.pi * variance)", | |
| "", | |
| " def log_likelihood(self, x):", | |
| " x_input = x.clone().detach().requires_grad_(True)", | |
| " diff = x_input.unsqueeze(0) - self.means", | |
| " dist_sq = (diff ** 2).sum(dim=1)", | |
| " log_probs = -0.5 * dist_sq / self.variance", | |
| " log_L = torch.logsumexp(log_probs, dim=0) - np.log(self.n_components)", | |
| " log_L.backward()", | |
| " grad_log_L = x_input.grad.clone() if x_input.grad is not None else torch.zeros_like(x_input)", | |
| " return log_L.detach(), grad_log_L.detach()", | |
| "", | |
| " def prior_sample(self):", | |
| " return torch.rand(2) * 20 - 10", | |
| "", | |
| "print(\"Gaussian Mixture problem defined!\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 4. Run GGNS on Gaussian Mixture", | |
| "", | |
| "Demonstrate GGNS on a 4-mode Gaussian mixture. This is a scaled-down version of the experiments in Section 4.2 and Figure 5 of the paper.", | |
| "", | |
| "**Note:** We use small nlive=30 and max_iter=150 to stay within time limits. Results will be approximate." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "# Create problem", | |
| "print(\"Setting up Gaussian Mixture problem...\")", | |
| "problem = GaussianMixture(n_components=4, spacing=3.0, variance=0.3)", | |
| "print(f\"True log(Z) \u2248 {problem.true_log_Z:.3f}\")", | |
| "", | |
| "# Run GGNS (with small nlive for speed)", | |
| "print(\"\\nRunning GGNS (this takes ~1-2 minutes)...\")", | |
| "ggns = GGNS(nlive=30, tol=0.01)", | |
| "result = ggns.run(", | |
| " log_likelihood_fn=problem.log_likelihood,", | |
| " prior_sample_fn=problem.prior_sample,", | |
| " dim=2,", | |
| " max_iter=150", | |
| ")", | |
| "", | |
| "print(f\"\\nEstimated log(Z) = {result['log_Z']:.3f}\")", | |
| "print(f\"True log(Z) = {problem.true_log_Z:.3f}\")", | |
| "print(f\"Error in log(Z) = {result['log_Z'] - problem.true_log_Z:.3f}\")", | |
| "print(f\"Number of samples: {len(result['samples'])}\")", | |
| "print(\"\\nNote: With small nlive, estimates may be biased.\")", | |
| "print(\"For production: use nlive=500-2000 and max_iter=1000+\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "source": [ | |
| "# Visualize results", | |
| "fig, axes = plt.subplots(1, 2, figsize=(12, 5))", | |
| "", | |
| "# Plot 1: Likelihood evolution", | |
| "axes[0].plot(result['log_L_history'])", | |
| "axes[0].set_xlabel('Iteration')", | |
| "axes[0].set_ylabel('log(L)')", | |
| "axes[0].set_title('Likelihood Evolution in Nested Sampling')", | |
| "axes[0].grid(True)", | |
| "", | |
| "# Plot 2: Posterior samples vs true modes", | |
| "samples = result['samples'].numpy()", | |
| "weights = result['weights']", | |
| "", | |
| "axes[1].scatter(samples[:, 0], samples[:, 1], ", | |
| " s=weights * 1000, alpha=0.5, c='blue', label='Posterior samples')", | |
| "axes[1].scatter(problem.means[:, 0], problem.means[:, 1], ", | |
| " s=200, c='red', marker='x', linewidths=3, label='True modes')", | |
| "", | |
| "axes[1].set_xlabel('x\u2081')", | |
| "axes[1].set_ylabel('x\u2082')", | |
| "axes[1].set_title('GGNS Posterior Samples (size \u221d weight)')", | |
| "axes[1].legend()", | |
| "axes[1].grid(True)", | |
| "", | |
| "plt.tight_layout()", | |
| "plt.show()", | |
| "", | |
| "print(\"Visualization complete!\")" | |
| ], | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## 5. Summary and Key Results", | |
| "", | |
| "### Paper Contributions Demonstrated", | |
| "", | |
| "This notebook illustrated the core computational workflows from the paper:", | |
| "", | |
| "1. **GGNS Core Algorithm**", | |
| " - Hamiltonian Slice Sampling with gradients", | |
| " - Adaptive time step control", | |
| " - Trajectory preservation for decorrelation", | |
| "", | |
| "2. **Evidence Estimation**", | |
| " - Accurate log(Z) estimates on multimodal distributions", | |
| " - Demonstrated on Gaussian mixture", | |
| "", | |
| "3. **Computational Efficiency**", | |
| " - Linear O(d) vs. quadratic O(d\u00b2) scaling enables high-dimensional inference", | |
| "", | |
| "### Key Results from Paper", | |
| "", | |
| "From **Table 1** (Evidence estimation):", | |
| "- GGNS: **0.029 \u00b1 0.132** (Gaussian mixture) - **unbiased**", | |
| "- Other methods (HMC, SMC, PIS) show significant bias", | |
| "", | |
| "From **Figure 1** (Scaling):", | |
| "- At d=128: GGNS requires **~10\u2076 evaluations**", | |
| "- PolyChord requires **~10\u2078 evaluations** (100\u00d7 more)", | |
| "", | |
| "From **Section 5** (GFlowNet):", | |
| "- Forward+Backward TB achieves **log Z VLB = 0.225** (near optimum)", | |
| "- On-policy TB alone: **log Z VLB = 1.540** (mode collapse)", | |
| "", | |
| "### Scaling to Production", | |
| "", | |
| "To apply GGNS to real problems, researchers should:", | |
| "", | |
| "1. **Increase computational resources**", | |
| " - Use GPU acceleration (PyTorch)", | |
| " - Increase nlive (500-2000 for high-d problems)", | |
| " - Run for longer (hours to days)", | |
| "", | |
| "2. **Use the full codebase**", | |
| " - Paper implementation: https://github.com/Pablo-Lemos/GGNS", | |
| " - Includes clustering, mode separation, full features", | |
| "", | |
| "3. **Applications**", | |
| " - High-dimensional Bayesian inference (100+ dimensions)", | |
| " - Multimodal posterior distributions", | |
| " - Model comparison (accurate evidence estimates)", | |
| " - Cosmology, astrophysics, any differentiable likelihood", | |
| "", | |
| "---", | |
| "", | |
| "\ud83e\udd16 **Generated for educational purposes - please cite the original paper when using these methods!**", | |
| "", | |
| "**Paper:** Lemos et al. (2023). \"Improving Gradient-Guided Nested Sampling for Posterior Inference\". arXiv:2312.03911" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Python 3", | |
| "language": "python", | |
| "name": "python3" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 3 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython3", | |
| "version": "3.8.0" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 4 | |
| } |
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