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Learning about a changing state

Benjamin Davies

TL;DR

The paper develops a tractable model of dynamic information acquisition for a long‑lived Bayesian agent who buys costly signals about a time‑varying Gaussian state. It derives closed‑form solutions for optimal myopic precisions under Brownian dynamics and extends the analysis to forward‑looking planning and to other Gaussian state processes such as OU and linear processes. Key contributions include the two‑stage learning rule (wait until information is valuable, then acquire to hit a target posterior variance), explicit expressions for the residual variance and optimal precisions, and comparative insights across processes and discounting. The results clarify how state dynamics and preferences shape information acquisition, with implications for real‑world settings like news consumption, vaccine testing, and concept drift in machine learning.

Abstract

A long-lived Bayesian agent observes costly signals of a time-varying state. He chooses the signals' precisions sequentially, balancing their costs and marginal informativeness. I compare the optimal myopic and forward-looking precisions when the state follows a Brownian motion. I also compare the myopic precisions induced by other Gaussian processes.

Learning about a changing state

TL;DR

The paper develops a tractable model of dynamic information acquisition for a long‑lived Bayesian agent who buys costly signals about a time‑varying Gaussian state. It derives closed‑form solutions for optimal myopic precisions under Brownian dynamics and extends the analysis to forward‑looking planning and to other Gaussian state processes such as OU and linear processes. Key contributions include the two‑stage learning rule (wait until information is valuable, then acquire to hit a target posterior variance), explicit expressions for the residual variance and optimal precisions, and comparative insights across processes and discounting. The results clarify how state dynamics and preferences shape information acquisition, with implications for real‑world settings like news consumption, vaccine testing, and concept drift in machine learning.

Abstract

A long-lived Bayesian agent observes costly signals of a time-varying state. He chooses the signals' precisions sequentially, balancing their costs and marginal informativeness. I compare the optimal myopic and forward-looking precisions when the state follows a Brownian motion. I also compare the myopic precisions induced by other Gaussian processes.
Paper Structure (22 sections, 9 theorems, 91 equations, 2 figures)

This paper contains 22 sections, 9 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. The agent's environment
  4. Evolution of $\{\theta(t)\}_{t\ge0}$
  5. Learning about $\{\theta(t)\}_{t\ge0}$
  6. Choices and payoffs
  7. Discussion of modeling assumptions
  8. Gaussian states and signals
  9. Quadratic payoffs
  10. Linear costs
  11. Countable signals
  12. Myopic preferences
  13. Optimal myopic precisions
  14. Optimal forward-looking precisions
  15. Other state-generating processes
  16. ...and 7 more sections

Key Result

Lemma 1

Let $n\in\mathbb{N}$ and let denote the variance of $\theta(t_n)$ that is not explained by the signals in $\mathcal{H}_n\setminus\{s_n\}$. Then the unique solution to eq:problem yields posterior variance

Figures (2)

  • Figure 1: Learning about $\{\theta(t)\}_{t\ge0}$
  • Figure 2: Optimal precisions and posterior variances

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Lemma 3
  • Proposition 3
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • ...and 8 more