Unfortunately this is a difficult question to address. Visibility depends on many factors (rain, snow, dust, fog, pollution....) It also depends on the colour of the light one is considering (red versus blue).
In general, there is a power-law relationship between visibility V and rain rate R:
V = a (1/R)^b
This says that visibility drops off as rain increases. The problem is that the constants "a" and "b" vary from study to study. For example, if the rain has more large drops than usual, then the visibility will be longer for a given rain rate. This variability depends on the rain type (convective versus widespread, or maritime versus continental).
Snow has a similar relationship, but it is even more variable, because the optical properties of snow are extremely changeable. This is a big issue, because aircraft icing depends on snow rates, but only visibility is commonly measured at airports. Estimating snow rates from visibility is highly desired but is only approximate. I know the Canadian and American transport authorities have different guidelines.
I cannot help with further resources. I have seen lots of isolated studies, finding a and b values for different circumstances, but never a nice summary. I thought there was a "standard" formula used in aviation but I cannot find it.