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I came across the concept of closed-form analytical solution while learning MCMC and got a bit confused. At this point, my understanding is that the problem does not have closed-form solution if it can not be expressed by explict mathmatical formula with specific parameters found through certain optimization of loss function.

If my understanding is correct, I guess methods like EM and sampling method (MCMC) are without closed-form analytical solution. Is my understanding regarding closed-form solution accurate?

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EM and MCMC are algorithms/procedures to find solutions, they're not expressions. But, yes, the solutions they lead are not in closed-form. See the definition of a closed-form expression from wikipedia:

...a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration..."

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