Areas of Interest

NOTE: I am not taking on any new students until the 2018--2019 academic year at the earliest.

Using mis-specified models

• Bayesian convergence under mis-specification: specific applications, predictive properties (see below under learning theory)
• Model checking to identify, measure, and if possible fix mis-specification, as in my paper with Gelman
  • Efficient implementation of double bootstrap, cross-validation, or other bias corrections to posterior predictive tests
  • Tests based on de-Baysed prediction intervals (a la Larry, following the "conformal predictors" crowd)
  • Tests that tell us how to change the model, not just that something is wrong (as with CSSR, or causal discovery algorithms)
• Ensemble methods (see below under growing ensemble)
• Bootstrap mis-specification tests for parametric regression based on non-parametric smoothers: computational implementation, power (see relevant chapter in ADAfaEPoV)

Networks: structure

• Cross-validation: theoretical properties of the "latin CV" of Dabbs and Junker
• Bootstrapping networks
  • adapting the "pigeonhole" bootstrap of Owen and Eckles
  • The "empirical graphon" (AG)
  • The snowball bootstrap of Eldardiry and Neville: This seems to over-sample high-degree nodes, but are there functionals it is good for? Are there corrections?
• Smoothing adjacency matrices and graph-sequence limits (BK, LW)
• Relation of network sufficient statistics to projectibility and prediction (AR)
  • Are the only projectible ERGMs dyadic-independence models?
  • Asymptotic distribution of MLE in projectible exponential families --- can the Gartner-Ellis result be extended to get a Gaussian distribution?
  • Algebraic characterization of projectibility in ERGMs
  • Ditto without exponential-family assumption (perhaps assuming completeness of sufficient statistics?)
• Model selection (CM)
  • For block models specifically
  • More generally for network models, especially when graphs are sparse
• Statistical approaches to discovering communities (more generally, blocks)
• Detecting network change (CG, AT, DA, LW)
  • Significance of fluctuations in network summary statistics
  • Testing for differences in higher-level network structure
• Effects of aggregating nodes on network inference (SM)

Networks: dynamics

• Causal inference of contagion/influence on networks
  • Use of community discovery (HW, MK, EM)
  • What is identified by random-neighbor assignment?
  • Bounds/partial identification (AT)
  • What is the "largest" parameter identified in the usual case?
  • Experimental design: when is it better to experiment on nodes vs. edges? (AR, VK)
  • Analogues to "genomic control" to measure typical size of pure-homophily effects?
  • Adaptation of "cryptic relatedness" measures from genetics?
• Use of social networks as sensor networks (DA)
• Distinguishing cultural from biological transmission (DA)
• Distributed learning and problem-solving (HF)
• Effects of network structure on institutional change (HF)

Unidentified models

• Mathematical construction of maximal identified parameter
• Application to social influence
• Application to macroeconomic models
• Connection to "partial identification" in Manski's sense?

Learning theory for stochastic processes

• Consistency and convergence of Bayesian nonparametrics for stochastic processes
  • More comprehensible ("primitive") sufficient conditions for convergence
  • Consistency/convergence of PDFAs with Ptiman-Yor priors
  • Consistency/convergence for infinite dynamic Bayesian networks
  • Proof of convergence in risk under Kullback-Leibler loss
  • Extension to more complicated index sets
  • Large deviations for location of posterior in space of distributions
  • Gaussian process approximations to posterior distribution (from expanding existing LDP?)
• Measuring dependence and effective sample size (DM, MS)
  • Estimating measures of weak dependence other than beta-mixing, along the lines of our estimation of beta-mixing coefficients
  • Purely finite-dimensional bounds on generalization error (as opposed to the current bounds, which invoke functionals of the infinite-dimensional distribution)
• Model complexity for time series (DM, MS)
  • Rademacher complexity of time-series models
    • Estimation of empirical Rademacher complexity
    • Other possible noise-correlation notions of complexity
  • Implicit constraints from stationarity
  • Complexity of general state-space models
  • Complexity of specific restricted forms of state-space model
• Bootstrap-type bounds on forecasting error (DM, RL)
• Validity of cross-validation for mixing processes, e.g., based on Kontorovich-Ramanan concentration of measure results
• Learning theory on mixtures of processes
  • Construction of Rademacher bounds for predictive risk
  • Other bounds on predictive risk
• Learning theory for infinite-memory prediction
  • Characterization of uniform asymptotic-equipartion-style convergence (perhaps VC dimension of the functions X^* -> P(X) ?)
  • Explicit risk bounds for same

Predictive-state reconstruction

• Improved algorithms for time series (KLK, SS)
• Classification of time series (KLK)
• Bootstrap theory for uncertainty estimates (GDM)
• BIC for tuning control settings
• Ensemblification by randomizing over hypothesis tests
• Automatic identification of order parameters: Given complexity field, what function of the immediate state best matches it?

Exponential families of stochastic processes

• How far can they be justified as asymptotic approximations from large deviations principles?
• Projectibility and the algebraic form of sufficient statistics (AR; see under network structure)

Regression

• Lebesgue smoothing (GMG, LW)
• Use of fused lasso to decide how much partial pooling to do in hierarchical models
• Using non-parametric smoothers to test parametric specifications (see above)
• Distribution of typical regression coefficients for random low-dimensional projections of sparse high-dimensional systems (i.e., what is the right null model for linear regression?)

Thomson's "sampling" model of psychological abilities

• Asymptotic probabilistic analysis
• What aspects are identifiable in pre-asymptotic regime? Discrimination from factor models
• Adaptation to discrete-choice models, e.g. ideal point models, NOMINATE (JG)

Large deviations

• LDP for stochastic automata
• LDP for adaptive-population processes
• Exponential-family connections (see under "exponential families" above)

Individual sequence learning

• Growing ensembles
  • Regret and risk under stationary sources (MS)
  • Regret bounds in terms of variation of losses (MS)
  • Tuning of epoch length, fixed share, weight of new model (MS)
  • Practical applications (AZJ, AC)
• When does low regret imply a generalization error bound?
• Working with infinite spaces of models
• Working with limited feedback

Networks of information flow in neural (and other) systems

• Remapping in fMRI (CG, EM)
• E-mail networks

Simulation-based inference

• Consistency conditions for indirect inference (LZ, MS)
• Indirect inference with non-parametric auxiliary models (SH)
• Indirect inference for network models (MF)
• Indirect inference for agent-based models
• Tractability of indirect inference with chaotic dynamics
• "Approximate Bayesian computation" with non-parametric summaries; what advantages, if any, does ABC offer over indirect inference?

Power-law distributions

• Consistency (and rate?) of Clauset's estimator of the tail cut-off
• Semi-parametric estimation, with non-parametric density estimation below threshold and power-law tail; properties
• Practical non-parametric density estimation with heavy tails (extending Markovitch and Krieger)
• Test of Yule-Simon model for citations (AC)
• Exact distributions for fluctuating feedback (NW)

Sufficiency and process characterization

Density estimation on graphical models

Neutral model of inquiry

• Flesh out calculations about life-span distribution of findings
• Compare to data from least-favorite field
• Modifications to basic model, e.g., initial finding inhibits replication but excites testing of related hypotheses