Multifactor Authentication

In this work, we consider a more general case of the probabilistic decision-making methodology, while a combination of the measurement results for the individual sensors is made similarly to the previous works by using the Bayes estimator. Since the outcomes of measurements have a probabilistic nature, the decision function is suitable for the maximum a posteriori probability solution.

In more detail, the decision function may be described as follows. At the input, it requires a conditional probability of the measured value from each sensor $P\left(z_{i} \mid H_{0}\right)$ and $P\left(z_{i} \mid H_{1}\right)$ together with a priori probabilities of the hypotheses $P\left(H_{0}\right)$ and $P\left(H_{1}\right)$. The latter values could be a part of the company's risk policy as they determine the degree of confidence for specific users. Then, the decision function evaluates the a posteriori probability of the hypothesis $P\left(H_{1} \mid Z\right)$ and validates that the corresponding probability is higher than a given threshold $P_{T H}$

The measurement-related conditional probabilities can be considered as independent random variables; hence, the general conditional probability is as follows:

Further, the total probability $P(Z)$ is calculated as

where $P\left(z_{i} \mid H_{J}\right), J \in\{0 ; 1\}$ are known from the sensor characteristics, while $P\left(H_{0}\right)$ and $P\left(H_{1}\right)$ are a priori probabilities of the hypotheses (a part of the company's risk policy).

Based on the obtained results, the posterior probability for each hypothesis $H_{J}, J \in\{0 ; 1\}$ can be produced as

For a comprehensive decision over the entire set of sensors, the following rule applies

As a result, the decision may be correct or may lead to an error. The FAR and FRR values could then be utilized for selecting the appropriate threshold $P_{T H}$ based on all of the involved sensors.