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A Bayesian sequential soft classification problem for a Brownian motion's drift

Steven Campbell, Yuchong Zhang

TL;DR

This work develops a soft-classification variant of Bayesian sequential testing for a Brownian drift and reformulates it as an optimal stopping problem with a loss $g$ on the posterior $\Pi_t$. The main contribution is a semi-explicit, free-boundary solution: the value function is $V_*(\pi)=2K^{-1}\Psi(\pi)+H_*(\pi)$ where $H_*$ is the convex envelope of $H$, and the continuation region is an interval $(A^*,B^*)$ with $A^*(K),B^*(K)$ determined by a pair of transcendental equations. The paper characterizes existence/uniqueness, derives asymptotic behavior of the boundaries as the information ratio $K=\alpha^2/(c\sigma^2)$ varies, and provides numerical illustrations for cross-entropy and $L_1$ penalties, comparing the soft-classification solution to the classic hard-decision problem. Overall, the results illuminate how observation cost and signal-to-noise trade through the information ratio to shape decision timing in soft-sequential settings.

Abstract

In this note we introduce and solve a soft classification version of the famous Bayesian sequential testing problem for a Brownian motion's drift. We establish that the value function is the unique non-trivial solution to a free boundary problem, and that the continuation region is characterized by two boundaries which may coincide if the observed signal is not strong enough. By exploiting the solution structure we are able to characterize the functional dependence of the stopping boundaries on the signal-to-noise ratio. We illustrate this relationship and compare our stopping boundaries to those derived in the classical setting.

A Bayesian sequential soft classification problem for a Brownian motion's drift

TL;DR

This work develops a soft-classification variant of Bayesian sequential testing for a Brownian drift and reformulates it as an optimal stopping problem with a loss on the posterior . The main contribution is a semi-explicit, free-boundary solution: the value function is where is the convex envelope of , and the continuation region is an interval with determined by a pair of transcendental equations. The paper characterizes existence/uniqueness, derives asymptotic behavior of the boundaries as the information ratio varies, and provides numerical illustrations for cross-entropy and penalties, comparing the soft-classification solution to the classic hard-decision problem. Overall, the results illuminate how observation cost and signal-to-noise trade through the information ratio to shape decision timing in soft-sequential settings.

Abstract

In this note we introduce and solve a soft classification version of the famous Bayesian sequential testing problem for a Brownian motion's drift. We establish that the value function is the unique non-trivial solution to a free boundary problem, and that the continuation region is characterized by two boundaries which may coincide if the observed signal is not strong enough. By exploiting the solution structure we are able to characterize the functional dependence of the stopping boundaries on the signal-to-noise ratio. We illustrate this relationship and compare our stopping boundaries to those derived in the classical setting.
Paper Structure (7 sections, 12 theorems, 52 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction and problem set-up
  2. Related Literature
  3. The free boundary problem
  4. Verification and characterization of the value function
  5. Asymptotic behavior in the information ratio
  6. Solution illustration and comparison with the classic problem
  7. Conclusion

Key Result

Theorem 2.1

The free boundary problem eqn:fb1-eqn:fb7 admits the trivial solution $V\equiv g$ with any $A= B\in(0,1)$, and a unique non-trivial solution $V$ in $C^2((0,1)\setminus\{A^*,B^*\})\cap C^1(0,1)$ with boundaries $A^*<B^*$ if and only if $\mathcal{A}g(\pi_0)<-K^{-1}$ for $\pi_0\in(0,1)$ from (G2). More where $V(\pi;A^*)$ is given by eqn:V.pi.A, and $A^*$ and $B^*$ are the unique solutions to the equa

Figures (2)

  • Figure 1: Visualization of the critical points of $H'$ for the cross-entropy loss (left). Illustration of the arguments of Lemma \ref{['lem:common.tangents.convex.func']} using an example of $H$ which defines $f_0$, $f_1$ (right). Visualized below the graph of $H$ is the unique common tangent.
  • Figure 2: Penalties $g$ induced by loss functions (left) and the associated optimal stopping boundaries as functions of $K$ (right).

Theorems & Definitions (26)

  • Example 1.2: Cross-entropy loss
  • Example 1.3: $L_1$ and $L_2$ losses
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 16 more