In many regression tasks, the target variable is a continuous value that is strictly greater than zero, meaning $y^{(i)} \in (0, \infty)$.

Applications of Positive Continuous Targets

Unlike count data (which must be integers), these targets can take on any fractional value but cannot be negative. Common real-world examples include:

• Reliability Engineering: Predicting the "time-to-failure" of a mechanical component or the lifespan of an electronic device.
• Finance & Insurance: Estimating the monetary size of an insurance claim, customer lifetime value (CLV), or real estate pricing, where costs are always positive and often exhibit large outliers.
• Healthcare: Modeling a patient's length of stay in a hospital or the time until a specific medical event occurs.
• Logistics & Transportation: Predicting the exact distance of a delivery route or the time it takes to complete a specific trip.

Definition of Gamma Distribution

The Gamma distribution is a continuous probability distribution defined exclusively for positive real numbers.

Because it is bounded at zero and can flexibly model heavily right-skewed data (where lower values are common but extreme high values occasionally occur), it is highly suitable for predicting positive continuous targets in regression. Real-world data like transaction costs or waiting times rarely follow a perfectly symmetric "bell curve" and are strictly non-negative, making the Gamma distribution a much better fit than a standard Gaussian (Normal) distribution.

If a continuous random variable $Y$ follows a Gamma distribution parameterized by a shape parameter $\alpha$ and a scale parameter $\rho$, its probability density function (PDF) is given by:

$$ p(y ; k,\rho) = \frac{1}{\Gamma(k)\left(\rho\right)^{k}} \left(y\right)^{k-1} \exp\left(-\frac{y}{\rho}\right) $$

Where:

• $y > 0$ is the continuous target value.
• $k > 0$ is the shape parameter, which controls the overall profile of the distribution.
• $\rho > 0$ is the scale parameter, which stretches or squeezes the distribution.
• $\Gamma(k)$ is the Gamma function, which serves as a normalizing constant to ensure the probabilities integrate to 1.

Conditioned Likelihood

To model this probabilistically, we assume that the target variable $y$, given the input $x$, follows a Gamma distribution parameterized by a shape $k > 0$ and a scale $\rho> 0$.

$$ y^{(i)} \mid x^{(i)},\theta \sim \text{Gamma}(k^{(i)}, \rho^{(i)}) $$