# Likelihood functions

## Likelihood functions

### Likelihood function

We want to estimate parameters. One way of looking into this is to look at the likelihood function:

$$L(\theta ; X)=P(X|\theta )$$

The likelihood function shows the chance of the observed data being generated, given specific parameters.

If this has high peaks then it provides information that $$\theta$$ is located in this region.

### IID

For multiple events, the likelihood function is:

$$L(\theta ; X)=P(X|\theta )$$

$$L(\theta ; X)=P(A_1 \land B_2 \land C_3 \land D_4…|\theta )$$

If the events are independent, that is the chance of a flip doesn’t depend on any other outcomes, then:

$$L(\theta ; X)=P(A_1|\theta ).P(B_2|\theta ).P(C_3|\theta ).P(D_4|\theta )...$$

If the events are identically distributed, the chance of flipping a head doesn’t change across flips (for example the heads side doesn’t get heavier over time) then:

$$L(\theta ; X)=P(A|\theta ).P(B|\theta ).P(C|\theta ).P(D|\theta )...$$

$$L(\theta ; X)=\prod_{i=1}^n P(X_i|\theta )$$