The blog post revisits William Benter's groundbreaking 1995 paper detailing the statistical model he used to successfully predict horse race outcomes in Hong Kong. Benter's approach went beyond simply ranking horses based on past performance. He meticulously gathered a wide array of variables, recognizing the importance of factors like track condition, jockey skill, and individual horse form. His model employed advanced statistical techniques, including Bayesian networks and meticulous data normalization, to weigh these factors and generate accurate probability estimates for each horse winning. This allowed him to identify profitable betting opportunities by comparing his predicted probabilities with publicly available odds, effectively exploiting market inefficiencies. The post highlights the rigor, depth of analysis, and innovative application of statistical methods that underpinned Benter's success, showcasing it as a landmark achievement in predictive modeling.
This blog post explores how to cheat at Settlers of Catan by subtly altering the weight distribution of the dice. The author meticulously measures the roll probabilities of standard Catan dice and then modifies a set by drilling small holes and filling them with lead weights. Through statistical analysis using p-values and chi-squared tests, he demonstrates that the loaded dice significantly favor certain numbers (6 and 8), giving the cheater an advantage in resource acquisition. The post details the weighting process, the statistical methods employed, and the resulting shift in probability distributions, effectively proving that such manipulation is possible and detectable through rigorous analysis.
HN users discussed the practicality and ethics of the dice-loading method described in the article. Some doubted its real-world effectiveness, citing the difficulty of consistently achieving the subtle weight shift required and the risk of detection. Others debated the statistical significance of the results presented, questioning the methodology and the interpretation of p-values. Several commenters pointed out that even if successful, such cheating would ruin the fun of the game for everyone involved, highlighting the importance of fair play over a marginal advantage. A few users shared anecdotal experiences of suspected cheating in Settlers, while others suggested alternative, less malicious methods of gaining an edge, such as studying probability distributions and optimal placement strategies. The overall consensus leaned towards condemning cheating, even if statistically demonstrable, as unsporting and ultimately detrimental to the enjoyment of the game.
This paper explores the relationship between transformer language models and simpler n-gram models. It demonstrates that transformers, despite their complexity, implicitly learn n-gram statistics, and that these statistics significantly contribute to their performance. The authors introduce a method to extract these n-gram distributions from transformer models and show that using these extracted distributions in a simple n-gram model can achieve surprisingly strong performance, sometimes even exceeding the performance of the original transformer on certain tasks. This suggests that a substantial part of a transformer's knowledge is captured by these implicit n-gram representations, offering a new perspective on how transformers process and represent language. Furthermore, the study reveals that larger transformers effectively capture longer-range dependencies by learning longer n-gram statistics, providing a quantitative link between model size and the ability to model long-range contexts.
HN commenters discuss the paper's approach to analyzing transformer behavior through the lens of n-gram statistics. Some find the method insightful, suggesting it simplifies understanding complex transformer operations and offers a potential bridge between statistical language models and neural networks. Others express skepticism, questioning whether the observed n-gram behavior is a fundamental aspect of transformers or simply a byproduct of training data. The debate centers around whether this analysis genuinely reveals something new about transformers or merely restates known properties in a different framework. Several commenters also delve into specific technical details, discussing the implications for tasks like machine translation and the potential for improving model efficiency. Some highlight the limitations of n-gram analysis, acknowledging its inability to fully capture the nuanced behavior of transformers.
To avoid p-hacking, researchers should pre-register their studies, specifying hypotheses, analyses, and data collection methods before looking at the data. This prevents manipulating analyses to find statistically significant (p<0.5) but spurious results. Additionally, focusing on effect sizes rather than just p-values provides a more meaningful interpretation of results, as does embracing open science practices like sharing data and code for increased transparency and reproducibility. Finally, shifting the focus from null hypothesis significance testing to estimation and incorporating Bayesian methods allows for more nuanced understanding of uncertainty and prior knowledge, further mitigating the risks of p-hacking.
HN users discuss the difficulty of avoiding p-hacking, even with pre-registration. Some highlight the inherent flexibility in data analysis, from choosing variables and transformations to defining outcomes, arguing that conscious or unconscious bias can still influence results. Others suggest focusing on effect sizes and confidence intervals rather than solely on p-values, and emphasizing the importance of replication. Several commenters point out that pre-registration itself isn't foolproof, as researchers can find ways to deviate from their plans or selectively report pre-registered analyses. The cynicism around "publish or perish" pressures in academia is also noted, with some arguing that systemic issues incentivize p-hacking despite best intentions. A few commenters mention Bayesian methods as a potential alternative, while others express skepticism about any single solution fully addressing the problem.
Linear regression aims to find the best-fitting straight line through a set of data points by minimizing the sum of squared errors (the vertical distances between each point and the line). This "line of best fit" is represented by an equation (y = mx + b) where the goal is to find the optimal values for the slope (m) and y-intercept (b). The blog post visually explains how adjusting these parameters affects the line and the resulting error. To efficiently find these optimal values, a method called gradient descent is used. This iterative process calculates the slope of the error function and "steps" down this slope, gradually adjusting the parameters until it reaches the minimum error, thus finding the best-fitting line.
HN users generally praised the article for its clear and intuitive explanation of linear regression and gradient descent. Several commenters appreciated the visual approach and the focus on minimizing the sum of squared errors. Some pointed out the connection to projection onto a subspace, providing additional mathematical context. One user highlighted the importance of understanding the underlying assumptions of linear regression, such as homoscedasticity and normality of errors, for proper application. Another suggested exploring alternative cost functions beyond least squares. A few commenters also discussed practical considerations like feature scaling and regularization.
The blog post "Determining favorite t-shirt color using science" details a playful experiment using computer vision and Python to analyze a wardrobe of t-shirts. The author photographs their folded shirts, uses a script to extract the dominant color of each shirt, and then groups and counts these colors to determine their statistically "favorite" t-shirt color. While acknowledging the limitations of the method, such as lighting and folding inconsistencies, the author concludes their favorite color is blue, based on the prevalence of blue-hued shirts in their collection.
HN commenters largely found the blog post's methodology flawed and amusing. Several pointed out that simply asking someone their favorite color would be more efficient than the convoluted process described. The top comment highlights the absurdity of using a script to scrape Facebook photos for color analysis, especially given the potential inaccuracies of such an approach. Others questioned the statistical validity of the sample size and the representativeness of Facebook photos as an indicator of preferred shirt color. Some found the over-engineered solution entertaining, appreciating the author's humorous approach to a trivial problem. A few commenters offered alternative, more robust methods for determining color preferences, including using color palettes and analyzing wardrobe composition.
An analysis of chord progressions in 680,000 songs reveals common patterns and some surprising trends. The most frequent progressions are simple, diatonic, and often found in popular music across genres. While major chords and I-IV-V-I progressions dominate, the data also highlights the prevalence of the vi chord and less common progressions like the "Axis" progression. The study categorized progressions by "families," revealing how variations on a core progression create distinct musical styles. Interestingly, chord progressions appear to be getting simpler over time, possibly influenced by changing musical tastes and production techniques. Ultimately, while common progressions are prevalent, there's still significant diversity in how artists utilize harmony.
HN users generally praised the analysis and methodology of the original article, particularly its focus on transitions between chords rather than individual chord frequency. Some questioned the dataset's limitations, wondering about the potential biases introduced by including only songs with available chord data, and the skewed representation towards Western music. The discussion also explored the subjectivity of music theory, with commenters highlighting the difficulty of definitively labeling certain chord functions (like tonic or dominant) and the potential for cultural variations in musical perception. Several commenters shared their own musical insights, referencing related analyses and discussing the interplay of theory and practice in composition. One compelling comment thread delved into the limitations of Markov chain analysis for capturing long-range musical structure and the potential of higher-order Markov models or recurrent neural networks for more nuanced understanding.
This blog post explains Markov Chain Monte Carlo (MCMC) methods in a simplified way, focusing on their practical application. It describes MCMC as a technique for generating random samples from complex probability distributions, even when direct sampling is impossible. The core idea is to construct a Markov chain whose stationary distribution matches the target distribution. By simulating this chain, the sampled values eventually converge to represent samples from the desired distribution. The post uses a concrete example of estimating the bias of a coin to illustrate the method, detailing how to construct the transition probabilities and demonstrating why the process effectively samples from the target distribution. It avoids complex mathematical derivations, emphasizing the intuitive understanding and implementation of MCMC.
Hacker News users generally praised the article for its clear explanation of MCMC, particularly its accessibility to those without a deep statistical background. Several commenters highlighted the effective use of analogies and the focus on the practical application of the Metropolis algorithm. Some pointed out the article's omission of more advanced MCMC methods like Hamiltonian Monte Carlo, while others noted potential confusion around the term "stationary distribution". A few users offered additional resources and alternative explanations of the concept, further contributing to the discussion around simplifying a complex topic. One commenter specifically appreciated the clear explanation of detailed balance, a concept they had previously struggled to grasp.
Seattle has reached a new demographic milestone: for the first time, half of the city's men are unmarried. 2022 Census data reveals that 50.6% of men in Seattle have never been married, compared to 36.8% of women. This disparity is largely attributed to the influx of young, single men drawn to the city's booming tech industry. While Seattle has long had a higher proportion of single men than the national average, this shift marks a significant increase and underscores the city's unique demographic landscape.
Hacker News commenters discuss potential reasons for the high number of unmarried men in Seattle, citing the city's skewed gender ratio (more men than women), the demanding work culture in tech, and high cost of living making it difficult to start families. Some suggest that men focused on career advancement may prioritize work over relationships, while others propose that the dating scene itself is challenging, with apps potentially exacerbating the problem. A few commenters question the data or its interpretation, pointing out that "never married" doesn't necessarily equate to "single" and that the age range considered might be significant. The overall sentiment leans towards acknowledging the challenges of finding a partner in a competitive and expensive city like Seattle, particularly for men.
Cross-entropy and KL divergence are closely related measures of difference between probability distributions. While cross-entropy quantifies the average number of bits needed to encode events drawn from a true distribution p using a coding scheme optimized for a predicted distribution q, KL divergence measures how much more information is needed on average when using q instead of p. Specifically, KL divergence is the difference between cross-entropy and the entropy of the true distribution p. Therefore, minimizing cross-entropy with respect to q is equivalent to minimizing the KL divergence, as the entropy of p is constant. While both can measure the dissimilarity between distributions, KL divergence is a true "distance" metric (though asymmetric), whereas cross-entropy is not. The post illustrates these concepts with detailed numerical examples and explains their significance in machine learning, particularly for tasks like classification where the goal is to match a predicted distribution to the true data distribution.
Hacker News users generally praised the clarity and helpfulness of the article explaining cross-entropy and KL divergence. Several commenters pointed out the value of the concrete code examples and visualizations provided. One user appreciated the explanation of the difference between minimizing cross-entropy and maximizing likelihood, while another highlighted the article's effective use of simple language to explain complex concepts. A few comments focused on practical applications, including how cross-entropy helps in model selection and its relation to log loss. Some users shared additional resources and alternative explanations, further enriching the discussion.
Francis Bach's "Learning Theory from First Principles" provides a comprehensive and self-contained introduction to statistical learning theory. The book builds a foundational understanding of the core concepts, starting with basic probability and statistics, and progressively developing the theory behind supervised learning, including linear models, kernel methods, and neural networks. It emphasizes a functional analysis perspective, using tools like reproducing kernel Hilbert spaces and concentration inequalities to rigorously analyze generalization performance and derive bounds on the prediction error. The book also covers topics like stochastic gradient descent, sparsity, and online learning, offering both theoretical insights and practical considerations for algorithm design and implementation.
HN commenters generally praise the book "Learning Theory from First Principles" for its clarity, rigor, and accessibility. Several appreciate its focus on fundamental concepts and building a solid theoretical foundation, contrasting it favorably with more applied machine learning resources. Some highlight the book's coverage of specific topics like Rademacher complexity and PAC-Bayes. A few mention using the book for self-study or teaching, finding it well-structured and engaging. One commenter points out the authors' inclusion of online exercises and solutions, further enhancing its educational value. Another notes the book's free availability as a significant benefit. Overall, the sentiment is strongly positive, recommending the book for anyone seeking a deeper understanding of learning theory.
This project explores probabilistic time series forecasting using PyTorch, focusing on predicting not just single point estimates but the entire probability distribution of future values. It implements and compares various deep learning models, including DeepAR, Transformer, and N-BEATS, adapted for probabilistic outputs. The models are evaluated using metrics like quantile loss and negative log-likelihood, emphasizing the accuracy of the predicted uncertainty. The repository provides a framework for training, evaluating, and visualizing these probabilistic forecasts, enabling a more nuanced understanding of future uncertainties in time series data.
Hacker News users discussed the practicality and limitations of probabilistic forecasting. Some commenters pointed out the difficulty of accurately estimating uncertainty, especially in real-world scenarios with limited data or changing dynamics. Others highlighted the importance of considering the cost of errors, as different outcomes might have varying consequences. The discussion also touched upon specific methods like quantile regression and conformal prediction, with some users expressing skepticism about their effectiveness in practice. Several commenters emphasized the need for clear communication of uncertainty to decision-makers, as probabilistic forecasts can be easily misinterpreted if not presented carefully. Finally, there was some discussion of the computational cost associated with probabilistic methods, particularly for large datasets or complex models.
The "inspection paradox" describes the counterintuitive tendency for sampled observations of an interval-based process (like bus wait times or class sizes) to be systematically larger than the true average. This occurs because longer intervals are proportionally more likely to be sampled. The blog post demonstrates this effect across diverse examples, including bus schedules, web server requests, and class sizes, highlighting how seemingly simple averages can be misleading. It explains that the perceived average is actually the average experienced by an observer arriving at a random time, which is skewed toward longer intervals, and is distinct from the true average interval length. The post emphasizes the importance of understanding this paradox to correctly interpret data and avoid drawing flawed conclusions.
Hacker News users discuss various real-world examples and implications of the inspection paradox. Several commenters offer intuitive explanations, such as the bus frequency example, highlighting how our perception of waiting time is skewed by the longer intervals between buses. Others discuss the paradox's manifestation in project management (underestimating task completion times) and software engineering (debugging and performance analysis). The phenomenon's relevance to sampling bias and statistical analysis is also pointed out, with some suggesting strategies to mitigate its impact. Finally, the discussion extends to other related concepts like length-biased sampling and renewal theory, offering deeper insights into the mathematical underpinnings of the paradox.
Autoregressive (AR) models predict future values based on past values, essentially extrapolating from history. They are powerful and widely applicable, from time series forecasting to natural language processing. While conceptually simple, training AR models can be complex due to issues like vanishing/exploding gradients and the computational cost of long dependencies. The post emphasizes the importance of choosing an appropriate model architecture, highlighting transformers as a particularly effective choice due to their ability to handle long-range dependencies and parallelize training. Despite their strengths, AR models are limited by their reliance on past data and may struggle with sudden shifts or unpredictable events.
Hacker News users discussed the clarity and helpfulness of the original article on autoregressive models. Several commenters praised its accessible explanation of complex concepts, particularly the analogy to Markov chains and the clear visualizations. Some pointed out potential improvements, suggesting the inclusion of more diverse examples beyond text generation, such as image or audio applications, and a deeper dive into the limitations of these models. A brief discussion touched upon the practical applications of autoregressive models, including language modeling and time series analysis, with a few users sharing their own experiences working with these models. One commenter questioned the long-term relevance of autoregressive models in light of emerging alternatives.
MIT's 6.S184 course introduces flow matching and diffusion models, two powerful generative modeling techniques. Flow matching learns a deterministic transformation between a simple base distribution and a complex target distribution, offering exact likelihood computation and efficient sampling. Diffusion models, conversely, learn a reverse diffusion process to generate data from noise, achieving high sample quality but with slower sampling speeds due to the iterative nature of the denoising process. The course explores the theoretical foundations, practical implementations, and applications of both methods, highlighting their strengths and weaknesses and positioning them within the broader landscape of generative AI.
HN users discuss the pedagogical value of the MIT course materials linked, praising the clear explanations and visualizations of complex concepts like flow matching and diffusion models. Some compare it favorably to other resources, finding it more accessible and intuitive. A few users mention the practical applications of these models, particularly in image generation, and express interest in exploring the code provided. The overall sentiment is positive, with many appreciating the effort put into making these advanced topics understandable. A minor thread discusses the difference between flow-matching and diffusion models, with one user suggesting flow-matching could be viewed as a special case of diffusion.
This post provides a gentle introduction to stochastic calculus, focusing on the Ito integral. It explains the motivation behind needing a new type of calculus for random processes like Brownian motion, highlighting its non-differentiable nature. The post defines the Ito integral, emphasizing its difference from the Riemann integral due to the non-zero quadratic variation of Brownian motion. It then introduces Ito's Lemma, a crucial tool for manipulating functions of stochastic processes, and illustrates its application with examples like geometric Brownian motion, a common model in finance. Finally, the post briefly touches on stochastic differential equations (SDEs) and their connection to partial differential equations (PDEs) through the Feynman-Kac formula.
HN users generally praised the clarity and accessibility of the introduction to stochastic calculus. Several appreciated the focus on intuition and the gentle progression of concepts, making it easier to grasp than other resources. Some pointed out its relevance to fields like finance and machine learning, while others suggested supplementary resources for deeper dives into specific areas like Ito's Lemma. One commenter highlighted the importance of understanding the underlying measure theory, while another offered a perspective on how stochastic calculus can be viewed as a generalization of ordinary calculus. A few mentioned the author's background, suggesting it contributed to the clear explanations. The discussion remained focused on the quality of the introductory post, with no significant dissenting opinions.
Kartoffels v0.7, a hobby operating system for the RISC-V architecture, introduces exciting new features. This release adds support for cellular automata simulations, allowing for complex pattern generation and exploration directly within the OS. A statistics module provides insights into system performance, including CPU usage and memory allocation. Furthermore, the transition to a full 32-bit RISC-V implementation enhances compatibility and opens doors for future development. These additions build upon the existing foundation, further demonstrating the project's evolution as a versatile platform for low-level experimentation.
HN commenters generally praised kartoffels for its impressive technical achievement, particularly its speed and small size. Several noted the clever use of RISC-V and efficient code. Some expressed interest in exploring the project further, looking at the code and experimenting with it. A few comments discussed the nature of cellular automata and their potential applications, with one commenter suggesting using it for procedural generation. The efficiency of kartoffels also sparked a short discussion comparing it to other similar projects, highlighting its performance advantages. There was some minor debate about the project's name.
This paper presents a simplified derivation of the Kalman filter, focusing on intuitive understanding. It begins by establishing the goal: to estimate the state of a system based on noisy measurements. The core idea is to combine two pieces of information: a prediction of the state based on a model of the system's dynamics, and a measurement of the state. These are weighted based on their respective uncertainties (covariances). The Kalman filter elegantly calculates the optimal blend, minimizing the variance of the resulting estimate. It does this recursively, updating the state estimate and its uncertainty with each new measurement, making it ideal for real-time applications. The paper derives the key Kalman filter equations step-by-step, emphasizing the underlying logic and avoiding complex matrix manipulations.
HN users generally praised the linked paper for its clear and intuitive explanation of the Kalman filter. Several commenters highlighted the value of the paper's geometric approach and its focus on the underlying principles, making it easier to grasp than other resources. One user pointed out a potential typo in the noise variance notation. Another appreciated the connection made to recursive least squares, providing further context and understanding. Overall, the comments reflect a positive reception of the paper as a valuable resource for learning about Kalman filters.
The blog post "Kelly Can't Fail" argues against the common misconception that the Kelly criterion is dangerous due to its potential for large drawdowns. It demonstrates that, under specific idealized conditions (including continuous trading and accurate knowledge of the true probability distribution), the Kelly strategy cannot go bankrupt, even when facing adverse short-term outcomes. This "can't fail" property stems from Kelly's logarithmic growth nature, which ensures eventual recovery from any finite loss. While acknowledging that real-world scenarios deviate from these ideal conditions, the post emphasizes the theoretical robustness of Kelly betting as a foundation for understanding and applying leveraged betting strategies. It concludes that the perceived risk of Kelly is often due to misapplication or misunderstanding, rather than an inherent flaw in the criterion itself.
The Hacker News comments discuss the limitations and practical challenges of applying the Kelly criterion. Several commenters point out that the Kelly criterion assumes perfect knowledge of the probability distribution of outcomes, which is rarely the case in real-world scenarios. Others emphasize the difficulty of estimating the "edge" accurately, and how even small errors can lead to substantial drawdowns. The emotional toll of large swings, even if theoretically optimal, is also discussed, with some suggesting fractional Kelly strategies as a more palatable approach. Finally, the computational complexity of Kelly for portfolios of correlated assets is brought up, making its implementation challenging beyond simple examples. A few commenters defend Kelly, arguing that its supposed failures often stem from misapplication or overlooking its long-term nature.
This blog post presents a different way to derive Shannon entropy, focusing on its property as a unique measure of information content. Instead of starting with desired properties like additivity and then finding a formula that satisfies them, the author begins with a core idea: measuring the average number of binary questions needed to pinpoint a specific outcome from a probability distribution. By formalizing this concept using a binary tree representation of the questioning process and leveraging Kraft's inequality, they demonstrate that -∑pᵢlog₂(pᵢ) emerges naturally as the optimal average question length, thus establishing it as the entropy. This construction emphasizes the intuitive link between entropy and the efficient encoding of information.
Hacker News users discuss the alternative construction of Shannon entropy presented in the linked article. Some express appreciation for the clear explanation and visualizations, finding the geometric approach insightful and offering a fresh perspective on a familiar concept. Others debate the pedagogical value of the approach, questioning whether it truly simplifies understanding for those unfamiliar with entropy, or merely offers a different lens for those already versed in the subject. A few commenters note the connection to cross-entropy and Kullback-Leibler divergence, suggesting the geometric interpretation could be extended to these related concepts. There's also a brief discussion on the practical implications and potential applications of this alternative construction, although no concrete examples are provided. Overall, the comments reflect a mix of appreciation for the novel approach and a pragmatic assessment of its usefulness in teaching and application.
Summary of Comments ( 45 )
https://news.ycombinator.com/item?id=44105470
HN commenters discuss Bill Benter's horse racing prediction model, praising its statistical rigor and innovative approach. Several highlight the importance of feature engineering and data quality, emphasizing that Benter's edge came from meticulous data collection and refinement rather than complex algorithms. Some note the parallels to modern machine learning, while others point out the unique challenges of horse racing, like limited data and dynamic odds. A few commenters question the replicability of Benter's success today, given the increased competition and market efficiency. The ethical considerations of algorithmic gambling are also briefly touched upon.
The Hacker News post titled "Revisiting the algorithm that changed horse race betting (2023)" linking to an annotated version of Bill Benter's paper has generated a moderate amount of discussion. Several commenters focus on the complexities and nuances of Benter's approach, moving beyond the simplified narrative often presented.
One compelling point raised is the crucial role of accurate data. Multiple comments emphasize that Benter's success wasn't solely due to a brilliant algorithm, but heavily reliant on obtaining and cleaning high-quality data, a task that required significant effort and resources. This highlights the often overlooked aspect of data integrity in machine learning successes. One commenter even suggests that Benter's real edge was his superior data collection and processing, rather than the algorithm itself.
Another key theme revolves around the idea of diminishing returns and the efficient market hypothesis. Commenters discuss how Benter's success likely influenced the market, making it more efficient and thus harder for similar strategies to achieve the same level of profitability today. This illustrates the dynamic nature of prediction markets and how successful strategies can eventually become self-defeating. The discussion touches on the constant need for adaptation and refinement in such environments.
Some commenters delve into the technical aspects of Benter's model, mentioning the challenges of overfitting and the importance of feature selection. They acknowledge the impressive nature of building such a system in the pre-internet era with limited computational power. The discussion around feature engineering hints at the depth and complexity of Benter's work, going beyond simply plugging data into an algorithm.
Finally, a few comments provide interesting anecdotes and context, like mentioning Benter's collaboration with Alan Woods and the broader landscape of quantitative horse racing betting. These comments enrich the discussion by providing a historical perspective and highlighting the collaborative nature of such endeavors.
Overall, the comments section offers valuable insights into the practical realities and complexities of applying quantitative methods to prediction markets, moving beyond the often romanticized narratives of algorithmic success. They emphasize the importance of data quality, the dynamic nature of markets, and the ongoing need for adaptation and refinement in the face of competition and changing conditions.