In many machine learning problems, you often need to compute expectations or probabilities involving complex, high-dimensional probability distributions. These computations typically require evaluating integrals that have no closed-form solution or are too expensive to calculate directly. For example, when working with a probabilistic model, you might want to find the expected value of a function with respect to a complicated distribution, or compute the probability of observed data under a given model. In practice, these integrals can be so challenging that direct computation becomes impossible, especially as the number of variables grows.

To build intuition for why this is so difficult, consider the geometry of high-dimensional spaces. In two or three dimensions, you can often visualize and compute areas or volumes with relative ease. However, as the number of dimensions increases, the volume of the space grows exponentially, and the region where the probability density is significant becomes vanishingly small relative to the whole space. This means that most of the space contributes almost nothing to the integral, and finding the "important" regions becomes extremely challenging. Even simple shapes like spheres behave unexpectedly: in high dimensions, almost all of the volume of a sphere is concentrated near its surface, making it hard to sample points that contribute meaningfully to an integral.

These geometric challenges have major implications for real-world machine learning tasks. In Bayesian inference, for instance, you need to compute posterior distributions that involve integrating over all possible parameter values — a task that quickly becomes intractable as the number of parameters grows. Similarly, in models such as Markov random fields or Boltzmann machines, you must evaluate partition functions, which require summing or integrating over an enormous number of possible configurations. In both cases, exact computation is out of reach, and you must turn to sampling methods to obtain approximate answers. Sampling allows you to estimate expectations and probabilities by generating representative samples from the distribution, making it possible to tackle problems that would otherwise be unsolvable.