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[@probnstat](/creator/twitter/probnstat)
"Physics-Informed Neural Networks (PINNs) are a game-changer. These models embed physical laws like partial differential equations directly into their loss function. This forces the AI to learn solutions that not only fit the data but also obey the fundamental laws of physics"  
[X Link](https://x.com/probnstat/status/1974175604963721339) [@probnstat](/creator/x/probnstat) 2025-10-03T18:11Z 71.3K followers, 153.9K engagements


"The Kolmogorov-Smirnov (K-S) test is a non-parametric test used to compare probability distributions. In machine learning it's a vital tool for detecting data drift checking if the input data for a model in production has statistically changed from the training data signaling a need for retraining. In real-life applications it's used in finance to test if stock returns follow a specific distribution and in manufacturing for quality control to ensure products meet specifications"  
[X Link](https://x.com/probnstat/status/1976300060330819794) [@probnstat](/creator/x/probnstat) 2025-10-09T14:53Z 71.3K followers, 46.9K engagements


"Random Matrix Theory (RMT) predicts universal patterns in the eigenvalues of matrices with random entries. In machine learning it's a crucial tool to distinguish signal from noise in high-dimensional data. By analyzing the spectrum of large covariance matrices it helps understand deep learning models. In real life RMT is fundamental to modeling energy levels in nuclear physics optimizing wireless communication channels and analyzing risk in financial markets"  
[X Link](https://x.com/probnstat/status/1977076174779277362) [@probnstat](/creator/x/probnstat) 2025-10-11T18:17Z 71.3K followers, 39.7K engagements


"Independent Component Analysis (ICA) is a statistical method for separating a mixed signal into its independent sources. It famously solves the "cocktail party problem" by isolating individual voices from one microphone. In machine learning it's used for blind source separation and feature extraction. Its key real-life application is in biomedical signal processing like separating distinct brain activity signals in EEG and fMRI data from noise. Image source:"  
[X Link](https://x.com/probnstat/status/1977365976023695861) [@probnstat](/creator/x/probnstat) 2025-10-12T13:29Z 71.3K followers, 33.7K engagements


"Quantitative finance uses mathematical models to price financial instruments and manage risk. In machine learning this is supercharged: algorithms are used for algorithmic trading fraud detection and credit scoring. In real life it powers derivatives pricing (options) enabling companies to hedge risk and is the engine behind sophisticated portfolio management and risk assessment for the global economy"  
[X Link](https://x.com/probnstat/status/1978089373758603423) [@probnstat](/creator/x/probnstat) 2025-10-14T13:23Z 71.3K followers, 15.2K engagements


"Neural networks were conceptually invented in 1943 by neurophysiologist Warren McCulloch and logician Walter Pitts. They proposed a simple mathematical model of a biological neuron that could take binary inputs and "fire" if the sum exceeded a threshold. They proved that networks of these artificial neurons could compute any logical function laying the theoretical groundwork for all of modern AI"  
[X Link](https://x.com/probnstat/status/1979495501214187817) [@probnstat](/creator/x/probnstat) 2025-10-18T10:31Z 71.3K followers, 43K engagements


"Jrgen Schmidhuber is a pioneering German AI researcher often called a "father of modern AI." His most crucial contribution was the co-invention of the Long Short-Term Memory (LSTM) network in 1997. LSTMs revolutionized how AI handles sequential data like speech and text becoming a cornerstone of the deep learning boom. His early work on competing neural networks also predated the principles behind modern GANs"  
[X Link](https://x.com/probnstat/status/1979868213249749161) [@probnstat](/creator/x/probnstat) 2025-10-19T11:12Z 71.3K followers, 7759 engagements


"TRUE or FALSE: You are offered a chance to play a game where a fair coin is tossed repeatedly until it lands on Heads for the first time. If the first Head appears on the k-th toss you are paid $2k$ dollars. The mathematically fair price to enter this game is infinite"  
[X Link](https://x.com/probnstat/status/1979989094785777697) [@probnstat](/creator/x/probnstat) 2025-10-19T19:12Z 71.3K followers, 9302 engagements


"MCMC (Markov Chain Monte Carlo) algorithms are used to sample from complex probability distributions that are otherwise intractable. In machine learning MCMC is the engine of Bayesian inference. It allows models to determine not just a single best parameter but a full distribution of plausible parameters which is crucial for quantifying uncertainty. Real-world applications include estimating risk in financial models inferring evolutionary trees in computational biology and simulating particle systems in physics"  
[X Link](https://x.com/probnstat/status/1974915778077204759) [@probnstat](/creator/x/probnstat) 2025-10-05T19:13Z 71.1K followers, 49.9K engagements


"Graph theory models networks of connected data. In real life it's the core of Google Maps finding the shortest route and social networks mapping friendships. In machine learning Graph Neural Networks (GNNs) work directly on this structure. They are used to predict molecular properties for drug discovery power recommendation engines and detect complex fraud in financial systems by analyzing the relationships between entities not just the entities themselves"  
[X Link](https://x.com/probnstat/status/1975207669012255177) [@probnstat](/creator/x/probnstat) 2025-10-06T14:33Z 71K followers, 22.7K engagements


"Functional analysis treats functions as points in an infinite-dimensional space. This is the mathematical backbone for non-parametric statistics and machine learning methods like Kernel SVMs and Gaussian Processes. These techniques learn complex flexible functions without assuming a fixed model structure. In real life functional analysis is fundamental to signal processing (analyzing audio/images) and is the language of quantum mechanics describing the state of physical systems"  
[X Link](https://x.com/probnstat/status/1975526133497303462) [@probnstat](/creator/x/probnstat) 2025-10-07T11:38Z 71.2K followers, 16.8K engagements


"Homotopy theory studies how shapes and paths can be continuously deformed into one another. In machine learning it's used in Topological Data Analysis (TDA) to understand the fundamental shape of data helping to verify if different neural network models are equivalent in some essential way. In real life it's crucial for robotics motion planning where it helps classify and find optimal paths for a robot to navigate around obstacles without collision"  
[X Link](https://x.com/probnstat/status/1975579049746731518) [@probnstat](/creator/x/probnstat) 2025-10-07T15:08Z 71.2K followers, 13K engagements


"Higgsfield Sora X achieves exceptional visual fidelity through a pioneering training methodology that significantly lowers operational expenses. Sora X uses a revolutionary training approach to create next-level video. #HiggsfieldSora2"  
[X Link](https://x.com/probnstat/status/1975657157128560863) [@probnstat](/creator/x/probnstat) 2025-10-07T20:19Z 70.9K followers, 2608 engagements


"Signal processing is used to filter analyze and extract information from signals like audio images and sensor readings. In machine learning it's a vital pre-processing step to clean data (e.g. remove noise from speech) and extract key features for models to learn from drastically improving accuracy. In real life it's the core technology behind noise-canceling headphones MP3/JPEG compression medical imaging (MRI/CT scans) and all wireless communications like Wi-Fi and 4G/5G. Image source:"  
[X Link](https://x.com/probnstat/status/1975838452970131931) [@probnstat](/creator/x/probnstat) 2025-10-08T08:19Z 71K followers, 31.3K engagements


"In a famous 2017 critique computer scientist Ali Rahimi called machine learning "alchemy." He argued that while the field achieves amazing results it often lacks a rigorous scientific understanding of why its powerful models work. He claimed progress relied on trial-and-error and folklore instead of fundamental theory. The quote was a powerful call for the community to better understand the powerful tools they were creating. Source:"  
[X Link](https://x.com/probnstat/status/1975901515446685928) [@probnstat](/creator/x/probnstat) 2025-10-08T12:30Z 71.2K followers, 15.6K engagements


"Kolmogorov's axioms are the three fundamental rules that provide the rigorous mathematical foundation for all of probability theory. They aren't an algorithm but are the bedrock that ensures any probabilistic machine learning model (e.g. Bayesian networks) is consistent and sound. In real life they are the basis for risk assessment in industries like insurance and finance allowing actuaries to build reliable models to calculate the likelihood of events and set premiums. Image Source:"  
[X Link](https://x.com/probnstat/status/1976211736169488636) [@probnstat](/creator/x/probnstat) 2025-10-09T09:02Z 71.2K followers, 16K engagements


"Empirical process theory is the mathematical foundation of statistical learning theory. It provides the tools (like VC-dimension) to prove that machine learning models trained on a finite dataset will generalize well to new unseen data. It essentially answers the question: "Why should a model that works on my data work on anyone else's" This ensures the reliability of ML systems used in real life from credit risk models in banking to medical diagnostic tools by bounding their error rate"  
[X Link](https://x.com/probnstat/status/1976579801532997988) [@probnstat](/creator/x/probnstat) 2025-10-10T09:25Z 71.2K followers, 13.1K engagements


"Convex optimization finds the guaranteed best solution to problems where the objective is a convex function. This is the mathematical engine behind many robust machine learning algorithms including Support Vector Machines (SVMs) logistic regression and Lasso. It ensures these models can be trained efficiently to a unique optimal solution. In real life it's essential for logistics (finding the cheapest routes) finance (portfolio optimization) and engineering design"  
[X Link](https://x.com/probnstat/status/1976616207840399530) [@probnstat](/creator/x/probnstat) 2025-10-10T11:50Z 71.2K followers, 23.5K engagements


"Self-Organizing Maps (SOMs) are unsupervised neural networks that visualize high-dimensional data on a 2D grid like a map. They excel at clustering and dimensionality reduction by preserving the "shape" of the original data. In machine learning they are used to discover hidden patterns in complex datasets. Real-world applications include customer segmentation in marketing analyzing gene expression data and creating insightful "poverty maps" from complex socioeconomic indicators. Lecture Notes:"  
[X Link](https://x.com/probnstat/status/1976929889706967050) [@probnstat](/creator/x/probnstat) 2025-10-11T08:36Z 71.2K followers, 8922 engagements


"Bernoulli Naive Bayes is a probabilistic classifier for binary feature data (e.g. present/absent). In machine learning it's a staple for text classification powering applications like spam filtering and sentiment analysis. It calculates the probability of a document belonging to a class based on the presence or absence of specific words assuming each word is independent. This makes it a fast and effective tool for real-life tasks like identifying spam or categorizing news articles"  
[X Link](https://x.com/probnstat/status/1977251962283606023) [@probnstat](/creator/x/probnstat) 2025-10-12T05:56Z 71K followers, 10.6K engagements


"Decision trees are flowchart-like models that make predictions through a series of conditional rules. In machine learning the algorithm learns the optimal feature-based conditions (e.g. "is age 30") to split the data. Each path from the root to a leaf represents a rule set for a prediction. This interpretability is key in real-life applications like medical diagnosis (symptom checks) and credit scoring where decisions need to be transparent and justifiable"  
[X Link](https://x.com/probnstat/status/1977424186747613504) [@probnstat](/creator/x/probnstat) 2025-10-12T17:20Z 71.2K followers, 15.4K engagements


"Neural PDEs use neural networks to solve the complex differential equations that govern the physical world. In machine learning this powers Physics-Informed Neural Networks (PINNs) which embed physical laws directly into the learning process. This revolutionizes real-life applications by drastically accelerating simulations for fluid dynamics (weather aerodynamics) drug discovery and financial modeling. Image Source:"  
[X Link](https://x.com/probnstat/status/1978393085689434585) [@probnstat](/creator/x/probnstat) 2025-10-15T09:30Z 71.2K followers, 14.7K engagements


"Gradient Boosting Machines (GBMs) create a single highly accurate model by sequentially training weak learners (typically decision trees). Each new tree is trained to correct the errors of the previous ones. This powerful ensemble approach makes them a top choice for real-life applications on tabular data such as credit scoring fraud detection and ranking search results on the web. Image source:"  
[X Link](https://x.com/probnstat/status/1978486844477337785) [@probnstat](/creator/x/probnstat) 2025-10-15T15:43Z 71K followers, 11.4K engagements


""Neural Networks for Babies" by Chris Ferrie and Dr. Sarah Kaiser is a colorful board book that introduces the core concept of AI to the youngest learners. Using simple bold illustrations it shows how a neural network like a little brain takes an input (like a picture of a cat) has a "neuron" that thinks about it and produces an output ("CAT"). It's not about math but about planting a seed of curiosity visually explaining the basic idea of how machines can learn to recognize patterns"  
[X Link](https://x.com/probnstat/status/1976224006492799126) [@probnstat](/creator/x/probnstat) 2025-10-09T09:51Z 71.3K followers, 24.8K engagements


"Game theory is the mathematical study of strategic decision-making. In machine learning it's the core concept behind Generative Adversarial Networks (GANs) where a "Generator" and a "Discriminator" network compete in a zero-sum game to produce realistic data. It's also foundational to multi-agent reinforcement learning. In real life it's used extensively in economics to model auctions and markets in politics to analyze strategies and in biology to study evolutionary dynamics"  
[X Link](https://x.com/probnstat/status/1976321423925973172) [@probnstat](/creator/x/probnstat) 2025-10-09T16:18Z 71.3K followers, 34.2K engagements


"Time series analysis models time-ordered data points to identify trends and seasonality. In machine learning it's the core of forecasting where models like LSTMs and Transformers learn from historical sequences to predict future events. This is vital in real life powering stock price prediction weather forecasting business demand planning and monitoring patient vital signs"  
[X Link](https://x.com/probnstat/status/1976895919116501403) [@probnstat](/creator/x/probnstat) 2025-10-11T06:21Z 71.3K followers, 29K engagements


"Graphons are mathematical objects that model the structure of massive networks. In machine learning they provide a powerful framework for analyzing and generating large graphs. They are used to estimate the underlying structure of a network predict missing links and understand the limits of graph algorithms. Real-life applications include modeling brain connectomes in neuroscience and understanding the large-scale structure of social networks. Image source:"  
[X Link](https://x.com/probnstat/status/1979117839815766527) [@probnstat](/creator/x/probnstat) 2025-10-17T09:30Z 71.3K followers, 50K engagements


"No camera No problem. Higgsfield Sora X Sketch-to-Video turns any sketch into high-resolution motion scenes ready for storytelling. #HiggsfieldSora2"  
[X Link](https://x.com/probnstat/status/1979464972444012590) [@probnstat](/creator/x/probnstat) 2025-10-18T08:30Z 71.3K followers, 2322 engagements


"Markov Chain Monte Carlo (MCMC) methods use a "random walk" from a Markov chain to draw samples from complex probability distributions. In machine learning this is the engine of Bayesian inference letting models find a distribution of plausible parameters to quantify uncertainty. In real life this powers Google's original PageRank algorithm (a giant Markov chain) and is used in physics to simulate particle systems"  
[X Link](https://x.com/probnstat/status/1979560098617032910) [@probnstat](/creator/x/probnstat) 2025-10-18T14:48Z 71.3K followers, 43.2K engagements


"Causal Inference is a hard but crucial area of machine learning that moves beyond correlation to determine why things happen. While standard models are great at prediction they can't answer "what if" questions like "What is the true impact of a marketing campaign" Its core difficulty is the "fundamental problem": we can never observe the counterfactual (what would have happened without the campaign). This requires specialized methods to isolate true cause from confounding variables which is vital for business strategy and scientific discovery. Image source:"  
[X Link](https://x.com/probnstat/status/1980006193419747744) [@probnstat](/creator/x/probnstat) 2025-10-19T20:20Z 71.3K followers, 13.3K engagements


"Statistical decision theory provides a framework for making optimal choices under uncertainty by minimizing a "loss function" that defines the cost of a wrong decision. In machine learning it's the formal basis for classification and regression guiding models to make predictions that minimize error. In real life it's used in medical diagnosis to choose the best treatment by weighing the risks of different outcomes and in finance to build optimal investment portfolios"  
[X Link](https://x.com/probnstat/status/1976248648699961804) [@probnstat](/creator/x/probnstat) 2025-10-09T11:29Z 71.2K followers, 12.2K engagements


"Commutative algebra studies algebraic structures like rings and fields. While its use in machine learning is highly theoretical it provides a deep structural understanding of certain statistical models through algebraic statistics. Its most significant real-life application is in cryptography and error-correcting codes. Modern encryption systems like elliptic-curve cryptography (ECC) which secures online transactions are built directly upon the principles of commutative algebra"  
[X Link](https://x.com/probnstat/status/1976511017388736810) [@probnstat](/creator/x/probnstat) 2025-10-10T04:52Z 71.2K followers, 14.2K engagements


"Knot theory classifies how loops can be tangled in space. In biology it's used to understand how protein folding creates molecular knots that determine a protein's function and how enzymes untangle DNA during replication. In deep learning it's an emerging tool in Topological Data Analysis (TDA) where the "knottedness" of data in high dimensions can be a powerful feature for models to learn from. Its principles are also applied in statistical mechanics"  
[X Link](https://x.com/probnstat/status/1976655693433430066) [@probnstat](/creator/x/probnstat) 2025-10-10T14:26Z 71.2K followers, 16.7K engagements


"The Beta distribution models uncertainty about a probability on a X X scale. In machine learning it's key in Bayesian inference to represent the probability of a parameter like a click-through rate. In protein folding it can model the distribution of torsion angles. In real life it powers A/B testing helping businesses determine which webpage version performs better by modeling the conversion rate for each as a probability distribution"  
[X Link](https://x.com/probnstat/status/1976735004135923740) [@probnstat](/creator/x/probnstat) 2025-10-10T19:42Z 71.2K followers, 13.3K engagements


"Causal inference moves beyond correlation to determine why things happen. In machine learning it helps build robust models that can predict the outcome of interventions (e.g. "what if we change the price") leading to fairer and more reliable decisions. In real life it's crucial for determining the true effectiveness of medical treatments in clinical trials evaluating economic policy impacts and measuring the real ROI of marketing campaigns. Image source:"  
[X Link](https://x.com/probnstat/status/1977023915836031390) [@probnstat](/creator/x/probnstat) 2025-10-11T14:50Z 71.2K followers, 30.9K engagements


"TRUE or FALSE: You flip a fair coin X times and it lands on Heads every single time. The probability of the next flip being Tails is now greater than 50%"  
[X Link](https://x.com/probnstat/status/1977048205503480148) [@probnstat](/creator/x/probnstat) 2025-10-11T16:26Z 71.3K followers, 27K engagements


"Machine learning helps navigate the vast mathematical landscapes of string theory finding patterns and potential solutions in complex geometries that are beyond human calculation. In real-world physics ML is indispensable. At particle accelerators like the LHC algorithms sift through petabytes of collision data to identify signals of new particles amidst background noise. It's also used to discover new materials with desired properties and to classify astronomical objects from telescope data"  
[X Link](https://x.com/probnstat/status/1977467789150761050) [@probnstat](/creator/x/probnstat) 2025-10-12T20:13Z 71.2K followers, 33.6K engagements


"TRUE or FALSE: There are two identical bags. Bag A contains two red marbles. Bag B contains one red marble and one blue marble. You randomly choose a bag and then randomly draw one marble from it. The marble you draw is red. Given that you drew a red marble the probability that you chose from Bag A is 1/2"  
[X Link](https://x.com/probnstat/status/1977617416902332908) [@probnstat](/creator/x/probnstat) 2025-10-13T06:08Z 71.2K followers, 10.6K engagements


"A stopping time is a rule for deciding when to halt a random process based only on information observed so far. In machine learning it's the principle behind early stopping where training halts when validation performance plateaus to prevent overfitting. In real life it's crucial in finance for executing stop-loss orders on stocks and in A/B testing to determine when enough data has been collected to declare a winner. Image source:"  
[X Link](https://x.com/probnstat/status/1977774130423591393) [@probnstat](/creator/x/probnstat) 2025-10-13T16:31Z 71.2K followers, 19.2K engagements


"Hedging is a risk management strategy to offset potential investment losses by taking an opposing position in a related asset. In real life an airline hedges against rising fuel prices by buying oil futures. In machine learning reinforcement learning (RL) is used to develop optimal dynamic hedging strategies. An RL agent learns when and how much to hedge by adapting to live market data minimizing risk more effectively than traditional static models"  
[X Link](https://x.com/probnstat/status/1977802115721630096) [@probnstat](/creator/x/probnstat) 2025-10-13T18:22Z 71.2K followers, 16K engagements


"The Ising model from physics shows how simple local interactions (like magnetic spins aligning) create global patterns. In machine learning it's the mathematical basis for Boltzmann Machines and other graphical models. Its principles are used for tasks like image denoising where noisy pixels are corrected to match their neighbors. It's also used to model social dynamics and protein folding"  
[X Link](https://x.com/probnstat/status/1978140733900259561) [@probnstat](/creator/x/probnstat) 2025-10-14T16:47Z 71.2K followers, 27.9K engagements


"The multivariate Gaussian distribution models elliptical clouds of data points across multiple dimensions capturing the correlation between variables. In machine learning it's the engine of Gaussian Mixture Models (GMMs) for clustering and anomaly detection. In real life it is vital in quantitative finance to model the joint movement of stock returns for portfolio optimization and in robotics for tracking object positions. Image source:"  
[X Link](https://x.com/probnstat/status/1978176718746562646) [@probnstat](/creator/x/probnstat) 2025-10-14T19:10Z 71.2K followers, 18.8K engagements


"TRUE or FALSE: Three prisoners A B and C are on death row. The governor has randomly chosen one to pardon. The warden knows who will be pardoned. Prisoner A asks the warden to name one of the other prisoners who will be executed. The warden truthfully says 'Prisoner B will be executed.' With this new information Prisoner A's probability of being the one pardoned is now 1/2"  
[X Link](https://x.com/probnstat/status/1978909932541087960) [@probnstat](/creator/x/probnstat) 2025-10-16T19:44Z 71.2K followers, 11.5K engagements


"Probability is the mathematical language of uncertainty. In machine learning it allows models to quantify their confidence in predictions which is essential for tasks like spam filtering and medical diagnosis. In real life it's the foundation of risk assessment used by insurance companies to set premiums in finance to model markets and in weather forecasting to predict the chance of rain"  
[X Link](https://x.com/probnstat/status/1979230452927074830) [@probnstat](/creator/x/probnstat) 2025-10-17T16:58Z 71.3K followers, 14.5K engagements


""Probability does not exist." This provocative quote is by the Italian statistician and probabilist Bruno de Finetti. It encapsulates the core idea of Subjectivism or Bayesianism a major school of thought in probability. De Finetti argued that probability is not an objective property of the world (like mass or temperature). Instead it's a measure of an individual's degree of belief about an uncertain event based on their knowledge and evidence. Essentially he meant that a "50% chance of rain" doesn't exist in the clouds themselves; it exists in the mind of the meteorologist who assigns that"  
[X Link](https://x.com/probnstat/status/1975270690736742857) [@probnstat](/creator/x/probnstat) 2025-10-06T18:43Z 71.3K followers, 245.7K engagements


"Stochastic Differential Equations (SDEs) model systems that evolve over time while being influenced by randomness. In machine learning they are the mathematical foundation for modern diffusion models like Stable Diffusion which generate images by learning to reverse a noise process. In real life SDEs are famously used in quantitative finance to model the random fluctuations of stock prices for option pricing (e.g. the Black-Scholes model) and for risk management"  
[X Link](https://x.com/probnstat/status/1975422566396559370) [@probnstat](/creator/x/probnstat) 2025-10-07T04:46Z 71.3K followers, 26.5K engagements


"A Markov process is a model where the future state depends only on the present not the past (the "memoryless" property). In machine learning it's the foundational framework for Reinforcement Learning (RL) defining how agents interact with an environment. In real life it powers Google's original PageRank algorithm simple text prediction (what's the next word) and models disease progression in healthcare. Image sources: - -"  
[X Link](https://x.com/probnstat/status/1977695494219665502) [@probnstat](/creator/x/probnstat) 2025-10-13T11:18Z 71.3K followers, 33K engagements


"Stochastic calculus is the math of random processes evolving over time. 🎲 In machine learning it's the engine behind modern diffusion models (like Stable Diffusion) which generate images by learning to reverse a precisely defined noise process. Its most famous real-life application is in quantitative finance powering the Black-Scholes model to price stock options by modeling their random fluctuations"  
[X Link](https://x.com/probnstat/status/1978511051533779335) [@probnstat](/creator/x/probnstat) 2025-10-15T17:19Z 71.3K followers, 22.8K engagements


"Measure theory provides the rigorous mathematical foundation for probability the language of uncertainty in machine learning. Integrals are essential for calculating key quantities like expected values and probabilities for continuous variables which is crucial for defining loss functions and building probabilistic models. In real life this framework is the bedrock of risk assessment in finance and insurance allowing for the precise modeling of uncertain outcomes"  
[X Link](https://x.com/probnstat/status/1978835551869616345) [@probnstat](/creator/x/probnstat) 2025-10-16T14:48Z 71.3K followers, 16.8K engagements


"Which time zone are you from"  
[X Link](https://x.com/probnstat/status/1979603632434893151) [@probnstat](/creator/x/probnstat) 2025-10-18T17:40Z 71.3K followers, 4727 engagements


"E-values are a modern alternative to p-values for hypothesis testing. Unlike p-values they allow for anytime-valid inference meaning you can continuously monitor a data stream and stop the test at any time without invalidating the results. In machine learning this is crucial for real-time A/B testing and monitoring models for data drift. In real life they make clinical trials more flexible and efficient allowing researchers to stop a trial as soon as significant evidence is found"  
[X Link](https://x.com/probnstat/status/1980187076156063868) [@probnstat](/creator/x/probnstat) 2025-10-20T08:19Z 71.3K followers, 33.1K engagements


"Quantum physics principles are fundamental to many real-life technologies including lasers MRI scanners and the transistors that power all modern electronics. In the realm of artificial intelligence quantum machine learning (QML) is an emerging field that harnesses quantum phenomena like superposition and entanglement. This allows for the processing of vast datasets and the solving of complex problems much faster than classical computers. QML has the potential to revolutionize areas such as drug discovery by simulating molecules optimizing financial models and enhancing pattern recognition"  
[X Link](https://x.com/probnstat/status/1974574485190914543) [@probnstat](/creator/x/probnstat) 2025-10-04T20:36Z 71.3K followers, 2.1M engagements


"Brownian motion the random movement of particles is a key model for random processes. In machine learning it's the conceptual basis for the noise added in diffusion models powerful generative AIs that create images by learning to reverse a random process. It's also fundamental to stochastic differential equations. In real life its most famous application is in quantitative finance where it's used to model the random fluctuations of stock prices for option pricing (Black-Scholes model)"  
[X Link](https://x.com/probnstat/status/1976375691047469506) [@probnstat](/creator/x/probnstat) 2025-10-09T19:54Z 71.3K followers, 39.4K engagements


"Gibbs sampling is an MCMC algorithm for sampling from complex distributions. In machine learning it's a key engine for Bayesian inference notably in Latent Dirichlet Allocation (LDA) for topic modeling. This is used in real life to discover hidden themes in customer reviews or scientific papers. It's also applied in computer vision for image restoration and in statistical physics. Image source:"  
[X Link](https://x.com/probnstat/status/1977661378828857373) [@probnstat](/creator/x/probnstat) 2025-10-13T09:03Z 71.3K followers, 35.1K engagements


"Percolation theory from physics models how things flow through a random network like a forest fire spreading. It's all about finding the "tipping point" where a connected path suddenly forms. In machine learning it's used to analyze the structure and robustness of neural networks. In real life it's crucial for modeling the spread of diseases and understanding how oil flows through porous rock"  
[X Link](https://x.com/probnstat/status/1978698202292592934) [@probnstat](/creator/x/probnstat) 2025-10-16T05:43Z 71.3K followers, 7370 engagements


"TRUE or FALSE: You draw two cards from a standard 52-card deck without replacement. The event 'the first card is a Spade' and the event 'the second card is an Ace' are independent events"  
[X Link](https://x.com/probnstat/status/1978719330767356287) [@probnstat](/creator/x/probnstat) 2025-10-16T07:07Z 71.3K followers, 7146 engagements


"ANOVA (Analysis of Variance) is a statistical test that compares the means of three or more groups to determine if at least one is significantly different. In machine learning it's a key tool for feature selection checking if a categorical feature meaningfully impacts a continuous target. In real life it's used to compare the effectiveness of different drugs on patient recovery times or to see if various fertilizers produce different average crop yields. Image Source:"  
[X Link](https://x.com/probnstat/status/1978750214929301529) [@probnstat](/creator/x/probnstat) 2025-10-16T09:09Z 71.3K followers, 37.4K engagements


"The homoscedasticity assumption in linear regression means the variance of the errors is constant across all levels of the predictors. This is crucial for reliable inference. While the model's coefficient estimates remain unbiased without it their standard errors become biased. This invalidates p-values and confidence intervals meaning you can't trust hypothesis tests about which features are truly significant. Image source:"  
[X Link](https://x.com/probnstat/status/1979454060836327488) [@probnstat](/creator/x/probnstat) 2025-10-18T07:46Z 71.3K followers, 141.5K engagements


"Are you enjoying my content here"  
[X Link](https://x.com/probnstat/status/1980327362056401266) [@probnstat](/creator/x/probnstat) 2025-10-20T17:36Z 71.3K followers, 8796 engagements