An example Formalization References \[ \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\E}{\mathbb{E}} \]
An example The example below is taken from [1]
Let \(X\) be a random variable with uniform distribution in \([0,10]\), \[ X \sim Uniform(0,10) \]
Consider the function \(h(x) = 10 e^{-2|x-5|}\). Suppose we want to calculate \(\E_X[h(X)]\). By definition,
\[\begin{align} \E_X[h(X)] &= \int_{0}^{10} h(x) f(x)dx \\ &= \int_{0}^{10} exp(-2|x-5|) dx \end{align}\] A straightforward way to do this is sampling \(X_i\) from the uniform(0,10) density and calculating the mean of \(10\cdot h(X_i)\)
Discrete random variables Analytical estimation of probabilities Empirical solution Intervention Continuous random variables Conditioning by observation Conditioning by intervention Joint distribution of \((Y,Z)\) Simulation of joint distribution \((Z,Y)\) References The example is taken from Chapter 17 ???. Let \(V = (X,Y,Z)\) represented by the following graph
Discrete random variables Let \(V = (X,Y,Z)\) have the following joint distribution
\[ \renewcommand{\vector}[1]{\mathbf{#1}} \newcommand{\matrix}[1]{\mathbf{#1}} \newcommand{\E}[1]{\mathbb{E}{\left(#1\right)}} \begin{align} X &\sim Bernoulli(1/2) \\ Y|X=x &\sim Bernoulli\left(\frac{e^{4x-2}}{1 + e^{4x-2}}\right) \\ Z|X=x, Y=y &\sim Bernoulli\left(\frac{e^{2(x+y)-2}}{1 + e^{2(x+y)-2}}\right) \end{align} \]
Obtaining data Fitting with pre-determined distribution The effects of sample size Goodness-of-fit Assuming Poisson distribution Assuming NB distribution The package MASS provides a function, fitdistr to fit an observation over discrete distribution using Maximum likelihood.
Obtaining data We first need to generate some data to fit. The rnegbin(n,mu,theta) function can be used to generate n samples of negative binomial with mean mu and variance mu + mu^2 / theta.
