likelihood

(136) Probability is not Likelihood. Find out why!!! - YouTube

• This is NOT bayes rule. It’s completely different
  • probability is: given a distribution, what is the the probability I’ll see this data?
  • likelihood is: given data that I know, what is the likelihood it came from this distribution?
• note that likelihood and probability have different conditionals (priors)
  • in probability, the distribution is fixed
  • in likelihood, the distribution is fixed The definition
• The likelihood function is the joint probability of observing the given data points, assuming that they are independent and identically distributed (i.i.d.)
• For a set of observations {x₁, x₂, …, xₙ}, the likelihood function is given by: $L(X|\theta )=f(x_1,x_2,\dots ,x_n|\theta )=f(x_1|\theta )\cdot f(x_1|\theta )\cdot \dots \cdot f(x_n|\theta )$
• Where:
  • L(θ) is the likelihood function
  • f(xᵢ | θ) is the probability density function - PDF (or PMF) of the i-th observation, given the parameter(s) θ
  • θ represents the parameter(s) of the distribution
• Note: calculating the log of the likelihood function is typically easier