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:

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