21 Oct '15 10:3611 edits

I have invented a new kind of probability density function but not sure what

let the notation we use here be:

density(x) means probability density of random continuous variable x

cumulative(x) means probability density of random variable x

The general equation for my new kind of probability density function is:

density( cumulative(h) = p ) = 1 where 0≤p≤1

and h is an observed value, and, in this case, necessarily the first and only observed value, of continuous random variable x in the sample space.

The equations reads as "the probability density of the cumulative of observed h equaling p is 1"

Note that this general equation is quite generic because it purposely doesn't specify what kind of continuous probability distribution cumulative(h) is actually of; it could be any kind of continuous probability distribution you like although, obviously, to actually use the equation, you must specify what kind of distribution.

But, for starters, I am not sure if this " density( cumulative(h) = p ) = 1 " can be defined as a prior probability density or a posterior probability density because, although it necessarily can only be defined after observing evidence h, I cannot see how it could come from the equation for posterior probability (which I take as being (H|E) = (E|H)(H) / (E) where E = evidence and H = hypothesis ) because that posterior probability equation seems to assume that there exists a prior probability ( probability density in this case ) and yet, in this case, I can tell you that no probability or probability density exists prior to observing h!

Instead, this " density( cumulative(h) = p ) = 1 " is meant to be taken as a first principle and thus doesn't require being derived from anything else thus doesn't require being derived from the equation for posterior probability.

So, can a probability still be defined as a "posterior probability" even if it cannot be derived from the equation for posterior probability! If not, does that mean that " density( cumulative(h) = p ) = 1 " is neither a prior nor a posterior probability! If so, what kind of probability would you call it if neither of those two kinds?

I also not sure if, in this case, whether I have defined a likelihood function or a probability that isn't a likelihood function. Which sort is it and why?

Any insight or opinion would be appreciated.

*kind*of probability it represents in this case so I wonder if someone here would form an option on this.let the notation we use here be:

density(x) means probability density of random continuous variable x

cumulative(x) means probability density of random variable x

The general equation for my new kind of probability density function is:

density( cumulative(h) = p ) = 1 where 0≤p≤1

and h is an observed value, and, in this case, necessarily the first and only observed value, of continuous random variable x in the sample space.

The equations reads as "the probability density of the cumulative of observed h equaling p is 1"

Note that this general equation is quite generic because it purposely doesn't specify what kind of continuous probability distribution cumulative(h) is actually of; it could be any kind of continuous probability distribution you like although, obviously, to actually use the equation, you must specify what kind of distribution.

But, for starters, I am not sure if this " density( cumulative(h) = p ) = 1 " can be defined as a prior probability density or a posterior probability density because, although it necessarily can only be defined after observing evidence h, I cannot see how it could come from the equation for posterior probability (which I take as being (H|E) = (E|H)(H) / (E) where E = evidence and H = hypothesis ) because that posterior probability equation seems to assume that there exists a prior probability ( probability density in this case ) and yet, in this case, I can tell you that no probability or probability density exists prior to observing h!

Instead, this " density( cumulative(h) = p ) = 1 " is meant to be taken as a first principle and thus doesn't require being derived from anything else thus doesn't require being derived from the equation for posterior probability.

So, can a probability still be defined as a "posterior probability" even if it cannot be derived from the equation for posterior probability! If not, does that mean that " density( cumulative(h) = p ) = 1 " is neither a prior nor a posterior probability! If so, what kind of probability would you call it if neither of those two kinds?

I also not sure if, in this case, whether I have defined a likelihood function or a probability that isn't a likelihood function. Which sort is it and why?

Any insight or opinion would be appreciated.