Taylor’s Formula

If $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots$,
$$a_n = \frac{f^{(n)}(0)}{n!}$$
with another base point
$$f(x) = f(b) + f’(b)(x-b) + \frac{f’’(b)}{2} (x-b)^2 + \frac{ f^{(3)}(b)}{3!}(x-b)^3 + \cdots$$
For example, $\sqrt{x}$ is not appropriate to expand at $0$ because it’s not differentiable at $0$. so use $b= 1$,
$$x^{\frac{1}{2}} = 1 + \frac{1}{2}(x-1) + \frac{ \left( \frac{1}{2} \right) \left( \frac{1}{2} - 1 \right) }{2!} (x-1)^2 + \cdots$$

likelihood function

어떤 probability distribution(parameter $\theta$)에서 sample $\mathbb x$를 뽑을 확률을 $p$라고 하면,
$$L(\theta | \mathbb x) = p (=p(\mathbb x | \theta) )$$
이다. 예를들어 Bernoulli distribution($Bin(n,\pi)$)은 다음과 같이 정의되는데,
$$f(x;\pi) = \pi ^ x (1-\pi)^{(1-x)} , x = 0 , 1, \pi \text{ is unknown parameter}$$
likelihood function은 다음과 같다.
$$L(\pi | x) = \frac{x!}{(n-x)!x!} \pi ^x (1-\pi)^{(n-x)}$$
계산상 편의를 이유로 Log를 취한 log likelihood가 자주 쓰인다.

Gibbs sampling

1. initial value $X^{(0)}$
2. update $j$th component by $p(x_j | x_1^{(i+1)}, \cdots , x_{j-1}^{(i+1)}, x_{j+1}^{(i)}, \cdots , x_n^{(i)})$
3. repeat

(from wikipedia)


cf. block gibbs sampling
update  $j$th component by $p(x_j | x_1^{(i)}, \cdots , x_{j-1}^{(i)}, x_{j+1}^{(i)}, \cdots , x_n^{(i)})$

words

reassess
preclude
correlate n.

contingent

salient
latent

privy  [|prɪvi]

endow
isotropic
superimpose
assess
cluttered
rigor

divergence theorem

$\iint\limits_S F \cdot dS = \iiint\limits_E \nabla F dV$

Euclidean algorithm

출처 : https://en.wikipedia.org/wiki/Euclidean_algorithm#/media/File:Euclidean_algorithm_1071_462.gif

wikipedia의 Euclidean Algorithm항목에 있는 gif animation인데 호제법을 이해하는데 이보다 뛰어난 그림을 본 적이 없다. 아주 직관적이다.

Pattern Recognition이

Clustering이랑 결국 같은건가?


그러니까..
3, 5, 7, ( ), 11 에서 괄호 안에 들어갈 숫자를 맞추는 작업이
clustering과 본질적으로 같은 문제인가?



Is Patter Recognition equal to clustering problem?
I mean, for example, if there is a series with an unknown number, is guessing that substantially equal to the problem of clustering?