Abduction, which is articulated by C.S. Peirce, is one of the forms of inference. Abduction has been researched not only in philosophy but also in artificial intelligence and information science. Finlay and Dix’s representation of abduction (1996) has almost the same meaning which is given by Peirce. On the other hand, Sawa and Gunji (2010) express three types of inference as operations of arrows on a simple triangular diagram.
In the present paper, we show that Sawa‐Gunji’s representation of abduction is consistent with Finlay‐Dix’s one, and synthesize the two representations. Both parameter estimation and abduction occupy a similar position on the synthesized representation, but they are not completely corresponding. We present “incomplete” parameter estimation as a sort of “simulated abduction”, since abduction has an intrinsic incompleteness, which means that abduction is formally equivalent to “the logical fallacy affirming the consequent”. In other words, a numerical aspect of abduction (i.e. the simulated abduction) is formalized as incomplete parameter estimation. The concept of simulated abduction is applied to parameter estimation of auto‐regressive models, and the effects of it is investigated. As a result of the numerical analyses, the distribution of the incompletely estimated parameter shows a power law of the slop ‐2 in the tail, although conventionally estimated parameter is normally distributed. The power law of the incompletely estimated parameter is based on the structure of ratio distribution. In other words, this result shows that the power law can arise when system observers premise a linearity of input and output data which are too small to estimate the system structure. We call the premise of the system observers “linearity bias”.
As an example of the cause of power law distributions, self‐organized criticality (SOC) has been known. These distributions are based on the mechanisms of the systems themselves, which have some organized interaction between their elements. On the other hand, power law distribution which is derived from the incomplete parameter estimation and the linearity bias is not based on a mechanism of system itself but on relationship between data on the system and observer of the data. Consequently, our research suggests that complexity expressed by a power law distribution can be derived from the incomplete parameter estimation which is a numerical aspect of abduction and is different from SOC mechanisms.
