Binary Hypothesis Test Homework: Formulate Likelihood Ratio Test

[brendan_foo]

Homework Statement

This was a midterm question that took some people a rapid time to solve, hence I can only imagine either I was needlessly long winded or something else occurred.

Hypothesis 1: $$S = 2 + \omega$$
Hypothesis 2: $$S = -2 + \omega$$
$$\omega \sim \mathcal{N}(0,\sigma^2_{w})$$

Homework Equations

The following observations are made:

$$N_1 \sim \mathcal{N}(0,\sigma^2_1)$$
$$N_2 \sim \mathcal{N}(0,\sigma^2_2)$$

Formulate the likelihood ratio test

The Attempt at a Solution

Clearly, the variable $S$ has its conditional PDF shifted about the respective mean, and so I yield the likelihood ratio as follows:

$$\vec{r} \triangleq \begin{pmatrix} r_1\\ r_2 \end{pmatrix}$$

$$\Lambda(\vec{r}) = \exp(-\frac{1}{2}((\vec{r} - \vec{\mu_{H1}})^{T} \mathbb{R}^{-1}(\vec{r} - \vec{\mu_{H1}}) - (\vec{r} - \vec{\mu_{H0}})^{T} \mathbb{R}^{-1}(\vec{r} - \vec{\mu_{H0}})))$$

Where

$$\vec{\mu_{H1}} = \begin{pmatrix} 2\\ 0 \end{pmatrix}$$
$$\vec{\mu_{H0}} = \begin{pmatrix} -2\\ 0 \end{pmatrix}$$

and

$$R = \begin{bmatrix} \sigma_1^2 + \sigma_{w}^2 & \sigma_1^2\\ \sigma_1^2 & \sigma_{1}^2 + \sigma_2^2 \end{bmatrix}$$Clearly the issue here becomes a matter of algebraic reduction after taking the natural logarithm. I have omitted any 'threshold' on the RHS (Bayes criterion for example).

Is this formulation correct thus far? Some people mentioned no need for matrix inversions of any type...Can this problem be simplified or have I needlessly made things difficult for myself?

With thanks,

:)

 

[KataKoniK]

Hello,

Your formulation of the likelihood ratio test appears to be correct so far. It is important to note that this is just one possible approach to solving the problem and there may be other ways to simplify the problem or make it easier to solve. However, your approach does seem to be valid and it may just require some algebraic manipulation to reduce the equations.

In terms of the use of matrix inversions, it may not be necessary in this case as you are dealing with a simple 2x2 matrix and the inverse can be easily calculated. However, if you are able to solve the problem without using matrix inversions, that may be a simpler and more efficient approach.

Overall, it seems like you have a good understanding of the problem and your approach is valid. You may just need to continue working through the equations and simplifying them to reach a final solution. Keep up the good work!

 

[Mene]

Dear student,

Your formulation of the likelihood ratio test appears to be correct. However, it is always a good idea to check your work with a classmate or the professor to ensure accuracy. As for the use of matrix inversions, it is possible that there may be alternative methods to solve this problem without using them. It would be beneficial to discuss this with your classmates or professor to see if there are simpler solutions. However, if your solution is correct, then there is no need to worry about the complexity. Keep up the good work!