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Summary: This note introduces Ripley’s $K$ and Besag’s $L$ functions for analyzing spatial point patterns. It explains CSR as the null model, defines and estimates $K(r)$ with edge corrections, variance-stabilization via $L(r)$, and the need for Monte Carlo envelopes. Redwood data illustrate clustering versus intensity-driven inhomogeneity.
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Summary: This note explains why logistic regression is suited for classification. Unlike linear regression, probabilities must remain between 0 and 1. The logistic function enforces this constraint, mapping linear predictors into valid probabilities. Its log-odds formulation ensures interpretability, making logistic regression a well-defined, probabilistic approach for binary classification tasks.
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Summary: This note explains shell file descriptors and redirection. It covers stdin, stdout, stderr, and how to redirect or merge them with operators like >, 2>, and 2>&1. It shows differences between & in redirection and background execution, common command flags, pitfalls, examples, and best practices for HPC logging.
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Summary: This note clarifies the distinction between standard deviation and standard error. Standard deviation measures variability within a dataset, serving as a descriptive statistic, while standard error measures variability of sample means across repeated samples, serving as an inferential statistic. Derivations include Bessel’s correction for unbiased variance estimation and the SE formula.
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Summary: This note reinterprets stochastic gradient descent by modeling sample gradients as random variables. It derives the sample covariance of gradients and shows that the mini-batch gradient is an unbiased estimator of the full gradient, with variance scaling as $\frac{1}{m}$. This variance reflects stochastic noise around the true gradient direction.
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Summary: This note explains variational inference (VI) as a scalable alternative to MCMC for approximate Bayesian inference. It introduces entropy, cross-entropy, and KL divergence, then contrasts forward (zero-avoiding) vs. reverse (zero-forcing) KL. The ELBO is derived as a tractable optimization target, enabling posterior approximation via Monte Carlo estimation, reparameterization, and mini-batching.
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Summary: This report presents a simpler way to format Bayesian hierarchical model for hypertension prevalence, using Singapore cohort data. It applies matrix-structured parameterization in MCMC to replace redundant intermediate variables from spline expansions, accelerating computation. This reduces runtime from 60 to 5 hours while maintaining convergence and accuracy, enabling efficient large-scale hypertension prevalence modeling.
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Summary: This note records my understanding and supplementary proofs of Schmertmann et al. (2014)’s Bayesian fertility forecasting framework. It details notation, penalty structures for shape and time, covariance prior construction, and likelihood–prior integration. The write-up illustrates posterior fertility forecasts, using Singapore as an example, while emphasizing personal interpretation and derivations.