Have you ever wondered why include the logarithm in the definition of log-likelihood?
The answer is simple: logarithm makes differentiation of products easier.
Let's see why!
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Although the derivative of a sum is the sum of derivatives, a similar property cannot be stated about the product of functions.
The derivative of a product is slightly more complicated: it is a sum of products.
The formula gets even more complicated when we have more functions in the product.
When potentially hundreds of terms are present, like in the likelihood function, computing this is not feasible.
This is where taking the logarithm can help us: it turns multiplication into addition!
So, instead of differentiating a product of ๐ functions, we have to deal with the sum!
Since the logarithm strictly increasing, it won't interfere with optimization either: the optimums are attained at the same place.
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