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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. $$\mathrm{owens\_t}(h,a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx$$