# What is the interpretation of log-loss value?

Does anyone have the interpretation of a log-loss value? Am I correct to assume that values closer to 0 and 1 are more likely to be an indication that the predicted value is incorrect?

Let $$D = \{(x_1, y_1), \ldots, (x_n, y_n)\}$$ be a set of i.i.d. observations where $$x_i$$ is some $$n$$ dimensional vector of independent variables and $$y_i$$ is a binary dependent variable. A common assumption is to assume $$y_i \sim Ber(f(x_i))$$ for some function $$f\colon X \to [0,1]$$. To model this assumption we can use some parameterized function $$h(\cdot; \theta)$$. Taking $$h(x;\theta) = \left(1 + e^{-\theta^T x}\right)^{-1}$$ with $$\theta \in \mathbb{R}^n$$ yields logistic regression. Taking $$h(x;\theta = \{w, W\}) = \left(1 + e^{-w^T\sigma(Wx)}\right)^{-1}$$ with $$W \in \mathbb{R}^{m \times n}$$, $$w \in \mathbb{R}^m$$, and $$\sigma$$ an elementwise sigmoid function yields a standard feed forward neural network for binary classification.
The log-loss arises when we ask how should we should choose the value of $$\theta$$ given our available data. The maximum likelihood estimate (MLE) for $$\theta$$ is given by \begin{align*} \theta^* &= \arg\max_{\theta}\left\{\prod_{i=1}^n P(y_i \mid h(x_i; \theta)) \right\}\\ &= \arg\max_{\theta}\left\{\prod_{i=1}^n h(x_i; \theta)^{y_i}(1 - h(x_i;\theta))^{1-y_i}\right\}\\ &= \arg\max_{\theta}\left\{\sum_{i=1}^n y_i\log(h(x_i;\theta)) + (1-y_i)\log(1-h(x_i;\theta))\right\}\\ &= \arg\min_{\theta}\left\{\sum_{i=1}^n -y_i\log(h(x_i;\theta)) - (1-y_i)\log(1-h(x_i;\theta))\right\}.\\ \end{align*}