Erdodic’s Wikipedia definition is very general (and confusing) — “time averages equal ensemble averages”? Here’s a simplified version for finite-state markov chain: Let $\{X_k\}_{k \geq 0}$ be a Markov chain on state $S$. Let $\pi\in \Delta(S)$. Let Let transition operatro be $P$, so $\pi^{t + 1} = P \pi^t$. Define stationary distribution $\bar \pi$ as $X_0 \sim \bar \pi$ where $P\bar\pi = \bar\pi$, the stationary process satisfies $$ (X_0, X_1, \dots) \overset{d}{=} (X_1, X_2, \dots) $$ Definition. The stationary process $\{X_k\}$ is ergodic if for every bounded function $f: S \to \mathbb{R}$ the following convergence hold a.s.: ...

Evgeny Kissin returned to Chicago Symphony Orchestra, this time with Andrey Boreyko, opened a short run of concerts built around a thoughtfully balanced program: Rimsky-Korsakov: Russian Easter Overture, Op. 36 Mozart: Piano Concerto No. 12 in A Major, K. 414 Rimsky-Korsakov: Suite from The Tale of Tsar Saltan, Op. 57 Scriabin: Piano Concerto in F-sharp Minor, Op. 20 Rating: ⭐️⭐️⭐️⭐️⭐️. The program is repeated in Chicago the upcoming weekends then next week in Boston. Kissin played Mozart K414 with breathtakingly excitement in elegance. The opening Allegro carried a faint trace of caution, as though he were testing the acoustic and the air of the hall and feeling his way into the space. Even in that slight reserve his articulation remained crystalline. And by the Andante he had fully arrived: the phrasing deepened, the tone softened into something inward and luminous, the music seemed to breathe with so much intimacy. The final Allegretto is buoyant, danced lightly—quick yet unhurried, delicate without fragility. ...

Side Note of Professor Luciano Pomatto from Caltech’s Theory Seminar Talk at UChicago. Kudos to Zizhe for the summary. Ben said it was a great model as it was also well received in the internal student’s discussion. Disclaimer: All intellectual and copy rights belong to the author of the paper. If you like the economic model, make sure to check out Professor Pomatto’s webpage for the OG paper. All mistakes are mine. Setup A decision maker acquires information, choose action, and output. ...

Some sidenotes from price theory iii: Definitions For distributions (pdf) $f_1(\cdot), f_2(\cdot)$: Monotone Likelihood Ratio (MLR): let ratio $$ \phi(l) = \frac{f_1(l)}{f_2(l)}. $$ $f_1$ dominate $f_2$ (in MLR sense) iff. $\phi(l)$ is weakly increasing in $l$. Intuition: Pointwise relative density shifts up Hazard Rate Dominance (HRD): define the hazard rate (aka conditional failure rate): $$ h_{f_i}(l) = \frac{f_i(l)}{1 - F_i(l)} \quad i =1, 2. $$ $f_1$ dominate $f_2$ (in HRD sense) iff. $h_{f_1}(l)\le h_{f_2}(l), \forall l$. ...

It’s very intellectually stimulating to read Professor Stiglitz’s book, or if you’re as lazy as me, playing his YouTube archive recording of the book launch at UChicago. Professor Stiglitz’s most provocative point is almost saying the invisible hand doesn’t exist. Making this claim at the University of Chicago is dangerous… But basically, Kenneth Arrow (also a Chicago veteran) proved the conditions under which Adam Smith’s invisible hand claim holds — and showed how narrow those conditions are. Stiglitz’s argument is that then economists cherry-picked Arrow’s conclusion and ignored his caveats. They took a theorem with heavy asterisks and preached it as gospel. ...

I found a very inspiring video from the digital marketing industry that has economic insights: The video is long. Here’s what I found interesting: The core claim is, micro-influencers (10K–100K followers) deliver positive ROI ~36% of the time. Mega-influencers lose money in 59% of campaigns. Follower count is a poor predictor of returns. Why: Micro influncer is “community based distribution”, like not ads but genuine communication. Engagement is ~4x higher than branded accounts, and you can activate ~100 micro-creators for the cost of one celebrity. ...

Jakub Hrůša and presents Songs of Love and Farewell Death with the CSO — look at the thoughtful program: Janáček Overture to From the House of the Dead Rachmaninov The Isle of the Dead R. Strauss Four Last Songs, with soprano Corinne Winters Wagner Prelude and Liebestod from Tristan and Isolde (Orchestral version) It is a very spiritual experience. My friend says “I feel like I’m floating and entering an ethereal state, and the music just ends there”. ...

Yu Long is our most famous conductor, famous for lot of things: leader of Chinese classical scene, a good Brahms interpreter, and his most popular meme: China’s classical music scene is growing — while young ppl flood into the concert hall to enjoy the music, we also want to show off our music taste by posting “repos” on Social Media (📕), sometimes a little bit overly mean and picky. ...

AlphaFold2, released by DeepMind in 2020 and made freely available in July 2021, uses deep learning + DNA sequence alignment to predict a protein’s 3D structure at near-experimental accuracy, and generalizes well out-of-sample. This solves a 50-year-old open problem in structural biology and won 2024 Nobel Prize in Chemistry. MOREOVER, DeepMind ran AlphaFold2 on essentially every known protein and release a free database lookup. Though note its key limitations are: AlphaFold predicts a single static “default” conformation and misses the flexibility, ligand-bound states, and allosteric changes that matter most for drug design; ~10% of high-confidence residues are still meaningfully misplaced; and it is pattern recognition over evolutionary data, not a physics-based simulation. ...

Professor Richard Holden from UNSW Sydney came to our theory seminar to give a talk about his paper [“Getting the Picture”](https://economics.uchicago.edu/sites/default/files/users/user337/Holden Getting the Picture.pdf). The paper’s idea of modeling an individual who has limited working memory originates from cognitive science. But gives a strong computer science flavor. Here’s my notes from student coffee seminar. We think there are certain computer science work can be done for it: Model Definitions Input: $P \in \mathcal P \subseteq \Omega^{MN}$ ...