Relative Likelihood Function

The relative likelihood function is

$R(\theta )=\frac{L(\theta )}{L(\hat{\theta })}$ where

• $L$ is the Likelihood Function
• $\hat{\theta }$ is the MLE of $\theta$

$0\le R(\theta )\le 1$ since $L(\hat{\theta })$ is maximized.

$R(\theta )$ is a way to normalize $L(\theta )$.

Example to Motivate Relative Likelihood Function

Example: Suppose a coin is tossed $200$ times and we observe $y=80$ heads and $\theta =P(heads)$.

• You saw in the Estimation example already, how the probabilities that come out are very small. Relative likelihood will be a way to normalize this Likelihood Function $L(\theta )$

To calculate Relative Likelihood, we obtain $R(0.7)=\frac{L(0.7)}{L(\hat{\theta })}=\frac{L(0.7)}{L(\frac{80}{200})}=\frac{\left(\genfrac{}{}{0ex}{}{200}{80}\right)0.7^{80}0.3^{120}}{\left(\genfrac{}{}{0ex}{}{200}{80}\right)0.4^{80}0.6^{120}}$

Why do we care about the relative likelihood when we already know how to estimate the optimal parameter $\hat{\theta }$ using Maximum Likelihood Estimation?

Well, sometimes, you want to give a reasonable range to your estimated parameter $\hat{\theta }$, i.e. assign upper and lower bounds

• Personal thought: This range is like quantifying your Uncertainty
• Notice that $R(\hat{\theta })=1$

We can define plausibility in terms of the Relative Likelihood Function:

• $R(\theta )\ge 0.5⟹$Very Plausible
• $R(\theta )\ge 0.1⟹$Plausible
• $R(\theta )<0.1⟹$Implausible
• $R(\theta )<0.01⟹$Very Implausible

We will explore more about Relative Likelihood when we discuss Confidence Interval, but see Confidence Interval, but see Interval Estimation where we talk about likelihood interval.

Related

• Interval Estimation