Algorithmic Inference - The General Inversion Problem Solving The Fisher Question

The General Inversion Problem Solving The Fisher Question

With insufficiently large samples, the approach: fixed sample – random properties suggests inference procedures in three steps:

1. Sampling mechanism. It consists of a pair, where the seed Z is a random variable without unknown parameters, while the explaining function is a function mapping from samples of Z to samples of the random variable X we are interested in. The parameter vector is a specification of the random parameter . Its components are the parameters of the X distribution law. The Integral Transform Theorem ensures the existence of such a mechanism for each (scalar or vector) X when the seed coincides with the random variable U uniformly distributed in .
Example. For X following a Pareto distribution with parameters a and k, i.e.

a sampling mechanism for X with seed U reads:

or, equivalently,

2. Master equations. The actual connection between the model and the observed data is tossed in terms of a set of relations between statistics on the data and unknown parameters that come as a corollary of the sampling mechanisms. We call these relations master equations. Pivoting around the statistic, the general form of a master equation is:
.

With these relations we may inspect the values of the parameters that could have generated a sample with the observed statistic from a particular setting of the seeds representing the seed of the sample. Hence, to the population of sample seeds corresponds a population of parameters. In order to ensure this population clean properties, it is enough to draw randomly the seed values and involve either sufficient statistics or, simply, well-behaved statistics w.r.t. the parameters, in the master equations.

For example, the statistics and prove to be sufficient for parameters a and k of a Pareto random variable X. Thanks to the (equivalent form of the) sampling mechanism we may read them as

respectively.
3. Parameter population. Having fixed a set of master equations, you may map sample seeds into parameters either numerically through a population bootstrap, or analytically through a twisting argument. Hence from a population of seeds you obtain a population of parameters.
Example. From the above master equation we can draw a pair of parameters, compatible with the observed sample by solving the following system of equations:

where and are the observed statistics and a set of uniform seeds. Transferring to the parameters the probability (density) affecting the seeds, you obtain the distribution law of the random parameters A and K compatible with the statistics you have observed.

Compatibility denotes parameters of compatible populations, i.e. of populations that could have generated a sample giving rise to the observed statistics. You may formalize this notion as follows:

Read more about this topic:  Algorithmic Inference

Famous quotes containing the words general, problem, solving, fisher and/or question:

    “A bill of rights is what the people are entitled to against every government on earth, general or particular, and what no just government should refuse, or rest on inference.”
    —Thomas Jefferson (1743–1826)

    “The great problem of American life [is] the riddle of authority: the difficulty of finding a way, within a liberal and individualistic social order, of living in harmonious and consecrated submission to something larger than oneself.... A yearning for self-transcendence and submission to authority [is] as deeply rooted as the lure of individual liberation.”
    —Wilfred M. McClay, educator, author. The Masterless: Self and Society in Modern America, p. 4, University of North Carolina Press (1994)

    “More than a decade after our fellow citizens began bedding down on the sidewalks, their problems continue to seem so intractable that we have begun to do psychologically what government has been incapable of doing programmatically. We bring the numbers down—not by solving the problem, but by deciding it’s their own damn fault.”
    —Anna Quindlen (b. 1952)

    “Children and old people and the parents in between should be able to live together, in order to learn how to die with grace, together. And I fear that this is purely utopian fantasy ...”
    —M.F.K. Fisher (1908–1992)

    “The question arises as to whether it is possible not to live in the world of men and still to live in the world.”
    —Louise Bernikow (b. 1940)