- #1
dIndy
I have a quadratic regression model ##y = ax^2 + bx + c + \text{noise}##. I also have a prior distribution ##p(a,b,c) = p(a)p(b)p(c)##. What I need to calculate is the likelihood of the data given solely the extremum of the parabola (in my case a maximum) ##x_{max} = M = -\frac{b}{2a}##. What I tried so far is:
$$p(y|M) = \int p(y|M,b,c)p(b,c|M)\,dbdc$$
I would like to rewrite this as a function of ##p(y|a,b,c)## and ##p(b,c|a) = p(b,c)##, substituting ##a## for ##M##. However, I'm not sure how to perform a change of variables for conditional variables.
What I've also tried is using Bayes' theorem to rewrite the likelihood:
$$p(y|M) = \frac{p(y)p(M|y)}{p(M)} = \frac{p(y)\int p(M,b,c|y)\,dbdc}{\int p(M,b,c)\,dbdc} $$
Then performing the substitution for ##M##:
$$ \frac{p(y)\int p(a,b,c|y)\det(J)\,dbdc}{\int p(a,b,c)\det(J)\,dbdc}$$
Using Bayes' theorem again:
$$ \frac{\int p(y|a,b,c)p(a,b,c)\det(J)\,dbdc}{\int p(a,b,c)\det(J)\,dbdc} = \frac{\int p(y|a,b,c)p(b,c)\det(J)\,dbdc}{\int p(b,c)\det(J)\,dbdc}$$
Can this be simplified further?
I've also got a far more challenging and general problem:
Given a non-linear regression model ## y_i = f(\theta,x_i) + \text{noise}##, with ##\theta## the vector of unknown parameters and ##x_i## the vector of dependent variables. I want to calculate the likelihood of the global maximum of ##f##. The problem here is that there is no closed form expression for ##x_{max}##.
$$p(y|M) = \int p(y|M,b,c)p(b,c|M)\,dbdc$$
I would like to rewrite this as a function of ##p(y|a,b,c)## and ##p(b,c|a) = p(b,c)##, substituting ##a## for ##M##. However, I'm not sure how to perform a change of variables for conditional variables.
What I've also tried is using Bayes' theorem to rewrite the likelihood:
$$p(y|M) = \frac{p(y)p(M|y)}{p(M)} = \frac{p(y)\int p(M,b,c|y)\,dbdc}{\int p(M,b,c)\,dbdc} $$
Then performing the substitution for ##M##:
$$ \frac{p(y)\int p(a,b,c|y)\det(J)\,dbdc}{\int p(a,b,c)\det(J)\,dbdc}$$
Using Bayes' theorem again:
$$ \frac{\int p(y|a,b,c)p(a,b,c)\det(J)\,dbdc}{\int p(a,b,c)\det(J)\,dbdc} = \frac{\int p(y|a,b,c)p(b,c)\det(J)\,dbdc}{\int p(b,c)\det(J)\,dbdc}$$
Can this be simplified further?
I've also got a far more challenging and general problem:
Given a non-linear regression model ## y_i = f(\theta,x_i) + \text{noise}##, with ##\theta## the vector of unknown parameters and ##x_i## the vector of dependent variables. I want to calculate the likelihood of the global maximum of ##f##. The problem here is that there is no closed form expression for ##x_{max}##.