This basic case with sampling with a uniform random variable suggests the existence of more complex estimators. What would be some good queries to get started with exploring and learning more about these more complex estimators, and what would be potential applications of estimators with no uniform random variables? Would a situation where a certain part of an image is appreciably more important be a good candidate for a more complex estimator?
kevintli
^ I'm also curious about the possibility of using non-uniform distributions for Monte Carlo estimation. I wonder if this would be desirable for images where certain regions have very high frequency, while other regions have low frequency? In those cases it could be advantageous to use a distribution that will sample more often from the high-frequency regions.
Another question I have is: how would you apply the basic Monte Carlo estimator for a function that is defined on (-infinity, infinity) if you need to have nonzero density for any x where f(x) != 0?
micahtyong
^ I think non-uniform distributions are absolutely used in practice for Monte Carlo algorithms (for reasons you mentioned). For example, perhaps a normal distribution would be a better proposal density function for the target distribution in the upper right! I think we're learning about uniform first since it's the quintessential unbiased estimator as we discuss in the next few slides.
This basic case with sampling with a uniform random variable suggests the existence of more complex estimators. What would be some good queries to get started with exploring and learning more about these more complex estimators, and what would be potential applications of estimators with no uniform random variables? Would a situation where a certain part of an image is appreciably more important be a good candidate for a more complex estimator?
^ I'm also curious about the possibility of using non-uniform distributions for Monte Carlo estimation. I wonder if this would be desirable for images where certain regions have very high frequency, while other regions have low frequency? In those cases it could be advantageous to use a distribution that will sample more often from the high-frequency regions.
Another question I have is: how would you apply the basic Monte Carlo estimator for a function that is defined on (-infinity, infinity) if you need to have nonzero density for any x where f(x) != 0?
^ I think non-uniform distributions are absolutely used in practice for Monte Carlo algorithms (for reasons you mentioned). For example, perhaps a normal distribution would be a better proposal density function for the target distribution in the upper right! I think we're learning about uniform first since it's the quintessential unbiased estimator as we discuss in the next few slides.