# How is Radon-Nikodym derivative different from the likelihood ratio?

I see that the Radon-Nikodym derivative is the ratio of probability measures, $$dP/dQ$$. How is this different, in general, from a likelihood ratio of two continuous distributions? I understand the RN-definition broadly applies for discrete/continuous/mixture densities, but beyond that is there any difference?

## 2 Answers

If $$dx$$ is Lebesgue measure, then it dominates both measures because they correspond with continuous random variables, and one of the properties of RN derivatives is $$\frac{dP}{dQ} = \frac{\frac{dP}{dx}}{\frac{dQ}{dx}}.$$ The numerator is the density of $$P$$, and the denominator is the density of $$Q$$. This is the second property on wikipedia.

So yes, the likelihood ratio is just a particular case. If these two measures were for discrete random variables, then you would replace $$dx$$ with the counting measure, and you would get a ratio of probability mass functions.

In Probability Theory, density functions are generally defined as Radon-Nikodym derivatives themselves, $$\frac{dP}{dQ}$$.

The likelihood function interprets these densities (R-N derivatives) as a function of the parameters, given some observed outcome. More explicitly, let $$X$$ be an absolutely continuous random variable. Then, $$\mathcal{L}(\theta|x\in X) = f(x|\theta) = \mathbb{P}(x\in X|\theta)$$ In other words, the likelihood function measures the probability of observing $$x$$ given the parameters $$\theta$$.

The likelihood ratio is meant to assess the goodness-of-fit of two statistical models (with different parameters) given the same set of observations $$x$$, not two entirely different distributions. More explicitly, let $$\Theta$$ be the set of all possible parameters, and consider some subsets $$\Theta_0, \Theta_1 \in \Theta$$. The likelihood ratio is then, $$\mathcal{L(\Theta_0,\Theta_1)} = -2\log\frac{\sup_{\Theta_0\in\Theta} \mathcal{L}(\theta)}{\sup_{\Theta_1\in\Theta}\mathcal{L}(\theta)}$$ which tests for the null hypothesis $$\theta\in\Theta_0$$.

• "Radon-Nikodym derivatives themselves" yes but then you should switch the notation instead of implicitly calling $Q$ Lebesgue measure – Taylor Apr 13 at 16:14
• Also, densities are not probability mass functions. They cannot be interepreted that way – Taylor Apr 13 at 16:14
• finally, the last expression is not the likelihood ratio, but it is a function of the likelihood ratio. The reason it is transformed is that when it is written in thsi way, it's asymptotically $\chi^2$ distributed--but this is irrelevant at the moment. – Taylor Apr 13 at 16:15