I am transforming an unscaled density function to log scale to avoid underflow issues.
BI was performing integration on this function on a grid of values before I used the log transormation, to build a grid based cumulative distribution function. Then, using a draw from uniform[0,1], I was choosing the the largest point on the grid which had a cumulative probability value smaller than the draw from the uniform. This worked fine as long as the univariate density function could be integrated.
With the transform to log-scale, I can't really get my head around this mechanism. The joint likelihood is so small that I can't back transform the density, so I have to perform the same uniform distribution based sampling on the log scale. Is this an established practice? Feedback and pointers would be appreciated.