# Misc Notes

## Fourier

differentials $$(\mathcal{F} f^{(n)})(s) = (2 \pi i s)^n \mathcal{F} f (s)$$ $$P(\frac{d}{dx}) = a_n(\frac{d}{dx})^n + ... + a_1 \frac{d}{dx} + a_0$$ $$(\mathcal{F} (P(\frac{d}{dx}) f))(s) = P(2 \pi i s) \mathcal{F} f (s) = (\sum_{i=0}^{n} a_i (2 \pi i s)^i) \mathcal{F} f (s)$$
duality & transformations $$\mathcal{F} f = (\mathcal{F}^{-1} f)^{-}$$ $$\mathcal{F} f^{-} = \mathcal{F}^{-1} f$$ shift $$f(t \pm b) \rightleftharpoons e^{\pm 2 \pi i s b} F(s)$$ stretch $$f(a t) \rightleftharpoons \frac{1}{|a|} F(\frac{s}{a})$$
coefficients $$c_n = \frac{1}{T} \int_{-T/2}^{T/2} e^{-2 \pi i n t / T} f(t) dt$$ Often put $\frac{1}{\sqrt T}$ into harmonic $\frac{1}{\sqrt T} e^{-2 \pi i n t / T}$

## Machine Learning

common distributions (less familiar ones)
Multinomial $$Mu(\mathbf{x} \mid n, \mathbf{\theta}) \triangleq {{n}\choose{x_1...x_K}} \prod_{j=1}^{K} \theta_{j}^{x_j}$$ where $${{n}\choose{x_1...x_K}} \triangleq \frac{n!}{x_1!...x_K!}$$ Rolling $K$ sided die once ("Multinoulli"): $$Cat(x \mid \mathbf{\theta}) \triangleq Mu(x \mid 1, \mathbf{\theta})$$ Gamma (for $T, a, b$ in $\mathbb{R}^+$) $$Ga(T \mid shape = a, rate = b) \triangleq \frac{b^a}{\Gamma(a)} T^{a-1} e^{-Tb}$$ where $$\Gamma(x) \triangleq \int_{0}^{\infty} u^{x-1} e^{-u} du$$ $$Exp(x \mid \lambda ) = Ga(x \mid 1, \lambda)$$ $$\chi^2(x \mid \nu) = Ga(x \mid \frac{\nu}{2}, \frac{1}{2})$$ Beta $$Beta(x \mid a, b) = \frac{1}{B(a,b)} x^{a-1} (1-x)^{b-1}$$ where $$B(a,b) \triangleq \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$$
transformations $$p_y(y) = p_x(x) \left\lvert \frac{dx}{dy} \right\rvert$$

Top