Probability Distributions
Random Variable is a variable that can take different values randomly.
Probability distributions describes how likely a random variable or a set of random variable is to each of its possible states. For discrete random variable this is defined using Probability Mass Function (PMF) while continuous ones are defined using Probability Density Function (PDF).
A Marginal probability distribution is a distribution on a subset of variables of the actual distribution defined using all variables.
Expectation / Variance
Expectation: Expected value of function $f(x)$ when $x$ is sampled from $P(x)$ is : $ \int P(x) f(x) dx $
Variance: $\mathbb{E} [ (f(x) - \mathbb{E}[f(x)])^2 ]$ ( second central moment)
Representation of probability distribution in terms of Moments
$n $ th moment is described as $ \mathbb{E} [ X^n ] $
Skweness can be described by third moment of standardized $X$: $ \mathbb{E}[((X-\mu) / \sigma)^3] $
Mixture of distributions
Gaussian mixture model models the our data $x$ in terms of $K$ gaussian distributions.
$ p(X) = \prod_{i} \sum_{k} \pi_k N (x_i | \mu_k, \Sigma_k)$
We can calculate the ideal parameters $\theta = {\mu, \Sigma, \pi}$ by using Expectation Maximization algorithm.
- Expectation: Calculate $ \mathbb{E} [ p(X, Z | \theta)]$
Bayes Theorem
Central Limit Theorem
Markov Chain and Stationary Distributions
Markov Chain: This is a chain of states, where each state depends on its previous state. The tranistion probability between states are defined by a transition matrix.
Initially, the probability of being in a particular state $s$ is $p_t(s)$. If we move to the next step based on the transition probability in the transition matrix, we will be at a state $s$ with a probability $p_{t+1}(s)$.
After a large number of such transitions, the probability distribution at state $t$ will the same as at $t+1$, at this point the markov chain is said to have reached a stationary distribution $\pi(s)$.
$\pi * P = \pi$
If we solve this equation, i.e find the left eigen vector of P with eigen value 1, we have found the stationary distribution.
Sampling form a distribution
Inverse Transform Sampling
Accept Reject Sampling
we want to find probability distribution $p(s)$ w.r.t a function $f(s)$
We have numerator $f(s)$ but the denominator ( normalizing constant ) which is the integral of $f$ over all possible values is intractable to calculate.
We can use another distribution $g(x)$ and then sample from $g$ and accept samples with a probability $f(x) / g(x)$ and reject otherwise. This actually gives us samples as if we sampled from the original distribution $p(s$)