A bit of context: I have some measurements $\{y_i\}$ of a cross-section for a nuclear reaction at different angles $\{\theta_i\}$. Each $y_i$ measurement is Gauss distributed around the same mean $\mu$, but with a standard deviation of its own $\{\sigma_i\}$. I have a model for the standard deviation as a function of the angle, with 3 parameters (A, B and C):
$\sigma(\theta) = A + B \cos(\theta) + C \cos(\theta^2)$
Like I said, I have a set of n measured data points $D = \{\ \{\theta_i\},\ \{y_i\},\ \{\sigma_i\}\ \}$. How do I do a Bayesian analysis to get the posterior P(A,B,C | D)? I already have an assumption for the priors $P_0(A)$, $P_0(B)$ and $P_0(C)$ for my parameters.
I'm totally lost. The theory I can find on Bayesian analysis on Gaussian distributed data deals with only one Gauss distributed variable and gives the posterior for the mean, not the standard deviation. And it doesn't go so deep as to assume it's a function of several parameters. Things get a bit fuzzy in my head when I try to draw parallels to my situation when I have several variables, each with its own $\sigma$. I need some help on how to approach this.
So I know the general Bayes Theorem for data D and parameter $\lambda$:
$P(\lambda | D) = \cfrac{P(D | \lambda)\ P_0(\lambda)}{\int{P(D | \lambda)\ P_0(\lambda)\ d\lambda}}$
Can I generalize it to 3 parameters like this?
$P(A, B, C | D) = \cfrac{P(D | A, B, C)\ P_0(A)\ P_0(B)\ P_0(C)}{\int\int\int{P(D | A, B, C)\ P_0(A)\ P_0(B)\ P_0(C) \ dA\ dB\ dC}}$
And what is the probability that I get my measured data? I don't have a single Gaussian distributed variable, I have several of them. I know for a single variable it is:
$G(y | \mu,\ \sigma)=\cfrac{1}{\sigma\sqrt{2 \pi}} \exp{-\cfrac{(y-\mu)^2}{2\sigma^2}}$
Then for a my data set $D = \{\ \{\theta_i\},\ \{y_i\},\ \{\sigma_i\}\ \}$, is the probability given A, B and C, the product of all Gaussians?
$P(D\ |\ A,\ B,\ C) = \prod_i^n\ G(y_i\ |\ \mu,\ \sigma(\theta_i)) $
where, again, my model says $\sigma(\theta_i) = A + B \cos(\theta_i) + C \cos(\theta_i^2)$
Can I use these equations to solve for the posterior $P(A,\ B,\ C\ |\ D)$? Thanks a lot!