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varphi(\Phi^{-1}(u)) \varphi(\Phi^{-1}(v))} \>, $$ where the numerator is the bivariate normal distribution with correlation $\rho$ evaluated at $\Phi^{-1}(u)$ and $\Phi^{-1}(v)$. … (x),\Phi(y))$ is $$ f(x,y) = \varphi(x) \varphi(y) c(\Phi(x), \Phi(y)) \> . $$ Note that by applying the Gaussian copula in the above equation, we recover the bivariate normal density. …

In Logit: $\Pr(Y=1 \mid X) = [1 + e^{-X'\beta}]^{-1} $ In Probit: $\Pr(Y=1 \mid X) = \Phi(X'\beta)$ (Cumulative standard normal pdf) In other way, logistic has slightly flatter tails. i.e the probit …

Alternatively, it can be expressed as $$a^i_j = \sigma(z^i_j) = \phi(z^i_j)$$. Where $\phi $is the Cumulative distribution function (CDF). See here for means of approximating this. …

Under regularity assumptions, we obtain the following asymptotic expansion for the cdf of the test statistic $T_n$: $$P(T_n\leq x)=\Phi(x)+n^{-1/2}\frac{1}{6}\gamma(2x^2+1)\phi(x)-n^{-1}x\Big(\frac{1}{ … 12}\kappa (x^2-3)-\frac{1}{18}\gamma^2(x^4+2x^2-3)-\frac{1}{4}(x^2+3)\Big)\phi(x)+o(n^{-1}),$$ where $\Phi(\cdot)$ is the cdf and $\phi(\cdot)$ is the pdf of the standard normal distribution. …

Instead, they are found (extracted) so as to minimize the angle $\phi$ between them. Cosine of that angle is the canonical correlation. …

The dispersion parameter $\phi$ indicates if we have more or less than the expected variance. … If $\phi=1$ this means we observe the expected amount of variance, whereas $\phi<1$ means that we have less than the expected variance (called underdispersion) and $\phi>1$ means that we have extra variance …

t) \phi(t) dt = \frac{-i}{2\pi} \int_{-\infty}^\infty t \exp(-i x t) \phi(t) dt.$$ For this to be well-defined, the last integral must converge absolutely; that is, $$\int_{-\infty}^\infty |t \exp(- … i x t) \phi(t)| dt = \int_{-\infty}^\infty |t| |\phi(t)| dt$$ must converge to a finite value. …

Intuitively, in its original form, VAEs sample from a random node $z$ which is approximated by the parametric model $q(z \mid \phi, x)$ of the true posterior. …

For $i=1,\dots,n$ independent observations, the distribution of each $y_i$ is an exponential family with density $$ f(y_i;\theta_i,\phi)=\exp\left(\frac{y_i\theta_i-\gamma(\theta_i)}{\phi}+\tau(y_i,\phi … )\right) = \alpha(y_i, \phi)\exp\left(\frac{y_i\theta_i-\gamma(\theta_i)}{\phi}\right) $$ Here, the parameter of interest (natural or canonical parameter) is $\theta_i$, $\phi$ is a scale parameter (known …

\left(\mathbf{x}^{\text {test}}\right)+b\right) \\ &= \text {sign}\left(\sum_{i =1}^{N}\alpha^{(i)}y^{(i)}\phi\left(x^{(i)}\right)^T\phi\left(\mathbf{x}^{\text {test}}\right)+b \right) \end{align*} The … Kernel trick One can observe that the optimization problem uses the $\phi\left(\mathbf{x}^{(i)}\right)$ only in the inner product $\phi\left(\mathbf{x}^{(i)}\right)^T \phi\left(\mathbf{x}^{(j)}\right) …

x_1^2, \ x_2^2) \ \begin{pmatrix} \sqrt{2}x_1'x_2' \\ x_1'^2 \\ x_2'^2 \end{pmatrix} \end{aligned} $$ k(\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} ) = \phi … (\mathbf{x})^T \phi(\mathbf{x'})$$ $$ \phi(\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}) =\begin{pmatrix} \sqrt{2}x_1x_2 \\ x_1^2 \\ x_2^2 \end{pmatrix}$$ Visualizing the feature map and the resulting boundary …

}\phi(w)\,\mathrm dw = \int_{-\infty}^\infty \Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw.$$ Now, $P\{X \leq Y\} = P\{X-Y \leq 0\}$ can be expressed in terms of $\Phi(\cdot)$ by noting that $X-Y \ … sim N(a,b^2+1)$, and thus we get $$\int_{-\infty}^\infty \Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw = \Phi\left(\frac{-a}{\sqrt{b^2+1}}\right)$$ which is the same as the result in whuber's answer …

It is defined as the arithmetic mean of the class-specific accuracies, $\phi := \frac{1}{2}\left(\pi^+ + \pi^-\right),$ where $\pi^+$ and $\pi^-$ represent the accuracy obtained on positive and negative …

We have then $$ \begin{align} F_Y (y) = P\left( |X|\leq \sqrt{y} \right) = P\left(-\sqrt{y} < X <\sqrt{y} \right) = \Phi\left(\sqrt{y}\right) - \Phi \left(- \sqrt{y}\right) \end{align} $$ where $\Phi … phi(\sqrt{y}) $$ where $\phi(.)$ is now the pdf of a standard normal variable and we have used the fact that it is symmetric about zero. …

The main idea is that you approximate $f$ by a mixture of continuous distributions $K$ (using your notation $\phi$), called kernels, that are centered at $x_i$ datapoints and have scale (bandwidth) equal …