0:07 Joint Likelihood: The goal is to determine the joint likelihood of a model's parameters given a dataset.
0:15 Independence Assumption: Typically, data points are assumed to be independent and identically distributed (IID). This implies each data point's position doesn't depend on others, and a single model describes all points.
0:29 Probability Product: The probability (and likelihood) of all independent data points is the product of probabilities (likelihoods) at each point.
0:40 Log Likelihood Defined: Log likelihood is the logarithm of the likelihood function. It transforms products into sums and exponents into products, which is useful for calculations.
1:04 Underflow Prevention: Multiplying many probabilities together can lead to underflow (values becoming too small to represent accurately). Using log likelihood avoids this issue.
1:20 Likelihood Comparison: In practice, likelihood is often used to compare different models. The key is determining which model has a greater likelihood.
1:35 Monotonic Property of Log: Because the logarithm is a monotonically increasing function, comparing log likelihoods preserves the same relationship as comparing likelihoods directly (i.e., if likelihood(model A) > likelihood(model B), then log-likelihood(model A) > log-likelihood(model B)). This makes log-likelihood a valid metric for model comparison.