Cognitive biases are mental shortcuts humans use in dealing with information and the environment, and which result in biased actions and behaviors (or, actions), unbeknownst to themselves. Biases take many forms, with cognitive biases occupying a central role that inflicts fairness, accountability, transparency, ethics, law, medicine, and discrimination. Detection of biases is considered a necessary step toward their mitigation. Herein, we focus on two cognitive biases - anchoring and recency. The recognition of cognitive bias in computer science is largely in the domain of information retrieval, and bias is identified at an aggregate level with the help of annotated data. Proposing a different direction for bias detection, we offer a principled approach along with Machine Learning to detect these two cognitive biases from Web logs of users' actions. Our individual user level detection makes it truly personalized, and does not rely on annotated data. Instead, we start with two basic principles established in cognitive psychology, use modified training of an attention network, and interpret attention weights in a novel way according to those principles, to infer and distinguish between these two biases. The personalized approach allows detection for specific users who are susceptible to these biases when performing their tasks, and can help build awareness among them so as to undertake bias mitigation.