How does the Brier Score break down to (Reliability - Resolution + Uncertainty)? The Wikipedia page states this in the decompositions section, and it is also stated in an older paper
I have never been able to understand these explanations, and I wonder what I am missing and if there is an easy way to understand this...
How does one get from this...
$$
BS = \frac{1}{N}\sum\limits _{t=1}^{N}\sum\limits _{i=1}^{R}(f_{ti}-o_{ti})^2
$$
To this...
$$
\begin{aligned}
    BS&=REL-RES+UNC \\
    BS&=\frac{1}{N}\sum\limits _{k=1}^{K}{n_{k}(\mathbf{f_{k}}-\mathbf{\bar{o}}_{\mathbf{k}})}^{2}-\frac{1}{N}\sum\limits _{k=1}^{K}{n_{k}(\mathbf{\bar{o}_{k}}-\bar{\mathbf{o}})}^{2}+\mathbf{\bar{o}}\left({1-\mathbf{\bar{o}}}\right)
\end{aligned}
$$

The Brier score (BS) is a measure of the accuracy of probabilistic predictions. It is calculated by taking the mean squared error between the predicted probabilities and the true outcomes.
The first equation you provided is the general form of the Brier score, which takes into account all of the predicted probabilities for all of the individual predictions that have been made.
The second equation you provided decomposes the Brier score into three components: the reliability (REL), the resolution (RES), and the uncertainty (UNC). These three components provide insight into the different factors that contribute to the overall Brier score, and allow you to analyze the performance of a probabilistic prediction system in more detail.
The two equations are equivalent.
The relative accuracy measures how close the predicted probabilities are to the observed frequencies. The resolution measures how well the predictions can distinguish between events with different frequencies. And the uncertainty measures the average predicted probability.
The second equation is obtained from the first one by introducing these three components and rearranging the terms. In particular, the relative accuracy is defined as:
$$ REL = \frac{1}{N}\sum\limits _{k=1}^{K}{n_{k}(\mathbf{f_{k}}-\mathbf{\bar{o}}_{\mathbf{k}})}^{2} $$
The resolution is defined as:
$$ RES = -\frac{1}{N}\sum\limits _{k=1}^{K}{n_{k}(\mathbf{\bar{o}_{k}}-\bar{\mathbf{o}})}^{2} $$
And the uncertainty is defined as:
$$ UNC = \mathbf{\bar{o}}\left({1-\mathbf{\bar{o}}}\right) $$
The Brier score is then obtained by adding these three components:
$$ BS = REL + RES + UNC $$
I hope this helps clarify the relationship between the two equations you provided.
