Introduction I intend to write a blog about knowledge in Mathematics, Programming, AI, Lean4. However, it’s difficult to write such blog better than an article with the same topic in a textbook. So, I change my intention into write a personal blog whose contents will focus more on my personal thoughts, philosophy and maybe few mathematical details only if it has a meaningful story behind. Bests, Minh Hai
Introduction Suppose we observe pairs $(X_1, Y_1), \ldots, (X_n, Y_n)$ where we want to estimate the regression function $$ m(x) = \mathbb{E}[Y \mid X = x]. $$The classical Nadaraya–Watson estimator is defined as $$ \hat{m}_n(x) = \frac{\sum_{i=1}^n K_h(x - X_i)\, Y_i}{\sum_{i=1}^n K_h(x - X_i)}, $$where $K_h(u) = h^{-1} K(u/h)$ is a kernel with bandwidth $h > 0$. The Recursive Version In my thesis, I studied a recursive variant suited for streaming data. Define the sequence of estimates: ...
Inline Math Einstein’s famous equation: $E = mc^2$ Display Math The Gaussian integral: $$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} $$Maxwell’s equations in differential form: $$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $$Code Snippet 1 2 3 4 5 6 7 8 def fibonacci(n: int) -> int: """Return the nth Fibonacci number.""" if n <= 1: return n a, b = 0, 1 for _ in range(2, n + 1): a, b = b, a + b return b