As machine learning practitioners, we’ve all been there – stuck deciding between linear regression and Poisson regression for our modeling needs. But what exactly are the key differences between these two techniques, and when should we opt for Poisson regression over its linear counterpart?
To answer this, let’s dive into the assumptions behind each model. In linear regression, we assume that the dependent variable (y) given the independent variable (x) follows a Gaussian distribution with a mean that can be linearly decomposed into our features. On the other hand, Poisson regression assumes that y|x follows a Poisson distribution, where the mean can be written as the exponential of a linear combination of our features.
So, what prevents us from using linear regression even when the underlying distribution y|x follows a Poisson distribution? The short answer is that linear regression can still work, but it might not be the best choice. In Poisson regression, the variance of the response variable is equal to its mean, which is not the case in linear regression. This means that linear regression might not accurately capture the variability in our data, leading to poor predictions.
But how can we demonstrate the superiority of Poisson regression in practice? One way is to experiment with real datasets that exhibit Poisson distributions. For instance, we could use the famous `Lahman` baseball dataset from Kaggle, which includes count data that can be modeled using Poisson regression. By comparing the performance of linear and Poisson regression on this dataset, we can see firsthand how Poisson regression outperforms its linear counterpart.
In conclusion, while linear regression can still be used when the underlying distribution is Poisson, it’s essential to understand the assumptions behind each model and choose the right tool for the job. By doing so, we can ensure that our models accurately capture the variability in our data and provide reliable predictions.