On the complexity of statistical reasoning

We show that basic problems in reasoning about statistics are NP-hard to even approximately solve. We consider the problem of detecting internal inconsistencies in a set of statistics. We say that a set of statistics is ε-inconsistent if one of the probabilities must be off by at least ε. For a positive constant ε, we show NP-hard to distinguish ε-inconsistent statistics from self-consistent statistics. This result holds when restricted to complete sets of pairwise statistics over Boolean domains. We next consider what may, be determined about distributions with a given (consistent) set of pairwise statistics over Boolean domains. We show it NP-hard to distinguish between the case that Pr(X_i∧X_j) is necessarily 0 and the case that Pr(X_i∧X_j) can have any value in [0, ½]. Similarly, we show it NP-hard to distinguish between the case that |Corr(X_i, X_j)|=-1 and the case that |Corr(X_i , X_j)| is unconstrained. Whereas the connection between PCP and hardness of approximations has been known since Feige et al. (1991), we introduce the application of “zero-knowledge” PCP´s as a tool for proving NP-hardness results for approximation problems