Revision of Probability

Read Chapter 5 of Mitchell.

Points covered:

1.

Nature of probability

2.

Prior beliefs and estimations

3.

Estimations by interpolation and back-off

4.

Disjoint and Mutually exclusive events

5.

Conditional probability and Independent events

6.

Rules of probability, product rule, sum rule, Bayes Theorem and the distributive rule.

7.

Sample spaces, finite and infinite

8.

Distributions over integers/Convergent series

9.

Binomial and Poisson distributions

10.

Continuous Vs Discrete distributions

11.

Normal distribution

12.

Monte-Carlo techniques

13.

Expectation

• Classical (r out of n equally likely ways) and Posterior (from relative frequency) methods of estimation.

• $P(\mbox{\rm Certain event}) = 1.0$

• Disjoint events: Eg. $P(\bigcup_i A_i) = \sum_i P(A_i)$

• Mutually exclusive events. P(A) = 1-P(B)

• Independent events: Eg. Coin tosses. P(A|B) = P(A)

• Product rule. $P(A \cap B) = P(A\vert B) P(B) = P(B\vert A) P(A)$

• Special case for independent events. $P(A \cap B) = P(A)P(B)$

• Sum rule. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Bayes rule. $P(H\vert D) = \frac{P(D\vert H) P(H)}{P(D)}$

• Distributive rule. If $\sum\limits_{i=1}^{n}P(A_i) = 1$ and $\forall i, j \le n: P(A_i \cap A_j) = 0$ then $P(B) = \sum_{i=1}^n P(B\vert A_i)P(A_i)$

• Sample spaces can be either finite or infinite. Give examples of either. Discuss countable and uncountable sample spaces.

• Find a scheme using which one can determine the probability of drawing each integer from the set of all integers.

1.

Read Rissanen (1983), ``A universal prior for integers'' (1st pass).

2.

Progressions and convergent series. Their use in determining distributions.

3.

Discrete case: The Binomial (Bernoulli) distribution. The probability of getting r successes P(r) in n trials where p = probability of getting one success and q = 1-p.

• $P(r) = \left( \begin{array}{c}n\\ r\\ \end{array} \right) p^rq^{n-r}$
• $\mu = E[X] = np$ where E[X] is the expected number of successes.
• $\sigma = \sqrt{npq}$

4.

A discrete case with infinite sample space: The Poisson distribution. If $\lambda$ is a positive number, then

• $P(r) = \lambda^re^{-\lambda}/r!$
• $\mu = \lambda$
• $\sigma = \sqrt{\lambda}$

For example, $P(r) = \frac{1}{er!}$ where $\mu = 1$.

1.

Discuss uncountable sample spaces, continuous variables and their distributions.

2.

The Normal (Gaussian) distribution. Use this when the variance ($\sigma^2$) of a binomial distribution $\ge 5$.

3.

$p(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2\sigma^2} (x-\mu)^2}$

4.

Expectation: The expected value of a function f(x) where x is a random variable taking values from a space X is:

$$\begin{eqnarray*}E[f(X)] &=& \sum_{x \in X} f(x)P(x) \qquad \mbox{(Discrete case... ...t_{-\infty}^{\infty} xf(x) dx \qquad \mbox{(Continuous case)}\\ \end{eqnarray*}$$

For example:

• Find the expected winning in a gamble of tossing an unbiased coin where it costs $10 to enter and the prize is $20. What if the entry fee was lower or higher and the prize was the same?

• Find the expected winning in a game where you get $\$2^k$ if you throw a head for the first time on the kth toss of an unbiased coin. Does it matter what the entry fee is? (The St. Petersburg paradox).

5.

The Central Limit Theorem: A sum of a large number of independent identically distributed random variables follows a distribution that is approximately Normal.