## The Unlucky broker

##### Abstract

This dissertation collects results of the work on the interpretation, characteri-
zation and quanti cation of a novel topic in the eld of detection theory -the
Unlucky Broker problem-, and its asymptotic extension. The same problem can be also applied to the context of Wireless Sensor
Networks (WSNs). Suppose that a WSN is engaged in a binary detection task.
Each node of the system collects measurements about the state of the nature
(H0 or H1) to be discovered. A common fusion center receives the observations
from the sensors and implements an optimal test (for example in the Bayesian
sense), exploiting its knowledge of the a-priori probabilities of the hypotheses.
Later, the priors used in the test are revealed to be inaccurate and a rened pair
is made available. Unfortunately, at that time, only a subset of the original data
is still available, along with the original decision. In the thesis, we formulate the problem in statistical terms and we consider
a system made of n sensors engaged in a binary detection task. A successive
reduction of data set's cardinality occurs and multiple re nements are required.
The sensors are devices programmed to take the decision from the previous
node in the chain and the available data, implement some simple test to decide
between the hypotheses, and forward the resulting decision to the next node.
The rst part of the thesis shows that the optimal test is very di cult to be
implemented even with only two nodes (the unlucky broker problem), because
of the strong correlation between the available data and the decision coming
from the previous node. Then, to make the designed detector implementable
in practice and to ensure analytical tractability, we consider suboptimal local
tests.
We choose a simple local decision strategy, following the rationale ruling the
optimal detector solving the unlucky broker problem: A decision in favor of H0
is always retained by the current node, while when the decision of the previous
node is in favor of H1, a local log-likelihood based test is implemented.
The main result is that, asymptotically, if we set the false alarm probability
of the rst node (the one observing the full data set) the false alarm probability
decreases along the chain and it is non zero at the last stage. Moreover, very
surprisingly, the miss detection probability decays exponentially fast with the
root square of the number of nodes and we provide its closed-form exponent, by
exploiting tools from random processes and information theory. [edited by the author]