How to find the likelihood given data?

Question

I have a textfile with the two columns

$$\mathbf{x}=(x_1,...,x_i)$$

$$\mathbf{y}=(y_1,...,y_i)$$

I want to use the following model for the data

$$y_i=A\sin\left(\frac{x_i}{B}\right)+C\epsilon_i,$$

where $\epsilon_i\sim N(0,1)$ and independent.

By guessing I found that $A=5.2, \ B=5.3$ and $C=1.0$ gives me a pretty good fit. Now I want to write a function in R-code that computes the likelihood function (the probability of observed data $y_1,\ldots,y_{i}$ given the observed values $x_1,...,x_i$ and the observed values for the parameters). But before I do that, I need to understand what's going on mathematically.

The posterior is given by

$$\pi(x_1,\ldots,x_i\mid y_1,\ldots,y_i)=\frac{\color{red}{\pi(y_1,\ldots ,y_i\mid x_1,\ldots,x_i)} \cdot \pi(x_1,\ldots,x_i)}{\pi(y_1,\ldots,y_i)},$$

where $\pi(y_1,\ldots,y_i\mid x_1,\ldots,x_i)$ is the likelihood. But how do I calculate the, posterior, evidence and the prior here? Any help is appreciated.

Answer 1

You've written down the wrong expression for the posterior to do Bayesian regression. Consider the $\vec{x}$ to be fixed by design. Thus, you don't need a prior for the $x$. Rather, you need to set a prior for A, B, and C. Define $r = y - \hat{y}$ in the usual way. The likelihood, based on the probability model for $\epsilon$ and given A, B, and C would be:

$$L(y_i | A, B, C)= \prod_{i=1}^n \phi\left(\left(y_i - A \sin(x_i/B)\right)/C\right)$$

where $\phi$ is the standard normal density.