# Theory

## Distributed Bayesian Reasoning Introduction

### Part of the Distributed Bayesian Reasoning series

Distributed Bayesian Reasoning is a kind of hypothetical opinion poll. It tells us not what people actually believe, but what they would believe if they knew more.

## The Deliberati Argument Model

In this article we introduce an argument model: a set of terms for analyzing arguments by naming their parts. There are various argument models in the academic literature on argumentation theory and related fields but none provide us with precise definitions for all the concepts behind our algorithms for improving for online conversations. So we will define those concepts here. Our model incorporates the basic ideas from the influential Toulmin model of argumentation first introduced in 1948.

## A Bayesian Account of Argumentation

### Part of the Bayesian Argumentation series

In this essay, I present an account of argumentation as the exchange of information between Bayesian rational agents. The basic idea of the Bayesian view of probability is that probabilities represent subjective degrees of belief. So if we know the beliefs of some rational “subject”, we can precisely define and measure various concepts relating to the quality of an argument in the mind of the subject. In other words we can objectively measure the subjective quality of an argument.

## Definition of Relevance

In the previous essay in this series, we introduced the basic ideas and terminology of Bayesian argumentation, including the concept of relevance.

## Necessity and Sufficiency

### Part of the Bayesian Argumentation series

Argument and Information In the previous essay in this series, we introduced the idea of relevance, and said that a premise is relevant to the conclusion iff $P(A \vert B) > P(A \vert \bar{B})$. Consider the argument (𝐴) this is a good candidate for the job because (𝐵) he has a pulse. Having a pulse may not be a very persuasive reason to hire somebody, but it is probably quite relevant, because if the candidate did not have a pulse, the subject would probably be much less likely to want to hire him.

## Informativeness and Persuasiveness

### Part of the Bayesian Argumentation series

Why Accept the Premise? In the previous essay in this series, we defined the ideas of necessity and sufficiency from the perspective of a Bayesian rational agent. If an argument is necessary, then if the subject were to reject the premise, they would decrease their acceptance of the conclusion. And if an argument is sufficient, then if the subject were to accept the premise, they would increase their acceptance of the conclusion.

## Bayesian Argumentation Definitions

### Part of the Bayesian Argumentation series

Bayesian Argumentation: Summary of Definitions Below is a summary of all the terms and equations defined in the essays in this series, followed by a detailed example. For an argument with premise 𝐵 and conclusion 𝐴, and a subject whose beliefs are represented by probability measure P… Relevant: The premise is relevant to the conclusion (or, the argument is relevant) iff $P(A \vert B) ≠ P(A \vert \bar{B})$ Otherwise, the premise is irrelevant to the conclusion (or, the argument is irrelevant) Irrelevance implies statistical independence of A and B.

## A Bayesian Inference Primer

“When you have eliminated the impossible, all that remains, no matter how improbable, must be the truth.” – Sherlock Holmes (Arthur Conan Doyle) For a long time Bayesian inference was something I understood without really understanding it. I only really got it it after reading Chapter 2 of John K. Kruschke’s textbook Doing Bayesian Data Analysis, where he describes Bayesian Inference as Reallocation of Credibility Across Possibilities I now understand Bayesian Inference to be essentially Sherlock Holmes’ pithy statement about eliminating the impossible quoted above, taken to its mathematical conclusion.

## Distributed Bayesian Reasoning Math

### Part of the Distributed Bayesian Reasoning series

In this article we develop the basic mathematical formula for calculating the opinion of the meta-reasoner in arguments involving a single main argument thread.

## The Meta-Reasoner

### Part of the Distributed Bayesian Reasoning series

In the Introduction to Distributed Bayesian Reasoning, we argue that the rules of Bayesian inference can enable a form of distributed reasoning. In this article we introduce the idea of meta-reasoner, which is the hypothetical fully-informed average juror. The meta-reasoner resembles the average juror in that it holds prior beliefs equal to the average beliefs of the participants, but it is fully-informed because it holds beliefs for every relevant sub-jury.