Understanding Log Likelihood: Why and How to Use It
0:00:09 Joint Likelihood: The goal is to determine the joint likelihood of a model's parameters given a set of data points.
0:00:15 Independence Assumption: Typically, it's assumed that each data point is independent and identically distributed (IID), meaning the location of one point doesn't depend on others, and one model describes all points.
0:00:29 Probability Product: Under the IID assumption, the probability of all points is the product of probabilities at each point; similarly, total likelihood is the product of individual likelihoods.
0:00:40 Log Likelihood Definition: Log likelihood is the logarithm of the likelihood. The log is used to turn products into sums and exponents into products simplifying calculations.
0:01:04 Preventing Underflow: Multiplying many probabilities (each less than 1) can quickly lead to underflow (approaching zero and becoming unrepresentable), making log likelihood essential.
0:01:16 Likelihood Comparison: Likelihood is usually used to compare the likelihood of one model with another. The important part is whether the likelihood of one model is greater than the other model.
0:01:35 Monotonically Increasing Function: The logarithm is a monotonically increasing function, preserving the "greater than" relationship when comparing likelihoods. Taking the logarithm of both sides does not change which likelihood is greater.
0:01:47 Log Likelihood Plot: Graphically, as likelihood increases, log likelihood also increases, maintaining relative comparison values.