True understanding is demonstrated by the ability to explain complex ideas in simple and accessible terms. I treat teaching as a reciprocal and evolving learning process centred on clarity, simplicity, and active engagement, whose central goal is to build confidence, critical thinking, and strong problem-solving skills rather than memorisation of redundant technical detail.
Two principles shape how I put this into practice. The first is research-led teaching: the questions, examples, and open problems I bring to the classroom are drawn directly from live research, so that students learn not only established results but how new knowledge is actually created, and can see where the field is heading. The second is experiment-based, learning-by-doing teaching: rather than presenting finished theory, I have students explore, form and test hypotheses, run simulations, and discover the underlying principles for themselves, so that intuition comes before formalism.
These two principles directly motivate my Software Demos. I turn each of my research papers, across mathematical statistics and AI for science, into a self-contained, interactive, in-browser demonstration, so that even a first-year student can grasp the core idea and intuition behind a state-of-the-art method (mixture-of-experts model selection, optimal-transport aggregation, likelihood-free inference, efficient transformers) by clicking, dragging, and watching the method run, well before meeting its full mathematical machinery. Making the frontier of research understandable to a beginner is, for me, the clearest test of both good teaching and genuine understanding.
A teaching portfolio should persuade with evidence and artefacts, not adjectives. Mine is built so you can operate the pedagogy yourself: the two principles below are not claims, they are wired into a catalogue of interactive demonstrations, a course designed by backward design, and a fully worked module. My unfair advantage is simple, most people teaching mathematical statistics or AI for science can only describe their method, I can show it running in your browser.
Research-led. The questions, examples, and open problems I bring to class are drawn from live research in mathematical statistics and AI for science, so students see not only established results but how new knowledge is made.
Experiment-based, learning by doing. Rather than presenting finished theory, I have students explore, form and test hypotheses, run simulations, and discover the underlying principle for themselves, so intuition arrives before formalism.
The lesson arc. Every concept follows the same path: an intuition demo, then the derivation, then a notebook, then an authentic assessment. The demo on the right is one such intuition step, a peer-reviewed result a first-year can manipulate by dragging.
A flagship course, Mathematical Statistics for AI in Science, designed backward from learning outcomes. Every module maps one outcome to an intuition demo and to an authentic assessment, so the alignment between what students should be able to do, what they practise, and how they are graded is explicit.
| Module | Learning outcome (students can ...) | Intuition demo | Authentic assessment |
|---|---|---|---|
| Inference foundations | state a likelihood and a prior, and read a posterior credible interval | Parameter balancing | derive and compute a posterior for a small model |
| Mixture models | fit a mixture and interpret its latent components | Dendrogram merging | cluster a dataset and justify the component count |
| Model selection | choose model order with BIC and the slope heuristic, and explain why it is consistent | BIC order selection, slope heuristic | select the order on real data and defend it |
| Mixtures of experts | explain why MoE are universal approximators and how fast they converge | Universal approximation, convergence rates | extend a demo to a new target function |
| Bayesian computation | run approximate Bayesian computation when the likelihood is intractable | GLLiM-ABC, ABC concentration | apply ABC to a new simulator |
| Calibration and UQ | test whether a 95% interval covers 95%, and repair it with random effects | Coverage (Figure 5) | run a leave-one-source-out coverage study |
| Deep learning for science | read a modern architecture (routing, token merging, vision-language) as a statistical model | PiToMe, CompeteSMoE, ExGra-Med | reproduce an efficiency and accuracy trade-off |
| Capstone (research-led) | reproduce a result from a live paper and explain it to a first-year | any of the 22 demos | reproduce a figure, plus a three-minute explainer |
The question "does a 95% interval actually cover 95%?" taught end to end. This is the lesson arc at the resolution of a single class, and it uses the real leave-one-source-out coverage result (Figure 5) from my parameter-balancing paper.
Students open the coverage demo: a fixed-effects Gaussian's 95% interval covers only about 69% of held-out measurements once data sources disagree. The gap is visible, not asserted.
Open the coverage demoWe derive the equal-tailed predictive interval and show that ignoring the between-source variance τ² makes it too tight. Leave-one-source-out cross-validation turns that omission into a measurable miscoverage.
On a small multi-source dataset, students fit fixed effects, then add source-level random effects, and watch empirical coverage move from about 0.69 back toward the nominal 0.95, family by family.
They reproduce the paper's Figure 5 coverage curves and write a short argument for when a fixed-effects interval can, and cannot, be trusted. Research-led assessment, not recall.
The demonstrations above are themselves the primary artefact of teaching effectiveness: each one was built to make a frontier result understandable to a first-year, which is, for me, the clearest test of both good teaching and genuine understanding. Full course history is in Teaching experience below. Detailed student evaluation summaries and peer-teaching feedback are available on request.
Every one of my Software Demos is built to be taught with. Each is a standalone, self-contained web page, so you can open it live in a lecture, drop its QR code on a slide for students to run on their phones, embed it in a slide deck or LMS (add ?embed=1 to the URL), or launch Presenter mode (fullscreen, then step through the built-in experiments with the arrow keys). Each carries a plain-language hook, a difficulty level, a Take the tour guided walkthrough, preset scenarios, and a Try this exercise box, so a first-year and a senior researcher can each enter at the right depth. Below they are grouped by theme with a suggested classroom use.
How models decide their own complexity (how many experts, which components) and why that is hard.
Intermediate Universal Approximation · Mixtures and Mixtures of Experts
Drag one slider up and watch a mixture model's curve snap onto whatever target shape you pick.
In class: In Mixture density mode, drag Components K from 2 up to 64 and watch the shaded red error band collapse as the teal curve hugs the dashed target.
Intermediate Spectrum-Preserving Token Merging (PiToMe)
Watch a Vision Transformer merge redundant tokens layer by layer while an energy score shields the informative ones.
In class: Set the policy to Naive ToMe, press Merge one layer repeatedly, and watch the pink arrows mark informative tokens being merged as the ToMe spectral distance climbs.
Intermediate HyperRouter: Efficient Sparse Mixture of Experts via a Frozen HyperNetwork
Train two token routers side by side and watch the ordinary one collapse onto a few experts while the frozen-hypernetwork one stays balanced.
In class: Click Run, then toggle 'Load-balancing aux (standard)' mid-training and watch the standard router's load bars level out and its collapse meter drop.
Intermediate BSL with Mixtures of Experts: a surrogate likelihood that bends
Watch a single stretched ellipse fail to trace a crescent-shaped cloud while a mixture of little ellipses bends right around it.
In class: Set K experts to 1 and click 'Classic BSL': confirm the mixture collapses back to the single amber ellipse and the two log-likelihood readouts match.
Advanced Mixture of Experts & Dendrogram Merging
Fit one over-stuffed mixture of experts, then drag a cut down a dendrogram and watch redundant experts merge until the right number appears.
In class: Set epsilon-contam to 0.40, then drag it back to 0 and watch AIC/BIC/ICL fall back toward K0 as the Laplace outliers disappear.
Advanced Bayesian Nonparametric Mixture of Experts · Merge-Truncate-Merge
Drag one "merge radius" slider and watch a Bayesian model collapse dozens of redundant experts down to the true count.
In class: Set True K₀ to 5 and drag the merge radius ω until the posterior mode K̂ lands on 5, then note how wide the credible spread is.
Advanced Convergence Rates for Parameter Estimation in Mixtures of Experts
Simulate a mixture of experts and read its parameter-recovery speed off a log-log plot, then add one surplus expert and watch the rate collapse.
In class: Set the model to Softmax-gated and toggle K0+1 then K0+2: read the amber slope exponent in the readout and confirm it drops each time you add a surplus expert.
Advanced Variable Selection with Missing-Not-at-Random Data
Crank up how often data go missing and watch naive methods misclassify points (red rings) while the MNAR-aware model holds.
In class: Set MNAR strength to 80%, then click through Complete-case, Mean-impute, and MNAR (unified): watch the ARI bars and count red rings for each handler.
Advanced CompeteSMoE: Effective Training of Sparse Mixture of Experts via Competition
Train two mixture-of-experts side by side and watch one router learn who wins while the other collapses onto a couple of experts.
In class: Click Run, then drag ω down to about 0.07: does the distilled router still reach high agreement with competition on only a few steps?
Advanced Model selection for mixtures of experts via the slope heuristic
Slide the penalty constant and watch the chosen number of experts tumble down a staircase until the data itself marks the right spot.
In class: In well-specified mode, drag Penalty kappa (#msel-kappa) from low to high and note the kappa value where the staircase makes its biggest single drop; that is the dimension jump.
Advanced Modifications of the BIC for order selection in finite mixture models
Slide the sample size up and watch each information criterion lock its minimum onto the true number of mixture components.
In class: Set the family to Laplace and slide n from 100 up to 10000: watch P(k-hat = k0) for BIC climb to 1 even though the components are non-differentiable at their location.
Advanced PSGaBloME: Penalized Model Selection for Softmax-Gated Mixture of Experts
Drag a penalty dial and watch a mixture-of-experts model shed complexity in a staircase, snapping to the best model.
In class: Set κ to 0, then drag it to 4 and watch the amber star fall down the staircase until it crosses the κ̂ = 2κmin line at the oracle.
Turning noisy, inconsistent data into calibrated uncertainty, and checking that the intervals are honest.
Intermediate Robust Bayesian Parameter Balancing
Drag noisy enzyme measurements and watch a Bayesian fit snap them into thermodynamically consistent parameters, then throw an outlier at it.
In class: Select Gaussian, click Add outlier a few times and read the Coverage number, then switch to Student-t and repeat: watch coverage recover.
Advanced GLLiM-ABC: Likelihood-Free Inference with Surrogate Posteriors
Click a map to place a hidden sound source, then watch four ways of matching simulations fight to recover its blurry, multi-armed posterior.
In class: Select Semi-auto (mean), then GLLiM-MW2, and compare where the amber posterior-mean square sits versus where the accepted dots land.
Advanced Revisiting concentration results for approximate Bayesian computation
Shrink the ABC tolerance toward zero and watch the posterior mass collapse: onto a point, two spikes, or a diamond it can never resolve.
In class: Switch to Finitely unident. and drag ε tolerance toward 0: watch the two spikes at ±θ₀ sharpen but never collapse into one.
Advanced On the large-sample limits of Bayesian model-evaluation statistics
Watch Bayesian criteria slide down a log-scale plot and lock onto a single limit line, unless the tempering schedule breaks the rule.
In class: Pick the 1/n schedule and slide n (bmel-n) up to 10^5: watch the gap-to-limit readout stay positive instead of closing, showing n*beta_n to infinity cannot be dropped.
Advanced The Asymptotic Law of the Minimum Empirical Risk
Reshape a risk valley from a sharp point to a flat plateau and watch the histogram of the best-fit error morph from a clean bell curve into a skewed one.
In class: In MER limit-law mode, set Plateau width w to 0 and hit Run to accumulate draws: confirm the histogram hugs the teal Gaussian, then drag w up to 0.4 and watch it detach and skew left.
Advanced Risk Bounds via the h-Lifted KL Divergence
Sharpen a fitted mixture until it dips to zero and watch the standard KL loss spike to infinity while the h-lifted loss stays finite.
In class: Push Sharpness to its maximum and Lift h to its minimum, then read the right panel: the standard KL risk reports infinity while KLh stays a finite number.
How large models route computation and compress representations without losing signal.
Intermediate StructSAM: Structure- and Spectrum-Preserving Token Merging for Segment Anything
Watch a patch grid collapse redundant cells while amber edges and the prompt box stay protected, keeping boundaries crisp.
In class: Switch to Naive ToMe, click Merge one layer repeatedly, and watch the amber boundary tokens get absorbed as the structure distance climbs; then switch back to StructSAM and repeat to see the boundaries survive.
Advanced Conformer Aggregation by Fused Gromov-Wasserstein Barycenter
Spin one 3D pose of a molecule: the naive coordinate average lurches, but the optimal-transport aggregate never moves.
In class: Drag Rotate from 0 to 360 and watch the naive-mean drift readout climb while the FGW barycenter drift stays 0.00.
Advanced Fragment-Aware, Fast Conformer Ensemble Transformer
Shuffle a molecule's 3D shapes and watch the averaged fingerprint hold still while the expensive all-pairs alignment vanishes.
In class: Drag Conformers (#facet-n) from 3 to 30 and compare the N(N-1)/2 pairwise FGW solves against the N+1 encode-plus-mean operations in the cost bars.
Advanced Extended Context Graph Alignment for Medical Vision-Language Models
Watch three views of a medical batch snap into agreement with one shared hub, and see crossed red matches turn green.
In class: Toggle #exgra-order from first to second order and watch how many red crossed matches turn green in the panel.
