# Eleisha's Segment 28: Gaussian Mixtures Models in 1D

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1. Draw a sample of 100 points from the uniform distribution $\displaystyle U(0,1)$ . This is your data set. Fit GMM models to your sample (now considered as being on the interval $\displaystyle -\infty < x < \infty$ ) with increasing numbers of components $\displaystyle K,$ at least $\displaystyle K=1,\ldots,5$ . Plot your models. Do they get better as K increases? Did you try multiple starting values to find the best (hopefully globally best) solutions for each $\displaystyle K$ ?
2. Multiplying a lot of individual likelihoods will often underflow. (a) On average, how many values drawn from U(0,1) can you multiply before the product underflows to zero? (b) What, analytically, is the distribution of the sum of $\displaystyle N$ independent values $\displaystyle \log(U)$ , where $\displaystyle U\sim U(0,1$ )? (c) Is your answer to (a) consistent with your answer to (b)?
1. Suppose you want to approximate some analytically known function $\displaystyle f(x)$ (whose integral is finite), as a sum of $\displaystyle K$ Gaussians with different centers and widths. You could pretend that $\displaystyle f(x)$ (or some scaling of it) was a probability distribution, draw $\displaystyle N$ points from it and do the GMM thing to find the approximating Gaussians. Now take the limit $\displaystyle N\rightarrow \infty$ , figure out how sums become integrals, and write down an iterative method for fitting Gaussians to a given $\displaystyle f(x)$ . Does it work? (You can assume that well-defined definite integrals can be done numerically.)