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skills/scvi-tools/references/theoretical-foundations.md

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+# Theoretical Foundations of scvi-tools
+
+This document explains the mathematical and statistical principles underlying scvi-tools.
+
+## Core Concepts
+
+### Variational Inference
+
+**What is it?**
+Variational inference is a technique for approximating complex probability distributions. In single-cell analysis, we want to understand the posterior distribution p(z|x) - the probability of latent variables z given observed data x.
+
+**Why use it?**
+- Exact inference is computationally intractable for complex models
+- Scales to large datasets (millions of cells)
+- Provides uncertainty quantification
+- Enables Bayesian reasoning about cell states
+
+**How does it work?**
+1. Define a simpler approximate distribution q(z|x) with learnable parameters
+2. Minimize the KL divergence between q(z|x) and true posterior p(z|x)
+3. Equivalent to maximizing the Evidence Lower Bound (ELBO)
+
+**ELBO Objective**:
+```
+ELBO = E_q[log p(x|z)] - KL(q(z|x) || p(z))
+       ↑                    ↑
+  Reconstruction          Regularization
+```
+
+- **Reconstruction term**: Model should generate data similar to observed
+- **Regularization term**: Latent representation should match prior
+
+### Variational Autoencoders (VAEs)
+
+**Architecture**:
+```
+x (observed data)
+    ↓
+[Encoder Neural Network]
+    ↓
+z (latent representation)
+    ↓
+[Decoder Neural Network]
+    ↓
+x̂ (reconstructed data)
+```
+
+**Encoder**: Maps cells (x) to latent space (z)
+- Learns q(z|x), the approximate posterior
+- Parameterized by neural network with learnable weights
+- Outputs mean and variance of latent distribution
+
+**Decoder**: Maps latent space (z) back to gene space
+- Learns p(x|z), the likelihood
+- Generates gene expression from latent representation
+- Models count distributions (Negative Binomial, Zero-Inflated NB)
+
+**Reparameterization Trick**:
+- Allows backpropagation through stochastic sampling
+- Sample z = μ + σ ⊙ ε, where ε ~ N(0,1)
+- Enables end-to-end training with gradient descent
+
+### Amortized Inference
+
+**Concept**: Share encoder parameters across all cells.
+
+**Traditional inference**: Learn separate latent variables for each cell
+- n_cells × n_latent parameters
+- Doesn't scale to large datasets
+
+**Amortized inference**: Learn single encoder for all cells
+- Fixed number of parameters regardless of cell count
+- Enables fast inference on new cells
+- Transfers learned patterns across dataset
+
+**Benefits**:
+- Scalable to millions of cells
+- Fast inference on query data
+- Leverages shared structure across cells
+- Enables few-shot learning
+
+## Statistical Modeling
+
+### Count Data Distributions
+
+Single-cell data are counts (integer-valued), requiring appropriate distributions.
+
+#### Negative Binomial (NB)
+```
+x ~ NB(μ, θ)
+```
+- **μ (mean)**: Expected expression level
+- **θ (dispersion)**: Controls variance
+- **Variance**: Var(x) = μ + μ²/θ
+
+**When to use**: Gene expression without zero-inflation
+- More flexible than Poisson (allows overdispersion)
+- Models technical and biological variation
+
+#### Zero-Inflated Negative Binomial (ZINB)
+```
+x ~ π·δ₀ + (1-π)·NB(μ, θ)
+```
+- **π (dropout rate)**: Probability of technical zero
+- **δ₀**: Point mass at zero
+- **NB(μ, θ)**: Expression when not dropped out
+
+**When to use**: Sparse scRNA-seq data
+- Models technical dropout separately from biological zeros
+- Better fit for highly sparse data (e.g., 10x data)
+
+#### Poisson
+```
+x ~ Poisson(μ)
+```
+- Simplest count distribution
+- Mean equals variance: Var(x) = μ
+
+**When to use**: Less common; ATAC-seq fragment counts
+- More restrictive than NB
+- Faster computation
+
+### Batch Correction Framework
+
+**Problem**: Technical variation confounds biological signal
+- Different sequencing runs, protocols, labs
+- Must remove technical effects while preserving biology
+
+**scvi-tools approach**:
+1. Encode batch as categorical variable s
+2. Include s in generative model
+3. Latent space z is batch-invariant
+4. Decoder conditions on s for batch-specific effects
+
+**Mathematical formulation**:
+```
+Encoder: q(z|x, s)  - batch-aware encoding
+Latent: z           - batch-corrected representation
+Decoder: p(x|z, s)  - batch-specific decoding
+```
+
+**Key insight**: Batch info flows through decoder, not latent space
+- z captures biological variation
+- s explains technical variation
+- Separable biology and batch effects
+
+### Deep Generative Modeling
+
+**Generative model**: Learns p(x), the data distribution
+
+**Process**:
+1. Sample latent variable: z ~ p(z) = N(0, I)
+2. Generate expression: x ~ p(x|z)
+3. Joint distribution: p(x, z) = p(x|z)p(z)
+
+**Benefits**:
+- Generate synthetic cells
+- Impute missing values
+- Quantify uncertainty
+- Perform counterfactual predictions
+
+**Inference network**: Inverts generative process
+- Given x, infer z
+- q(z|x) approximates true posterior p(z|x)
+
+## Model Architecture Details
+
+### scVI Architecture
+
+**Input**: Gene expression counts x ∈ ℕ^G (G genes)
+
+**Encoder**:
+```
+h = ReLU(W₁·x + b₁)
+μ_z = W₂·h + b₂
+log σ²_z = W₃·h + b₃
+z ~ N(μ_z, σ²_z)
+```
+
+**Latent space**: z ∈ ℝ^d (typically d=10-30)
+
+**Decoder**:
+```
+h = ReLU(W₄·z + b₄)
+μ = softmax(W₅·h + b₅) · library_size
+θ = exp(W₆·h + b₆)
+π = sigmoid(W₇·h + b₇)  # for ZINB
+x ~ ZINB(μ, θ, π)
+```
+
+**Loss function (ELBO)**:
+```
+L = E_q[log p(x|z)] - KL(q(z|x) || N(0,I))
+```
+
+### Handling Covariates
+
+**Categorical covariates** (batch, donor, etc.):
+- One-hot encoded: s ∈ {0,1}^K
+- Concatenate with latent: [z, s]
+- Or use conditional layers
+
+**Continuous covariates** (library size, percent_mito):
+- Standardize to zero mean, unit variance
+- Include in encoder and/or decoder
+
+**Covariate injection strategies**:
+- **Concatenation**: [z, s] fed to decoder
+- **Deep injection**: s added at multiple layers
+- **Conditional batch norm**: Batch-specific normalization
+
+## Advanced Theoretical Concepts
+
+### Transfer Learning (scArches)
+
+**Concept**: Use pretrained model as initialization for new data
+
+**Process**:
+1. Train reference model on large dataset
+2. Freeze encoder parameters
+3. Fine-tune decoder on query data
+4. Or fine-tune all with lower learning rate
+
+**Why it works**:
+- Encoder learns general cellular representations
+- Decoder adapts to query-specific characteristics
+- Prevents catastrophic forgetting
+
+**Applications**:
+- Query-to-reference mapping
+- Few-shot learning for rare cell types
+- Rapid analysis of new datasets
+
+### Multi-Resolution Modeling (MrVI)
+
+**Idea**: Separate shared and sample-specific variation
+
+**Latent space decomposition**:
+```
+z = z_shared + z_sample
+```
+- **z_shared**: Common across samples
+- **z_sample**: Sample-specific effects
+
+**Hierarchical structure**:
+```
+Sample level: ρ_s ~ N(0, I)
+Cell level: z_i ~ N(ρ_{s(i)}, σ²)
+```
+
+**Benefits**:
+- Disentangle biological sources of variation
+- Compare samples at different resolutions
+- Identify sample-specific cell states
+
+### Counterfactual Prediction
+
+**Goal**: Predict outcome under different conditions
+
+**Example**: "What would this cell look like if from different batch?"
+
+**Method**:
+1. Encode cell to latent: z = Encoder(x, s_original)
+2. Decode with new condition: x_new = Decoder(z, s_new)
+3. x_new is counterfactual prediction
+
+**Applications**:
+- Batch effect assessment
+- Predicting treatment response
+- In silico perturbation studies
+
+### Posterior Predictive Distribution
+
+**Definition**: Distribution of new data given observed data
+
+```
+p(x_new | x_observed) = ∫ p(x_new|z) q(z|x_observed) dz
+```
+
+**Estimation**: Sample z from q(z|x), generate x_new from p(x_new|z)
+
+**Uses**:
+- Uncertainty quantification
+- Robust predictions
+- Outlier detection
+
+## Differential Expression Framework
+
+### Bayesian Approach
+
+**Traditional methods**: Compare point estimates
+- Wilcoxon, t-test, etc.
+- Ignore uncertainty
+- Require pseudocounts
+
+**scvi-tools approach**: Compare distributions
+- Sample from posterior: μ_A ~ p(μ|x_A), μ_B ~ p(μ|x_B)
+- Compute log fold-change: LFC = log(μ_B) - log(μ_A)
+- Posterior distribution of LFC quantifies uncertainty
+
+### Bayes Factor
+
+**Definition**: Ratio of posterior odds to prior odds
+
+```
+BF = P(H₁|data) / P(H₀|data)
+     ─────────────────────────
+     P(H₁) / P(H₀)
+```
+
+**Interpretation**:
+- BF > 3: Moderate evidence for H₁
+- BF > 10: Strong evidence
+- BF > 100: Decisive evidence
+
+**In scvi-tools**: Used to rank genes by evidence for DE
+
+### False Discovery Proportion (FDP)
+
+**Goal**: Control expected false discovery rate
+
+**Procedure**:
+1. For each gene, compute posterior probability of DE
+2. Rank genes by evidence (Bayes factor)
+3. Select top k genes such that E[FDP] ≤ α
+
+**Advantage over p-values**:
+- Fully Bayesian
+- Natural for posterior inference
+- No arbitrary thresholds
+
+## Implementation Details
+
+### Optimization
+
+**Optimizer**: Adam (adaptive learning rates)
+- Default lr = 0.001
+- Momentum parameters: β₁=0.9, β₂=0.999
+
+**Training loop**:
+1. Sample mini-batch of cells
+2. Compute ELBO loss
+3. Backpropagate gradients
+4. Update parameters with Adam
+5. Repeat until convergence
+
+**Convergence criteria**:
+- ELBO plateaus on validation set
+- Early stopping prevents overfitting
+- Typically 200-500 epochs
+
+### Regularization
+
+**KL annealing**: Gradually increase KL weight
+- Prevents posterior collapse
+- Starts at 0, increases to 1 over epochs
+
+**Dropout**: Random neuron dropping during training
+- Default: 0.1 dropout rate
+- Prevents overfitting
+- Improves generalization
+
+**Weight decay**: L2 regularization on weights
+- Prevents large weights
+- Improves stability
+
+### Scalability
+
+**Mini-batch training**:
+- Process subset of cells per iteration
+- Batch size: 64-256 cells
+- Enables scaling to millions of cells
+
+**Stochastic optimization**:
+- Estimates ELBO on mini-batches
+- Unbiased gradient estimates
+- Converges to optimal solution
+
+**GPU acceleration**:
+- Neural networks naturally parallelize
+- Order of magnitude speedup
+- Essential for large datasets
+
+## Connections to Other Methods
+
+### vs. PCA
+- **PCA**: Linear, deterministic
+- **scVI**: Nonlinear, probabilistic
+- **Advantage**: scVI captures complex structure, handles counts
+
+### vs. t-SNE/UMAP
+- **t-SNE/UMAP**: Visualization-focused
+- **scVI**: Full generative model
+- **Advantage**: scVI enables downstream tasks (DE, imputation)
+
+### vs. Seurat Integration
+- **Seurat**: Anchor-based alignment
+- **scVI**: Probabilistic modeling
+- **Advantage**: scVI provides uncertainty, works for multiple batches
+
+### vs. Harmony
+- **Harmony**: PCA + batch correction
+- **scVI**: VAE-based
+- **Advantage**: scVI handles counts natively, more flexible
+
+## Mathematical Notation
+
+**Common symbols**:
+- x: Observed gene expression (counts)
+- z: Latent representation
+- θ: Model parameters
+- q(z|x): Approximate posterior (encoder)
+- p(x|z): Likelihood (decoder)
+- p(z): Prior on latent variables
+- μ, σ²: Mean and variance
+- π: Dropout probability (ZINB)
+- θ (in NB): Dispersion parameter
+- s: Batch/covariate indicator
+
+## Further Reading
+
+**Key Papers**:
+1. Lopez et al. (2018): "Deep generative modeling for single-cell transcriptomics"
+2. Xu et al. (2021): "Probabilistic harmonization and annotation of single-cell transcriptomics"
+3. Boyeau et al. (2019): "Deep generative models for detecting differential expression in single cells"
+
+**Concepts to explore**:
+- Variational inference in machine learning
+- Bayesian deep learning
+- Information theory (KL divergence, mutual information)
+- Generative models (GANs, normalizing flows, diffusion models)
+- Probabilistic programming (Pyro, PyTorch)
+
+**Mathematical background**:
+- Probability theory and statistics
+- Linear algebra and calculus
+- Optimization theory
+- Information theory