A probabilistic forecast represents an estimation of the respective probabilities for all the possible future outcomes of a random variable. In contrast to single-valued forecasts, such as median time-series forecasts or quantile forecasts, the probability forecast represents a probability density function. Probabilistic forecasts can be applied to numerous domains ranging from weather forecasting to sports betting, but they are especially useful for supply chain optimization.

Single-valued forecasts tend to focus on mean or median situations which do not properly reflect the extreme situations described above. Indeed, the problem does not lie in the quality of the forecasts – which might be more or less accurate – but rather in the very definition of the forecasts themselves: a mean forecast does not address the question of extreme events. This issue is traditionally addressed in supply chain through safety stock analysis. However, this analysis usually relies on strong assumptions, such as a normal distribution of the demand, which are highly inaccurate when it comes to supply chain.

In contrast, probabilistic demand forecasts offer the possibility to implement prioritized ordering policies which outperform more classic ordering policies because they exploit the precise structure of the probabilities estimated for future demand.

We can model future demand at time $t+h$ for each forecasting horizon $h$ as follows: $$y_{t+h}=g_h(\mathbf{y}_t)+\epsilon_{t+h}$$ Where:

- $g_h$ is a forecasting model specific of the horizon $h$
- $\epsilon_{t+h}$ denotes the model error

At this point, $g_h$ is still a single-valued forecasting model. We can adjust this definition into a probabilistic forecast by considering the following: $$Y_{t+h}=G_h(\mathbf{y}_t)$$ Where $G$ returns not a single value $y_{t+h}$, but a random variable $Y_{t+h}$ with an explicit density distribution, that is $P(y_{t+h}\leq y | \mathbf{y}_t)$. While this would go beyond the scope of the present entry, it can be noted that probabilistic forecasting can be reduced to the estimation of the cumulative distribution function.

Unless some specific assumptions can be made about the probability distribution of the demand, histograms introduce a degree of numerical approximation when representing the demand. Indeed, many probability distributions are unbounded, with non-zero probabilities found for arbitrarily large values. Such distributions cannot be perfectly represented through finite histograms. Similarly, by construction, the histogram averages the fine structure of the distribution within each bucket.

While histograms require significantly more computing resources in order to be processed compared to single values (i.e. non-probabilistic forecasts), processing histograms for very large numbers of time-series (or more) is well within the range of capabilities of modern computing systems. In practice, histograms can be designed to contain a sufficient number of buckets to make sure that the numerical imprecisions introduced through the discretization of the demand is negligible compared to the uncertainty of the forecast itself.