## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 3 (1994), 1212-1221.

### Two-Armed Dirichlet Bandits with Discounting

#### Abstract

Sequential selections are to be made from two independent stochastic processes, or "arms." At each stage we choose which arm to observe based on past selections and observations. The observations on arm $i$ are conditionally i.i.d. given their marginal distribution $P_i$ which has a Dirichlet process prior with parameter $\alpha_i, i = 1, 2$. Future observations are discounted: at stage $m$, the payoff is $a_m$ times the observation $Z_m$ at that stage. The discount sequence $A_n = (a_1, a_2,\cdots, a_n, 0,0,\cdots)$ is a nonincreasing sequence of nonnegative numbers, where the "horizon" $n$ is finite. The objective is to maximize the total expected payoff $E(\sum^n_1a_iZ_i)$. It is shown that optimal strategies continue with an arm when it yields a sufficiently large observation, one larger than a "break-even observation." This generalizes results of Clayton and Berry, who considered two arms with one arm known and assumed $a_m = 1 \forall m \leq n$.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 3 (1994), 1212-1221.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325626

**Digital Object Identifier**

doi:10.1214/aos/1176325626

**Mathematical Reviews number (MathSciNet)**

MR1311973

**Zentralblatt MATH identifier**

0818.62067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L05: Sequential design

Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

**Keywords**

Sequential decisions two-armed bandits one-armed bandits Dirichlet bandits Dirichlet process prior

#### Citation

Chattopadhyay, Manas K. Two-Armed Dirichlet Bandits with Discounting. Ann. Statist. 22 (1994), no. 3, 1212--1221. doi:10.1214/aos/1176325626. https://projecteuclid.org/euclid.aos/1176325626