What does a "closed-form solution" mean? I have come across the term "closed-form solution" quite often. What does a closed-form solution mean? How does one determine if a close-form solution exists for a given problem? Searching online, I found some information, but nothing in the context of developing a statistical or probabilistic model / solution. 
I understand regression very well, so if any one can explain the concept with reference to regression or model-fitting, it will be easy to consume. :)
 A: 
"An equation is said to be a closed-form solution if it solves a given
  problem in terms of functions and mathematical operations from a given
  generally accepted set. For example, an infinite sum would generally
  not be considered closed-form. However, the choice of what to call
  closed-form and what not is rather arbitrary since a new "closed-form"
  function could simply be defined in terms of the infinite sum."
  --Wolfram Alpha

and 

"In mathematics, an expression is said to be a closed-form expression
  if it can be expressed analytically in terms of a finite number of
  certain "well-known" functions. Typically, these well-known functions
  are defined to be elementary functions—constants, one variable x,
  elementary operations of arithmetic (+ − × ÷), nth roots, exponent and
  logarithm (which thus also include trigonometric functions and inverse
  trigonometric functions). Often problems are said to be tractable if
  they can be solved in terms of a closed-form expression." -- Wikipedia

An example of a closed form solution in linear regression would be the least square equation
$$\hat\beta=(X^TX)^{-1}X^Ty$$
A: Most estimation procedures involve finding parameters that minimize (or maximize) some objective function. For example, with OLS, we minimize the sum of squared residuals. With Maximum Likelihood Estimation, we maximize the log-likelihood function. The difference is trivial: minimization can be converted to maximization by using the negative of the objective function.  
Sometimes this problem can be solved algebraically, producing a closed-form solution. With OLS, you solve the system of first order conditions and get the familiar formula (though you still probably need a computer to evaluate the answer). In other cases, this is not mathematically possible and you need to search for parameter values using a computer. In this case, the computer and the algorithm play a bigger role. Nonlinear Least Squares is one example. You don't get an explicit formula; all you get is a recipe that you need to computer to implement. The recipe might be start with an initial guess of what the parameters might be and how they might vary. You then try various combinations of parameters and see which one gives you the lowest/highest objective function value. This is the brute force approach and takes a long time. For example, with 5 parameters with 10 possible values each you need to try $10^5$ combinations, and that merely puts you in the neighborhood of the right answer if you're lucky. This approach is called grid search. 
Or you might start with a guess, and refine that guess in some direction until the improvements in the objective function is less than some value. These are usually called gradient methods (though there are others that do not use the gradient to pick in which direction to go in, like genetic algorithms and simulated annealing). Some problems like this guarantee that you find the right answer quickly (quadratic objective functions). Others give no such guarantee. You might worry that you've gotten stuck at a local, rather than a global, optimum, so you try a range of initial guesses. You might find that wildly different parameters give you the same value of the objective function, so you don't know which set to pick.  
Here's a nice way to get the intuition. Suppose you had a simple exponential regression model where the only regressor is the intercept:
\begin{equation}
E[y]=\exp\{\alpha\}
\end{equation}
The objective function is 
\begin{equation}
Q_N(\alpha)=-\frac{1}{2N} \sum_i^N \left( y_i - \exp\{\alpha\} \right)^2
\end{equation}
With this simple problem, both approaches are feasible. The closed-form solution that you get by taking the derivative is $\alpha^* = \ln \bar y$. You can also verify that anything else gives you a higher value of the objective function by plugging in $\ln (\bar y + k) $ instead. If you had some regressors, the analytical solution goes out the window.
A: I think that this website provides a simple intuition, an excerpt of which is:

A closed-form solution (or closed form expression) is any formula that
  can be evaluated in a finite number of standard operations. ... A
  numerical solution is any approximation that can be evaluated in a
  finite number of standard operations. Closed form solutions and
  numerical solutions are similar in that they both can be evaluated
  with a finite number of standard operations. They differ in that a
  closed-form solution is exact whereas a numerical solution is only
  approximate.

A: Looking for lay terms or the painful verbiage that rigorously defines the meaning? I'll presume lay terms as the other can be found everywhere. Let's say you wanted the closed form solution of the square root of 8. The closed form solution is 2 * (2)^1/2 or two times the square root of two. This is in contrast to the non-closed form solution 2.8284. (see wikipedia square root of 2 to see than at 69 decimal places it is accurate to within 1/10,000) One is absolutely defined in mathematical terms whereas the other is not. A closed form solution provides an exact answer and one that is not closed form is an approximation, but you can get a non closed form solution as close as to a closed form solution as you want. Sounds counter intuitive, but if you need it more accurate, then just grind out a little bit more computations. 
