3

The Wikipedia entry for Probability Density Function states that the PDF "describes the relative likelihood for this random variable to take on a given value." Two questions:

  1. Does that mean that the ratio of two points reflects the difference in probability? For instance, on a standard normal distribution f(0.0)=0.3989, while f(1.0)=0.2420. Does this mean that an outcome of 0.0 is 1.65 times more likely than an outcome of 1.0? (I know that the actual probability of any exact value is zero.)

  2. If one can compare points in this way, can you then compare points from two different densities? For instance, on a standard normal distribution f(1.0)=0.2420, whereas f(1.0) for a Student's-T distribution with 1 degree of freedom is 0.1592. Does this mean an outcome of 1.0 is 1.52 times more likely on a standard normal distribution than a T distribution?

Mountains
  • 159
  • 1
    What distinction are you making between "probability" and "likely"? There is a nasty trap lurking here for the unwary. For instance, in the first case let a difference of lengths in meters have a standard normal distribution: then $f(0)=0.3989$ but $f(0)=0.003989$ when the lengths are expressed in centimeters. Does this mean $0$ cm is $100$ times less "likely" than $0$ meters? For some more information about interpreting PDFs, take a look at the closely related questions http://stats.stackexchange.com/questions/4220 and http://stats.stackexchange.com/questions/14483. – whuber May 08 '13 at 18:36
  • In my case, I actually don’t care about the actual values for probability or likelihood. I am evaluating two models that are identical except that they use two different probability density functions.

    For example, if the variable at issue returns a value of 1.0, and f(1.0) from one PDF is 0.3, and from the second PDF is 0.2, I am wondering if the first model “wins” to a degree of 1.5. The 1.5 would be considered relative and would be compared to calculations at other observations.

    It is not all that complicated, and there are no scaling issues since the models use identical data.

     – Mountains May 08 '13 at 19:02
  • My point is not about scaling at all. What I am trying to emphasize is that if your interpretation depends on an arbitrary choice such as which units of measure to use, then you had better watch out, because that interpretation may be meaningless. Even the interpretation of "relative" is subtle. It sounds like you are trying to compare models based on (formal) likelihood functions. That's legitimate provided the models are nested. If they are not, the comparison could be meaningless. – whuber May 08 '13 at 19:13

0 Answers0