# Is likelihood also defined as ratio of pdfs

My understanding of likelihood is that it is pdf except that it is a function of parameters rather than observations (as in link). I was reading link. Can likelihood be defined as ratio of probabilities? They call such term as likelihood ratio. When one considers ML estimation, do you maximize likelihood or likelihood ratio? If it is likelihood ratio what is the intuition of ML estimation maximizing likelihood ratio? If it is likelihood then I can understand that we are maximizing pdf over all possible parameters given a set of observations. This seems intuitive. Finally, please let me know how they obtained (5) in link. Is it through GLRT or differentiating likelihood ratio?

Likelihood is defined as probability density function (or probability mass function) evaluated on some data $$X$$,

$$\mathcal{L}(\theta|X) = \prod_{i=i}^N f_\theta(X_i)$$

we maximize it to find parameter $$\theta$$, such that makes observing the data "most likely".

The linked paper describes likelihood-ratio test, a test that compares two likelihoods, by dividing one by another,

$$\Lambda(X) = \frac{\mathcal{L}(\theta_1 | X)}{\mathcal{L}(\theta_1|X)}$$

We use it combined with some threshold $$c$$, to make the decision that if $$\Lambda(X)>c$$ then $$\mathcal{L}(\theta_1 | X)$$ has better fit to $$X$$.

The linked paper discusses likelihood-ratio test, as they say in the abstract (bolded by me):

In this letter, we develop a robust voice activity detector (VAD) for the application to variable-rate speech coding. The developed VAD employs the decision-directed parameter estimation method for the likelihood ratio test.

When you maximize the likelihood using some kind of optimizer, then you evaluate the likelihood on different values of $$\theta$$ and choose the one that maximizes likelihood. This is equivalent for comparing $$\theta_i$$ and $$\theta_j$$ pairs of values using likelihood ratios, but usually the second approach would be less efficient because of the number of pairwise comparisons you would need to make. As I understood (after briefly looking at the paper), the authors propose some method of using likelihood ratios for estimating parameters.

• Thanks Tim. I just want to adapt the mentioned equations a bit to context. $$\Lambda(X) = \frac{\mathcal{L}(\lambda_S | X)}{\mathcal{L}(X)}$$ Here $\lambda_S$ is unknown and it needs to be estimated by ML. This helps in evaluating decision rule for any given $X$. My specific question is that when you estimate $\lambda_S$ do you maximize $\Lambda(X)$ or $\mathcal{L}(\lambda_S | X)$? Please note that denominator has no unknown parameters. Numerator is case "with Speech" and denominator is "without speech". $\lambda_S$ is variance of speech. – Vinay Jan 26 '19 at 18:41
• @Vinay this seems to be another question, so please ask it as another question. – Tim Jan 26 '19 at 19:18