## 1 Introduction
In contrast, we focus on another big data analytics task--namely, association rule mining [22], which discovers interesting knowledge in the form of association rules \(A\Rightarrow C\) revealing associative relationships between (i) shopper market baskets \(A\) and \(C\) of frequently purchased merchandise items or (ii) collections \(A\) and \(C\) of frequently co-located events. By applying association rule mining to valuable big market basket data, data scientists can help shop owners/managers find interesting or popular patterns that reveal customer purchase behaviour. The research problem of association rule mining usually consists of two key steps:  
1. Mining of frequent patterns [23], and
2. Formation of association rules (by using the mined frequent patterns as antecedents and consequences of the rules). Since the introduction of the frequent pattern mining problem [22], numerous studies [24, 25, 26] have been conducted (e.g., mining frequent patterns from traditional databases of shopper market basket transactions). With these traditional databases of precise data, users definitely know whether an item is present in (or is absent from) a transaction. In contrast, data in many real-life applications are riddled with uncertainty [27, 28, 29, 30, 31]. It is partially due to inherent measurement inaccuracies, sampling and duration errors, network latencies, or intentional blurring of data to preserve anonymity. As such, users are usually uncertain about the presence or absence of items. As a concrete example, a physician may highly suspect (but cannot guarantee) that a coughing patient suffers from the Middle East respiratory syndrome (MERS). The uncertainty of such suspicion can be expressed in terms of existential probability (e.g., a 60% likelihood of suffering from the MERS). With this notion, each item in a transaction \(t_{j}\) in databases containing precise data can be viewed as an item with a 100% likelihood of being present in \(t_{j}\). To find frequent patterns from uncertain data, several uncertain data mining algorithms (e.g., UF-growth [32], tube-growth [33], and BLIMP-growth [34] algorithms) have been proposed. For many real-life applications, users look for all frequent patterns. However, for some other real-life applications, users may have some particular phenomena in mind on which to focus the mining (e.g., a physician may want to find only those patients who are suffering from MERS). To avoid waiting for a long period of time for numerous patterns out of which only a tiny fraction may be interesting to the users, constrained pattern mining [35]--which aims to find those patterns that satisfy the user-specified constraints--is in demand. For example, we previously [36] exploited a special class of constraints called SAM constraints. Such exploitation helps reduce the search space when mining patterns satisfying user-specified SAM constraints. However, many commonly used constraints (e.g., sum(_X.Expenses_) \(<\) 300 CHF, which finds every combination \(X\) of items with a total cost less than 300 CHF) do not belong to the class of SAM constraints. In the current article, we explore another class of constraints, called anti-monotone (AM) constraints, to which commonly used constraints belong. We explore two sub-classes of anti-monotone constraints: (i) the frequency constraint; and (ii) non-frequency AM constraints. In this article, our key contribution is our data analytic algorithm called MrCloud--which uses the MapReduce model in cloud environments for managing, querying, and processing uncertain big data. More specifically, MrCloud manages transactions of uncertain big data, allows users to query these big data by specifying anti-monotone constraints expressing their interests, and processes the user-specified constraints to discover useful information and knowledge in the form of frequent patterns from the uncertain big data.