School Notes

CSC411 Tutorial 1

Notation for probability:

Random var X represents outcomes or states of the world.

p(x) means the probability that X = x.

Sample space is the space of all possible outcomes. Could be discrete, continuous, or mixed.

p(x) is the probability mass, or density function. It just assigns a number to each value of X, each state, meaning how likely that state is to occur.

Rules

Sum rule: p(x) = sum over all values of y of p(x, y) p(x1) = sum x2 (sum x3 (sum x4 ... p(x1, x2, x3, ..., xn)))

Chain rule: P(x) = P(x|y)P(y)

Bayes' Rule: p(x|y) = (p(y|x)p(y)) / p(x)

Gives you a way of "reversing" conditional probabilities.

Independence

Two r.v.s are said to *independent iff their joint distribution factors p(x,y) = p(x)p(y). This means p(x) = p(x|y) and p(y) = p(y|x)

Two r.v.s. are conditionally independent given a third r.v. iff p(x,y|z) =

Continuous R.V.s

Outcomes are real values. The probability density functions define the distribution of those outcomes.

The probability mass in [a,b] is given by the integral from a to b of the probability density function.

Summarizing

Often useful to give summaries of distribution without defining whole distribution.

Mean: integral over x of (x dot p(x) dx) Variance: integral (x^2*f(x) dx) - mu^2

Recall Bernouli distribution = p(x | mu) = mu^x * (1 - mu)^(1-x), where mu is the probability of flipping heads.

Multinomial distribution: X in {1, 2, ..., K} p(x1, x2, ..., x_K | mu) = product from k=1 to K of (mu_k ^ x_k) p(X = k | mu) = mu_k For a single observation, considered a die toss. Marginal distribution: p(x_k | mu) = mu_k ^ x_k (1 - mu_k) ^ (1 - x_k) Mean of x_k: mu_k Variance of x_k: mu_k (1 - mu_k)

Normal (Gaussian) Distribution: p(x | mu, sigma) = (1 / (sqrt(2pi)sigma) exp (- (1 / 2sigma^2) * (x - mu)^2)

When you deal with multiple input variables, the formula changes to the multivariate distribution.

If we're given the variance of some data D with a Gaussian distribution, we can estimate the mean using a likelihood function.

This is the p(D|mu) = product from i=1 to N of p(x^i | mu, sigma)

We want to maximize this expression (the value that is most likely to occur). To do this, we take the log of the expression, use basic calculus to differentiate w.r.t. mu, set derivative to 0, and then solve for the sample mean.