is the unknown parameter.)As mentioned above, in ideal situations one has all three ingredients: a good model, good data, and good parameters. In this case the only remaining difficulty is a direct one, namely to solve the equations of the model with the given data and parameters to obtain the result. This type of situation pervades undergraduate homework exercises in applied mathematics and physics, and also accurately describes many mature areas of engineering (e.g. civil engineering or mechanical engineering) in which the model, data, and parameters are all well understood. One could also classify pure mathematics as being the quintessential example of this type of situation, since the models for mathematical foundations (e.g. the ZFC model for set theory) are incredibly well understood (to the point where we rarely even think of them as models any more), and one primarily works with well-formulated problems with precise hypotheses and data.However, there are many situations in which one or more ingredients are missing. For instance, one may have a good model and good data, but the parameters of the model are initially unknown. In that case, one needs to first solve some sort of inverse problem to recover the parameters from existing sets of data (and their outcomes), before one can then solve the direct problem. In some cases, there are clever ways to gather and use the data so that various unknown parameters largely cancel themselves out, simplifying the task. (For instance, to test the efficiency of a drug, one can use a double-blind study in order to cancel out the numerous unknown parameters that affect both the control group and the experimental group equally.) Typically, one cannot solve for the parameters exactly, and so one must accept an increased range of error in one's predictions. This type of problem pervades undergraduate homework exercises in statistics, and accurately describes many mature sciences, such as physics, chemistry, materials science, and some of the life sciences.Another common situation is when one has a good model and good parameters, but an incomplete or corrupted set of data. Here, one often has to clean up the data first using error-correcting techniques before proceeding (this often requires adding a mechanism for noise or corruption into the model itself, e.g. adding gaussian white noise to the measurements). This type of problem pervades undergraduate exercises in signal processing, and often arises in computer science and communications science.In all of the above cases, mathematics can be utilised to great effect, though different types of mathematics are used for different situations (e.g. computational mathematics when one has a good model, data set, and parameters; statistics when one has good model and data set but unknown parameters; computer science, filtering, and compressed sensing when one has good model and parameters, but unknown data; and so forth). However, there is one important situation where the current state of mathematical sophistication is only of limited utility, and that is when it is the model which is unreliable. In this case, even having excellent data and knowledge of parameters may lead to error or a false