If you are confused about likelihood functions and maximum likelihood estimation, this little diagram might help. 1/8 
Consider a probabilistic model f(x; θ) (top left). If you set the model parameter θ (top left, black horizontal line), you get a probability distribution over x (lower left). In this case, x is a continuous variable, so we get a probability density function (PDF). 2/8
If instead you set the value of x (top left, vertical blue line), you get a function of θ (top right): this is a likelihood function, noted ℒ(θ|x). 3/8
A likelihood function is *not* a probability distribution (it does not necessarily integrate to 1). It is a measure of how likely each possible θ value is after you observe the data x. 4/8
You can also think of the likelihood function as a measure of how well the model fits the observed data x, for each possible value of θ. 5/8
Training algorithms typically search for the parameter value θ^ that maximizes the likelihood of the observed data: this is called maximum likelihood estimation (MLE). 6/8
Maximizing the log likelihood is equivalent to maximizing the likelihood since the log is a stricly increasing function, but it is often easier (because the log turns products into sums). 7/8
The max value L^ of the likelihood function is used in theoretical information criteria like the BIC or AIC which are useful for model selection: these metrics prefer models that fit the data well (high L^), but they penalize models with many parameters. Hope this helps! :) 8/8
