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3.12 Probability-Related Functions

3.12.1 Normal Cumulative Distribution Functions

The error function erf is related to the standard normal cumulative distribution function \(\Phi\) by scaling. See section normal distribution for the general normal cumulative distribution function (and its complement).

R erf(T x)
error function, also known as the Gauss error function, of x

R erfc(T x)
complementary error function of x

R Phi(T x)
standard normal cumulative distribution function of x

R inv_Phi(T x)
standard normal inverse cumulative distribution function of p, otherwise known as the quantile function

R Phi_approx(T x)
fast approximation of the unit (may replace Phi for probit regression with maximum absolute error of 0.00014, see (Bowling et al. 2009) for details)

3.12.2 Other Probability-Related Functions

real binary_log_loss(int y, real y_hat)
Return the log loss function for for predicting \(\hat{y} \in [0,1]\) for boolean outcome \(y \in \{0,1\}\). \[ \mathrm{binary\_log\_loss}(y,\hat{y}) = \begin{cases} -\log \hat{y} & \text{if } y = 0\\ -\log (1 - \hat{y}) & \text{otherwise} \end{cases} \]

real owens_t(real h, real a)
Return the Owen’s T function for the probability of the event \(X > h\) and \(0<Y<aX\) where X and Y are independent standard normal random variables. \[ \mathrm{owens\_t}(h,a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]