Combined Study Materials Part 1

BIO 229 – Combined Study Materials Created on July 12, 2026 I've compiled the following key topics and notes to support your review. Prompt: In the Raft consensus protocol, how can a leader serve linearizable read-only requests without appending a no-op entry, and what assumptions and mechanisms (e.g., ReadIndex and leader lease) ensure safety under clock drift and potential leader changes? Answer: Raft’s ReadIndex uses a heartbeat round-trip to a majority to confirm the leader’s authority and a commit index fence, after which reads at or beyond that index are linearizable. With the leader-lease optimization, bounded clock skew and continuous majority heartbeats let the leader serve reads without a quorum round-trip, remaining safe because the lease must expire before any new leader can commit. Q: In a difference-in-differences design with staggered treatment adoption and heterogeneous treatment effects across cohorts and time, why can the two-way fixed effects estimator assign negative implicit weights to some cohort-time ATTs, and how do the Callaway–Sant’Anna (2021) or Sun–Abraham (2021) approaches mitigate this issue? Answer: TWFE uses already-treated units as controls, and with heterogeneity the residualization induces negative weights on some cohort-time effects. Callaway–Sant’Anna and Sun–Abraham estimate cohort-time ATTs using neveror not-yet-treated groups and then aggregate with transparent, nonnegative weights. Q: In sparse variational Gaussian Process regression under covariate shift, how do kernel misspecification and inducing-point placement jointly affect posterior uncertainty calibration, and which quantitative diagnostics can reveal the resulting miscalibration? Answer: Kernel misspecification and inducing points concentrated near the training manifold typically yield overconfident predictive variances on shifted inputs. Miscalibration can be diagnosed via proper scoring rules (e.g., negative log predictive density), reliability/calibration curves for predictive intervals, and conditional coverage tests across covariate strata. Q: In staggered-adoption difference-in-differences with dynamic, cohort-specific treatment effects, why do two-way fixed effects (TWFE) regressions yield biased average treatment effect estimates, and how do Sun–Abraham or Callaway–Sant’Anna estimators reweight cohort-time cells to recover interpretable event-study effects? Answer: TWFE aggregates heterogeneous effects using nonconvex (often negative) weights induced by comparisons to already-treated units, causing contamination bias. Cohort-time estimators restrict comparisons to not-yet-treated controls and aggregate with nonnegative, interpretable weights to identify dynamic, cohort-specific ATTs. Question: In Bayesian nonparametrics, how do the base measure and concentration parameter of a Dirichlet process prior jointly influence posterior clustering and the closed-form posterior predictive under a conjugate likelihood, and how does this compare to the Pitman–Yor process in terms of inducing power-law cluster sizes? Explanation: In a Dirichlet process, a larger concentration parameter yields more, smaller clusters while the base measure sets the prior location/shape of cluster atoms; under conjugacy, the posterior predictive follows a Pólya urn form. The Pitman–Yor process adds a discount parameter that produces power-law cluster-size distributions with more small clusters, while retaining a similar predictive scheme. Prompt: In a longitudinal cohort with time-varying confounders affected by prior treatment and informative dropout, how do marginal structural models use stabilized inverse probability weights that combine treatment and censoring models to recover unbiased causal hazard ratios, and under what identification assumptions does this hold? Solution: Stabilized weights are formed as the product of treatment and censoring weights based on the observed history, yielding a pseudo-population in which treatment and dropout are independent of measured time-varying confounders. Consistency of the estimated causal hazard ratios requires correct specification of the weight models, positivity, and no unmeasured confounding. Prompt: In difference-in-differences designs with staggered treatment adoption, why can the two-way fixed effects estimator yield biased treatment effects under heterogeneous treatment dynamics, and which alternative estimators address this problem? Answer: Two-way fixed effects implicitly use already-treated units as controls and assign negative weights to some cohort-time contrasts, so heterogeneous effects over cohorts/time induce bias; alternatives like Callaway–Sant’Anna and Sun–Abraham estimate cohort- and event-time-specific effects and aggregate with proper weights to avoid negative weighting. Q: In a staggered-adoption difference-in-differences design with an absorbing treatment, unit and time fixed effects, and heterogeneous dynamic treatment effects across cohorts and event time, under what conditions does the two-way fixed effects (TWFE) coefficient equal a convex, nonnegative-weighted average of cohort-by-event-time ATT, and what estimator should be used to consistently recover cohort-by-time ATT when those conditions fail? Answer: Only when treatment effects are constant across cohorts and event times (no heterogeneity or dynamics) does TWFE yield a convex-weighted average; otherwise it mixes contaminated comparisons with possibly negative weights. Use cohort- and event-time-specific estimators such as Sun–Abraham or Callaway–Sant’Anna to recover unbiased ATT. Question: In a binary instrument–binary treatment potential outcomes framework, under what precise assumptions does the Wald estimand (E[Y|Z=1]?E[Y|Z=0])/(E[D|Z=1]?E[D|Z=0]) identify the Local Average Treatment Effect (LATE) for compliers, and what empirical pattern might suggest violations of monotonicity? Explanation: It identifies LATE if the instrument is relevant, independent of (Y(0), Y(1), D(0), D(1)), satisfies exclusion (Z affects Y only through D), and monotonicity holds (D(1) ? D(0) almost surely). Monotonicity violations may appear as subgroup-specific first stages with opposite signs or a weak net first stage that masks offsetting complier/defier responses across covariate strata. Q: In Bayesian nonparametrics, how does the stick-breaking representation of a Dirichlet process DP(?, G0) induce the exchangeable partition structure of the Chinese restaurant process, and how does the concentration parameter ? quantitatively affect the expected number of clusters as a function of sample size n? Answer: Stick-breaking yields GEM(?) weights that generate exchangeable partitions with CRP predictive probabilities (join an existing cluster with probability proportional to its size or start a new one with probability proportional to ?). Consequently, the expected number of occupied clusters after n samples is E[K_n] ? ? H_n ? ? log n + O(?), increasing roughly logarithmically in n. Question: In a high-dimensional linear model y = X? + ? with p ? n, a debiased Lasso for ?_j uses a nodewise Lasso to estimate the j-th column of the precision matrix ?; under subGaussian design and errors, restricted eigenvalue conditions, and sparsity s, what rate condition on s, n, and p ensures asymptotic normality, what is the asymptotic variance in terms of ? and ?^2, and how does under-penalizing the nodewise Lasso affect inference validity? Explanation: Asymptotic normality holds if s log p / n ? 0 (often strengthened to s log p / ?n ? 0 for uniform coverage), with asymptotic variance ?^2 ?_jj. Under-penalizing the nodewise Lasso leaves non-negligible bias in the orthogonal score, breaking Neyman orthogonality and yielding anti-conservative inference. Prompt: In a Cox proportional hazards model with left-truncated survival data and a timedependent covariate Z_i(t), how is the partial likelihood constructed to account for delayed entry, and why does the unspecified baseline hazard not affect estimation of ?? Answer: At each event time t, the risk set includes only individuals whose entry time is ? t and who remain at risk, with covariates evaluated at t, so the partial likelihood is formed from event-wise hazard ratios over these dynamic risk sets. The baseline hazard cancels between numerator and denominator in each term, leaving ? estimable from relative risks alone. Q: In a staggered-adoption difference-in-differences design with cohort- and time-varying treatment effects, why can the two-way fixed effects estimator yield a negatively weighted average of group-time effects, and which estimators consistently recover ATT(g,t) under parallel trends? Solution: Because TWFE uses later-treated cohorts as controls for earlier-treated cohorts, heterogeneous effects induce implicit comparisons that assign negative weights to some cohort-time effects, biasing the aggregate; interactionweighted/event-study estimators and group-time ATT frameworks (e.g., Sun & Abraham, 2021; Callaway & Sant’Anna, 2021) consistently estimate ATT(g,t) under parallel trends. Prompt: In a difference-in-differences design with staggered adoption and heterogeneous treatment effects, why can the two-way fixed effects estimator yield negatively weighted averages of group-time ATT, and how do estimators like Callaway–Sant’Anna (2021) or Sun–Abraham (2021) address this? Explanation: TWFE implicitly uses already-treated units as controls for earlier-treated units, so with heterogeneous effects it mixes contaminated comparisons and induces negative implicit weights on some group-time effects. Callaway–Sant’Anna and Sun–Abraham estimate group-time ATT using appropriate never-treated or not-yet-treated controls and then aggregate with nonnegative weights, yielding interpretable averages. Question: In a difference-in-differences design with staggered treatment adoption and heterogeneous, dynamic effects, why can two-way fixed effects OLS produce inconsistent estimates of the average treatment effect, and which estimators recover unbiased group-time ATT under standard parallel trends assumptions? Explanation: TWFE pools comparisons across cohorts and periods with non-convex (often negative) implicit weights that mix already-treated units into the control, biasing estimates when effects vary over time or across groups. Estimators like Sun and Abraham (2021) interaction-weighted event studies and Callaway and Sant’Anna (2021) group-time ATT with not-yet-treated controls recover unbiased effects under parallel trends. Q: In a Bayesian hierarchical normal model with varying intercepts, under what regimes of the group-level variance does a non-centered parameterization improve HMC/NUTS mixing relative to a centered parameterization, and why? Solution: Non-centering improves mixing when the group-level variance is small or weakly identified, because it mitigates the Neal’s funnel geometry that induces strong curvature and step-size pathologies. When the variance is large and well-identified, the centered parameterization typically mixes better as the posterior is closer to Gaussian in centered coordinates. Prompt: In persistent homology, for filtrations induced by tame functions f,g: X -> R on a compact triangulable space X, how does the algebraic stability theorem bound the bottleneck distance d_B(Dgm(f), Dgm(g)) in terms of the L_infinity norm ||f - g||_?, and what structural condition on the persistence modules is required for this bound to hold? Explanation: The algebraic stability theorem states d_B(Dgm(f), Dgm(g)) <= ||f - g||_?, i.e., the persistence diagram map is 1-Lipschitz under L_infinity perturbations. This bound requires the associated persistence modules to be q-tame (finite rank over any interval), ensuring well-defined interleavings and induced matchings. Q: In the Dwork–Lynch–Stockmeyer model of partial synchrony, how does the FLP impossibility result constrain the design of Byzantine fault-tolerant consensus protocols like PBFT, and which minimal timing and quorum assumptions ensure liveness without sacrificing safety under up to f < n/3 Byzantine faults? Answer: FLP rules out deterministic consensus in full asynchrony, so PBFT preserves safety under arbitrary delays but attains liveness only after eventual synchrony—post–Global Stabilization Time with timeouts exceeding message delays. With n ? 3f + 1 replicas and 2f + 1 quorum intersection, view changes converge and progress is guaranteed without violating safety. Question: In a staggered-adoption difference-in-differences design with heterogeneous treatment effects across cohorts and event time, how does the Sun and Abraham (2021) interaction-weighted estimator correct the negative-weighting bias of two-way fixed effects, and what minimal assumptions ensure consistent identification of dynamic effects? Solution: It estimates cohort-by-relative-time event-study coefficients using not-yet-treated and never-treated units as controls, yielding convex combinations of cohort-specific effects rather than implicit negative weights. Identification requires cohort-specific parallel trends, no anticipation (exogeneity of treatment timing), and treatment censoring after adoption. Prompt: In a binary treatment–binary instrument model with treatment effect heterogeneity and two-sided noncompliance, under what conditions does two-stage least squares (2SLS) identify the Local Average Treatment Effect (LATE) for compliers, and how do monotonicity and the exclusion restriction interact to ensure point identification? Explanation: If the instrument is relevant, independent of potential outcomes and treatments, satisfies the exclusion restriction (no direct effect on outcomes), and monotonicity holds (no defiers), then 2SLS identifies the LATE for compliers; monotonicity defines the complier group while the exclusion restriction removes direct channels, jointly delivering point identification of the complier-average causal effect. Q: In a staggered-adoption difference-in-differences design with heterogeneous treatment effects over event time, how can dynamic treatment effects be identified without bias from negative weights, and what estimator structure should be used to obtain consistent group-time ATTs? Solution: Under cohort-specific parallel trends, identify group-time average treatment effects by comparing each treated cohort to never- or not-yet-treated units in the same period. Estimate dynamics using interactionweighted/group-time ATT estimators (e.g., stacked event studies) that aggregate cohort-specific effects with nonnegative weights, avoiding TWFE’s negative-weight bias. Question: In L1-regularized linear regression with standardized predictors and a full-rank design matrix, why is the coefficient path ?(?) piecewise linear in the penalty parameter ?, and how does this enable the LARS algorithm to compute the full solution path efficiently? Explanation: By the KKT conditions, between knots where the active set changes, the equicorrelation set remains fixed and ?(?) evolves affinely in ?, producing linear segments. LARS exploits this by moving along these linear directions and updating only when a variable enters or leaves the active set. Question: In staggered-adoption difference-in-differences with cohort- and time-heterogeneous treatment effects, why can the two-way fixed effects (TWFE) estimator assign negative weights to some 2×2 comparisons, and which estimators mitigate this bias? Solution: Because TWFE uses already-treated units as controls and aggregates all cohort-time contrasts into a single estimate, heterogeneity induces implicit weights that can be negative, biasing the overall effect. Estimators like Sun–Abraham’s interaction-weighted event-study and Callaway–Sant'Anna’s group-time ATT use not-yet-treated or never-treated controls to yield unbiased, interpretable averages. Prompt: In staggered-adoption difference-in-differences with two-way fixed effects (TWFE), how do heterogeneous treatment effects across cohorts and time induce negative weighting and contamination from already-treated units, and which estimators correct these biases while preserving a causal interpretation under appropriate parallel-trends assumptions? Explanation: Heterogeneity makes the TWFE coefficient a non-convex average that assigns negative weights and uses already-treated groups as controls, biasing estimates. Cohort-time estimators such as Callaway-Sant'Anna and SunAbraham compare each cohort only to not-yet-treated (or never-treated) units, restoring causal identification under cohort-specific parallel trends. Question: In a staggered-adoption difference-in-differences design with heterogeneous treatment effects, why can the two-way fixed effects (TWFE) estimator assign negative weights to some cohort-time effects, and how do Sun–Abraham or Callaway–Sant’Anna estimators adjust identification to recover unbiased group-time ATTs? Explanation: TWFE compares already-treated units to not-yet-treated units, inducing non-convex (sometimes negative) weights on cohort-time effects when treatment timing and effects vary, which biases the estimand. Sun–Abraham and Callaway–Sant’Anna re-specify comparisons to use appropriate never- or not-yet-treated controls and cohort-timespecific weighting, isolating uncontaminated group-time ATTs. Q: In a staggered-adoption difference-in-differences design with heterogeneous or timevarying treatment effects, why can the two-way fixed effects estimator be biased, and how does the Goodman–Bacon decomposition identify the source and sign of that bias? Answer: TWFE uses already-treated units as controls for later-treated units, so with heterogeneity it averages 2x2 DiD contrasts with possibly negative weights, distorting both magnitude and sign. The Goodman–Bacon decomposition breaks the estimator into these 2x2 components and their weights, revealing which comparisons (and negative weights) drive the bias. Q: In a difference-in-differences design with staggered treatment adoption and heterogeneous, dynamic treatment effects, how do the Goodman–Bacon decomposition weights bias the twoway fixed effects estimator, and which estimators consistently identify group-time ATT under parallel trends? Explanation: TWFE averages many 2×2 DiD contrasts with potentially negative weights and overuses already-treated groups as controls, yielding biased estimates when effects vary across cohorts or over time; estimators like Sun–Abraham’s interaction-weighted event study and Callaway–Sant’Anna’s group-time ATT recover unbiased effects under parallel trends. Q: In a difference-in-differences design with staggered adoption and heterogeneous treatment effects, why can the two-way fixed effects estimator assign negative weights to some cohorttime contrasts, and how do estimators like Sun–Abraham or Callaway–Sant’Anna avoid this problem? Explanation: With staggered timing, TWFE uses already-treated units as controls for later-treated units, so heterogeneity induces implicit comparison weights that can be negative and bias the aggregate effect. Sun–Abraham and Callaway–Sant’Anna estimate cohort- and event-time–specific effects and then aggregate them with nonnegative, design-based weights to avoid negative weighting. Q: In a staggered-adoption difference-in-differences setting with treatment effect heterogeneity across cohorts and over time, why can a two-way fixed effects (TWFE) estimator place negative implicit weights on some cohort-time average treatment effects, and which estimators avoid this problem while maintaining identification under parallel trends? Answer: TWFE compares earlier- and later-treated units so that already-treated units act as controls, yielding a regression coefficient that is a weighted average of cohort-time ATTs with potentially negative weights when effects vary. Group-time estimators such as Sun and Abraham (2021) and Callaway and Sant’Anna (2021) use never- or notyet-treated controls and cohort-specific parallel trends to produce nonnegative, interpretable aggregations. Question: In a difference-in-differences design with staggered adoption and heterogeneous treatment effects across cohorts and event time, why can the two-way fixed effects (TWFE) estimator deliver a negatively weighted average of cohort-specific ATT, and how do the Callaway–Sant’Anna and Sun–Abraham estimators avoid this problem? Solution: TWFE implicitly uses already-treated units as controls for later-treated units, creating comparison cells whose weights can be negative when effects vary over cohorts/event time; CS and SA estimate ATT by cohort and event time, restrict comparisons to not-yet-treated controls, and then aggregate with nonnegative, design-based weights. Q: In staggered-adoption difference-in-differences designs with cohort- and calendar-time heterogeneous treatment effects, why can the standard two-way fixed effects estimator be biased, and which estimators and diagnostics mitigate this issue? Answer: TWFE mixes comparisons of already-treated to not-yet-treated units and assigns negative or non-convex weights when effects vary across cohorts/time, so its coefficient can be a contaminated average rather than the target ATE. Use cohort-time-specific estimators (Sun–Abraham, Callaway–Sant’Anna, or Borusyak–Jaravel–Spiess imputation) and assess pre-trends and weight decomposition diagnostics. Question: In panel data with staggered treatment adoption and heterogeneous treatment effects, why does the two-way fixed effects (TWFE) difference-in-differences estimator become a non-convex (sometimes negatively weighted) average of cohort-by-time ATT parameters, and how do group-time ATT estimators address this? Solution: TWFE uses already-treated units as controls and imposes a single coefficient, so heterogeneity induces implicit comparisons that subtract some effects, yielding negative weights and a non-convex average. Group-time ATT estimators compare each treated cohort only to not-yet-treated controls and then aggregate with explicit, nonnegative, interpretable weights. Question: In a staggered-adoption difference-in-differences design with heterogeneous and dynamic treatment effects, how does the interaction-weighted (event-study) estimator address the negative-weight and contamination problems of two-way fixed effects, and what cohortspecific parallel-trends and anticipation assumptions are required for identification? Solution: By estimating cohort-by-event-time average treatment effects and aggregating them with nonnegative, interpretable weights that exclude already-treated units from control risk sets, it avoids negative weights and contamination. Identification requires cohort-specific parallel trends in untreated potential outcomes and a limitedanticipation assumption specifying zero pre-treatment effects outside a defined anticipation window. Q: In a difference-in-differences design with staggered treatment adoption and heterogeneous treatment effects across cohorts and time, why can a two-way fixed effects (TWFE) regression produce a negatively weighted average of group-time effects, and which estimators recover unbiased aggregate effects under parallel trends? Answer: Because already-treated units serve as controls for later-treated units, TWFE implicitly compares treated to treated and aggregates group-time ATTs with weights that can be negative when effects vary, biasing the coefficient. Use interaction-weighted/event-study estimators like Sun and Abraham (2021) or group-time ATT aggregation as in Callaway and Sant'Anna (2021) to obtain unbiased aggregates under parallel trends. Question: In a 1+1D conformal field theory with a conserved U(1) current deformed by a marginally irrelevant J·J interaction (Luttinger liquid), how do finite-size corrections to the charged entanglement entropy of a single interval depend on the Luttinger parameter K and the U(1) twist angle ?, and how is this encoded in the scaling of charged Rényi moments on a cylinder? Answer: Charged Rényi moments on the cylinder exhibit a universal quadratic dependence on the twist, log Z_n(?) ? (K/n) ?^2 log[(L/?a) sin(??/L)], set by the U(1) current two-point function. Consequently, the charged entanglement entropy scales linearly with K and shows O(1/L) finite-size corrections fixed by that correlator and the cylinder conformal map. Good luck with your exam; I hope this guide proves useful. --- End of Document 1 ---