# Influence tests I: ideal composite hypothesis tests, and causal semimeasures

## TL;DR

This paper explores ideal composite hypothesis tests using universal enumerable semimeasures, proposing a new influence testing framework for discrete time series that can infer causality even with halting information, contrasting with Bayesian and classical methods.

## Contribution

It introduces a novel influence testing approach based on universal semimeasures and generalized structural equations, capable of inferring causality with halting information transmission.

## Key findings

- Ideal tests using universal semimeasures are proposed.
- Instantaneous causality can be inferred with halting information.
- Contrasts with Bayesian and classical causality methods.

## Abstract

Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete time series is defined using generalized structural equations. Several ideal tests are introduced, and it is argued that when Halting information is transmitted, in some cases, instantaneous cause and consequence can be inferred where this is not possible classically. The approach is contrasted with Bayesian definitions of influence, where it is left open whether all Bayesian causal associations of universal semimeasures are equal within a constant. Finally the approach is also contrasted with existing engineering procedures for influence and theoretical definitions of causation.