Im reading this paper: Uncertainty in Deep Learning and in it (page4), the softmax loss is defined as

\begin{align*} E(X,Y) = -\frac{1}{N} \sum^N_{n=1} \log(\hat{p}_{n,c_n}), \end{align*}

where $c_n \in \{1,...,D\}$ is the class label for input $n$, and

$\hat{p}_{nd} = \exp(\hat{y}_{nd})/(\sum_{d'} \exp(\hat{y}_{nd'})$ is the element-wise softmax-function applied to model prediction vector $\hat{y}$.

Where does the class label $d$ actually show up in the softmax calculation? I.e. what does $\hat{y}_{nd}$ or $\hat{p}_{n,c_n}$ respectively shall mean?


  • $\begingroup$ The $y_i$ is either 0 or 1. $\endgroup$ May 24, 2020 at 10:44
  • $\begingroup$ yes Sir, thats correct for (binary) classification $\endgroup$
    – MJimitater
    May 24, 2020 at 10:52
  • $\begingroup$ There are $D$ classes. $\hat{y}_{nd}$ for $d \in \{1,\dots,D\}$ are the probabilities for the $n$th sampling unit, and $\hat{y}_{n,c_n}$ is the largest probability (of the $D$ probabilities). $\endgroup$
    – papgeo
    May 24, 2020 at 13:09
  • $\begingroup$ Thanks @papgeo for your answer. Your explaination of $\hat{y}_{nd}$ makes sense, thank you! And you mean $\hat{p}_{n,c_n}$ to be the largest probability? $\endgroup$
    – MJimitater
    May 24, 2020 at 15:21
  • $\begingroup$ If you formulate those insights into an answer, Im willing to accept that. $\endgroup$
    – MJimitater
    May 24, 2020 at 15:22

1 Answer 1


We are given $N$ input-output pairs: $\{(\textbf{x}_1,y_1),\dots,(\textbf{x}_N,y_N)\}$. The $\textbf{x}$'s are input vectors, also known as covariates or predictors, and the $y$'s are the labels. The problem here is to learn how to predict a class label given the vector of inputs $\textbf{x}$.

$\hat{y}_{nd}$ is what brings the $\textbf{x}$'s into the computations. It is also known as the linear predictor and it is given by $\hat{y}_{nd} = \beta_{0d} + \beta_{1d} x_{1n} + \dots + \beta_{pd} x_{pn}$, where the $\beta$'s are regression coefficients (to be estimated) and $x_{1n}, \dots, x_{pn}$ are the elements of the input vector $\textbf{x}_n$.

$\hat{p}_{n,c_n}$ is the probability of the class that sampling unit $n$ belongs to.


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