0:00:09 Joint Likelihood: Calculation of the model's parameters given a set of data points, assuming each data point is independent and identically distributed.
0:00:29 Probability and Likelihood Products: The probability/likelihood of all data points is the product of the probabilities/likelihoods at each individual point.
0:00:40 Log Likelihood Defined: Log likelihood is the logarithm of the likelihood, which transforms products into sums and exponents into products. This is mathematically advantageous.
0:01:04 Preventing Underflow: Multiplying many probabilities can lead to underflow, where the number becomes too small for the computer to represent properly. Log likelihood avoids this issue by converting products into sums.
0:01:20 Comparative Nature of Likelihood: Likelihood is usually used to compare different models. Only the relative values matter.
0:01:35 Monotonically Increasing Function: Logarithm is a monotonically increasing function, meaning it never decreases. Therefore, the greater than sign stays the same when taking the logarithm of both sides of an inequality.
0:01:57 Comparing Likelihoods: When comparing two likelihoods to determine which is greater, one can instead compare their log likelihoods because the logarithm preserves the relationship. This greatly simplifies calculations and avoids numerical issues.