# Bayesian Estimation of Decision Models

Bayesian analysis has many practical advantages that that make it a formidable tool for the estimation of (nonlinear) model of decision-making. Here, I provide a tutorial for Bayesian estimation of such models using R and Stan (though the Stan code could easily be used based on other platforms such as Python, or indeed directly launched from the command prompt). Feel free to use these notes as you see fit. Below, I provide some indications about the different sections that are available. To keep checking back, though, as this is work in progress.

PLEASE NOTE: This is the first version I post, and it will inevitably contain some (hopefully small) mistakes, inconsistencies between section, and possibly some passages that are difficult to understand. If you do find such errors or have questions, do not hesitate to contact me.

If you use code from these notes in your work, or if you build upon the exposition in this tutorial, please cite the notes as follows:

Vieider, Ferdinand M. (2024). Bayesian Estimation of Decision Models. URL: https://fvieider.quarto.pub/bstats/

## Overview

Below, I provide a brief overview of the sections that are currently available. I also provide a brief description, that hopefully will allow you to assess where to start.

### Inntroduction

The introduction contains a concise discussion of the practical merits of Bayesian statistics, particularly when used to estimate decision-making models. You best start here if you are curious about Bayesian statistics, but you do not know its practical advantages or you are hesitant about usinng them.

### Bayesian Foundations

This Chapter provides a gentle technical introduction to Bayesian analysis using conjugate analysis, i.e. purely relying on distributions for which a closed-form posterior can be obtained. I focus mostly on the important and eminently tractable bernoulli/binomial–Beta distribution, and on the Gaussian. The chapter develops valuable intuition for later parts. If you are unsure whether to read this ask yourself whether you know how to derive a gaussian posterior with unknown mean and variance. If the answer is no, then you should definitely read this part.

### Aggregate models

This section gives a gentle introduction to the use of Stan code to analyse decision-making data. I start from linear regressions, and quickly move on from there to the estimation of aggregate representative-agent models. I then cover inndividual-level (fixed effects) estimation, and how to make it more efficient. There is also a first treatment of regressions with individual-level parameters as dependent variables. If you have never coded a model in Stan, you should read this.

### Hierarchical modelling

COMING SOON!

The chapter will cover a wide variety of topics (hence the delay):

- why hierarchical estimation?

- linear models with hierarchical intercepts

- hierarchical intercepts and hierarchical slopes

- simple nonlinear hierarchies

- covariance matrices for multi-parameter models

- regression in hierarchical models

- meta-analysis as hierarchical analysis

- hierarchies and error clustering

- hierarchies of 3+ levels

- cross--classification