Notes

The do-calculus uses graphical rules to identify causal effects from observational data when interventions are not available.

Causal trees partition covariate space to estimate conditional average treatment effects (CATE) with honesty to avoid overfitting.

PBFT tolerates arbitrary (including malicious) faults by requiring 3f+1 replicas and 2f+1 quorums with authenticated messages across pre-prepare, prepare, and commit phases.

Under network partitions, a system must choose between immediate consistency and availability; designs position along this trade-off with specific guarantees.

CRDTs converge automatically under concurrent, out-of-order updates by designing operations that commute (or using causal metadata to resolve order).

Paxos gets a cluster to agree on a value despite crashes and message delays by using quorums and epochs (proposal numbers) to ensure safety.

Raft decomposes consensus into leader election, log replication, and safety; its design emphasizes understandability without sacrificing performance.

MLT uses Markov Chain Monte Carlo to explore important light paths by mutating existing paths, dramatically reducing variance for difficult lighting (caustics, small lights).

Optimize an image so its deep features match the content of one image and the style (Gram statistics) of another, producing artworks that blend both.

Perceptual losses compare deep feature activations rather than pixels, aligning optimization with human perception for tasks like style transfer and super-resolution.

Retinex explains how we perceive stable surface colors under changing illumination by comparing reflectance ratios across spatial scales.

The rendering equation expresses outgoing radiance as emitted light plus reflected incident light integrated over directions; path tracing solves it via Monte Carlo integration.

Transformers replace recurrence and convolution with self-attention, letting models focus on the most relevant tokens anywhere in a sequence and enabling parallel training that scales.

Autodiff computes exact derivatives of programs by chaining local derivatives through a computational graph; reverse mode powers deep learning by yielding gradients at constant cost per parameter.

BBVI estimates noisy but unbiased gradients of the ELBO using Monte Carlo and generic score-function estimators, enabling variational inference without model-specific algebra.

Under assumptions (causal sufficiency, faithfulness, acyclicity), algorithms can infer aspects of causal structure from observational data via conditional independencies or scores.

Contextual bandits choose actions based on features; algorithms like LinUCB and Thompson sampling with linear models achieve regret scaling with feature dimension.

Train a network to reverse a gradual noising process; sample by iteratively denoising from pure noise to data.

The Dirichlet process defines a prior over discrete distributions with countably infinite support, enabling models to grow complexity with data (e.g., infinite mixture models).

Double ML uses orthogonal moments and cross-fitting to estimate low-dimensional causal/structural parameters while controlling bias from high-dimensional nuisance components learned by ML.

EM maximizes likelihoods with latent variables by alternating between inferring hidden structure (E-step) and optimizing parameters given that structure (M-step).

GANs pit a generator against a discriminator in a minimax game so the generator learns to produce samples indistinguishable from real data.

Message-passing GNNs are at most as powerful as the 1-WL isomorphism test; architecture choices affect expressivity and oversmoothing.

Flows transform a simple base distribution into a complex target using a sequence of invertible maps; log-likelihoods are exact via the change-of-variables formula.

Probably Approximately Correct (PAC) learning quantifies how many samples suffice to learn a hypothesis with small error and high confidence, governed by capacity measures like VC dimension.

PAC-Bayes bounds provide tight, data-dependent generalization guarantees for randomized predictors via a trade-off between empirical loss and KL divergence to a prior.

Bandit algorithms balance exploration and exploitation to minimize cumulative regret; UCB and Thompson sampling achieve logarithmic regret under standard assumptions.

Learn the score (the gradient of log density) at multiple noise levels and sample with Langevin dynamics or SDE solvers.

SHAP attributes a model’s prediction to features using Shapley values from cooperative game theory, satisfying axioms like efficiency, symmetry, and additivity.

Variational inference turns Bayesian inference into optimization: choose a tractable family qϕ(z|x) and fit it to p(z|x) by maximizing the ELBO (equivalently, minimizing KL(q||p)).

VQ-VAE replaces continuous latents with a learned discrete codebook, improving fidelity and avoiding posterior collapse via commitment and codebook losses.

People perceive attractive products as easier to use; aesthetics can mask usability issues—or buy you goodwill to fix them.

System designs reflect the communication structures of the organizations that build them; change the org if you want to change the architecture.

People tend to prefer things they’ve seen before; repeated, non-annoying exposure increases liking up to saturation.

Teams learn and innovate when members feel safe to take interpersonal risks—asking questions, admitting mistakes, and offering ideas without fear of ridicule.

Beyond a point, more options increase anxiety, delay decisions, and reduce satisfaction; good design curates and guides rather than dumping choices.

With the axiom of choice, a 3D ball can be partitioned into finitely many non-measurable pieces and reassembled into two balls identical to the original.

Any sufficiently strong, consistent formal system cannot prove all true statements about arithmetic (first theorem) and cannot prove its own consistency (second theorem).

HoTT interprets types as spaces and equalities as paths; the univalence axiom identifies equivalent types, enabling structural reasoning and machine-checked proofs.

Is every efficiently checkable problem also efficiently solvable? Most believe P ≠ NP, implying inherent limits on algorithmic efficiency.

Benders splits a hard problem into a master over complicating variables and subproblems over the rest; iteratively add feasibility/optimality cuts from subproblem duals.

Branch-and-bound solves discrete optimization by recursively partitioning the search space and pruning subproblems using bounds from relaxations.

Convex problems have no bad local minima; duality and KKT conditions provide certificates of optimality and efficient algorithms.

Differential privacy (DP) limits how much any single individual can affect an output by adding calibrated noise; guarantees compose gracefully across analyses.

FHE lets you compute on encrypted data without decrypting; the result decrypts to the same output as computing on plaintext, enabling private cloud computation.

Entropy quantifies uncertainty; channel capacity and coding theorems show how reliably we can communicate over noisy channels.

Interior-point methods follow a central path defined by barrier functions to solve large LP/QP/SDP problems efficiently with polynomial-time guarantees.

Optimal transport measures the cost to move one distribution into another; Wasserstein distances provide geometry-aware losses for ML and statistics.

Decoherence explains the emergence of classical behavior: environmental entanglement suppresses interference between pointer states, making superpositions effectively classical mixtures.

MPC lets parties compute a function on their private inputs without revealing them, learning only the output.

Aggregated data can reverse trends present within groups; stratification and causal reasoning resolve the apparent contradiction.

There is no algorithm that decides for every program–input pair whether it halts; undecidability arises from self-reference and diagonalization.

Zero-knowledge proofs let a prover convince a verifier a statement is true without revealing anything beyond its truth.

No local hidden-variable theory can reproduce all quantum predictions: certain correlations violate Bell inequalities, revealing nonlocality or the failure of local realism.

At low temperatures and high magnetic fields, electrons form correlated states with quantized Hall conductance at fractional values and quasiparticles with anyonic statistics.

AdS/CFT equates a gravity theory in (d+1)-dimensional Anti–de Sitter space with a conformal field theory on its d-dimensional boundary; bulk geometry ↔ boundary quantum dynamics.

In disordered interacting systems, eigenstates can fail to thermalize; local memory persists indefinitely, violating the eigenstate thermalization hypothesis.

Yang–Mills theory generalizes electromagnetism to noncommuting (nonabelian) gauge symmetries; its self-interacting gauge fields explain phenomena like asymptotic freedom and confinement in QCD.

Quantum amplitudes arise by summing contributions from all possible paths, each weighted by e^{i S[path]/ħ}; classical physics emerges from stationary action via interference.

Quantum error correction encodes logical qubits into entangled physical qubits so local noise can be detected and corrected without measuring the quantum information itself.

Renormalization group (RG) explains how a system’s description changes as you zoom in or out; fixed points of that zooming flow determine universal behavior like critical exponents near phase transiti

When the lowest-energy state does not share the symmetry of the laws, massless Nambu–Goldstone modes appear; with gauge fields, the Higgs mechanism gives masses to gauge bosons while preserving gauge

Hawking radiation seems thermal and featureless, suggesting information loss; modern holographic ideas indicate unitarity is preserved, with the Page curve recovering late-time purity.

Under no-arbitrage with geometric Brownian motion and continuous hedging, option prices satisfy the Black–Scholes PDE; the delta hedge replicates payoffs and eliminates risk.

The Bass model explains adoption curves via innovators (external influence) and imitators (word-of-mouth), producing an S-shaped cumulative adoption over time.

The Kelly criterion prescribes the fraction of capital to bet that maximizes long-run exponential growth, not short-run win rate.

When a product’s value grows with the number of users, early advantages can snowball; compatibility and standards shape market structure and welfare.

Empirical evidence suggests volatility is “rough” (Hurst H≈0.1), better captured by fractional stochastic processes than classical diffusions, improving fits to implied volatility surfaces.

If the host always opens a goat door and offers a switch, switching doubles your chance of winning: from 1/3 to 2/3.

Platforms must balance cross-side externalities; optimal pricing often subsidizes one side to attract the other, affecting adoption and competition.

Small, structured perturbations can reliably fool neural networks; robustness requires training objectives and architectures aligned with worst-case perturbations.