Probability distributions

relevancy.edu

January 23, 2014

tcausey@zulily

What are probability distributions?

- Descriptions of data
generating processes

- Functions

- Probability mappings for
possible outcomes in
sample space

Two classes of distributions

- Probability density functions (PDFs):

outcomes are real numbers
(continuous)

- Probability mass functions (PMFs):

outcomes are discrete or
non-numeric

Descriptions

Where do distributions come from?

- Many are empirically derived

- Someone recorded many observations

- Derived the function that
best fit the observed frequencies

Student's t distribution

William Sealy Gosset
(working with Karl Pearson)

Yields of barley
while brewing at Guinness

Approximates Gaussian
distribution with small samples

Functions

PDF in its generic form:

f(x) = P(X = x)

(probability of observing a value
of x for a random variable X)

Functions

That f(x) takes a different form
depending on the distribution.

Gaussian distribution:

Note there are three non-constants:

sigma, x, and mu

Functions

f(x) = P(X = x)

Output of the function is a probability
(i.e., a positive real number between
0 and 1, inclusive)

Integral over the entire
sample space must equal 1.

(Or, for a PMF, the area must equal 1)

Probability

Really, the P(X = x) is infinitesimally small, ~ 0
for continuous PDFs

Often give probabilities by integrating
over an interval

i.e., what's the probability
of the interval
10.01 <= x <= 10.02

Parameters

Distributions have one or more parameters

Given the parameters, you can output
a probability for any value of x.

Gaussian distribution (aka "normal distribution"):

parameters are mean (mu) and variance (sigma^2)

Why?

Why even bother using a PDF?

Why not just use data?

(aka empirical PDFs/CDFs)

Parametric PDFs map
probabilities to all
outcomes, not just
observed outcomes

Things to consider

1) Do you already have data?

2) Do you have substantive knowledge
that suggests your data will follow
a (family of) distribution(s)?

3) Can you reasonably fit a

- Do my data approximate some
known physical process?

- Are most of the data in the
middle of the possible values,
trailing off evenly as values get smaller/bigger?

- Is there some kind of exponential decay process?

- Can my data take on any value?

Can my data be non-negative?

Fitting a distribution to data

- Two (classes of) question(s) to answer

- What family of distribution do my data approximate?

- What parameters describe that distribution?

Maximum likelihood estimation

Asked another way, what are the
parameters that maximize the likelihood
of observing the data I have in front of me?

The product of the probabilities
produced by a PDF with a given
(set of) parameter(s) theta

Prob X1 and X2 and X3 ... and Xn

Maximum likelihood estimation

The set of parameters theta-hat
that maximize the likelihood of
the data are called the
maximum likelihood estimator (MLE)

Requires calculating product
of many small floating point values,
so usually maximize the log likelihood

(or minimize the negative log likelihood)

Common distributions

Uniform
Bernoulli
Gaussian ('normal')
Beta-binomial
Exponential
Weibull

Uniform distribution

Dice rolls.
All outcomes are equally likely.

PMF: f(x) = 1/n

Has only one parameter,
the number of discrete outcomes.

PDF: f(x) = 1/(b - a)

Where all outcomes are on the interval [a, b]

Often used as a 'non-informative prior'

Gaussian ('normal')

Often a good first stop.

Two parameters, mu and sigma.

Bernoulli

Coin flips.

PDF:

Has only one parameter,
p (P(k = 1))

Binomial distribution

Series of independent Bernoulli trials

PMF:

Probability of k successes

Parameters are n (number of trials)
and p (probability of a success)

Used when you have a dichotomous
discrete outcome

Beta distributionDistribution over probabilities

where B is the Beta function

Two parameters, alpha and beta (we all know and love them)
Note how parameters change shape  and allow for increased/decreased uncertainty
around expectation

Exponential

PDF: λ e^(−λx)

Rate parameter lambda
Events that follow a Poisson process

Weibull

Survival analysis
Demography
Industrial engineering (mechanical failure times)
When k = 1, it's the exponential distribution

Multivariate distributions

Joint probability of two or more outcomes
PDF where Sigma is the covariance matrix

Dimensionality becomes an issue
Can treat as univariate distribution while holding
constant other variables

Other common distributions

Gamma
Multivariate normal
Beta-binomial
Wishart

Fun one:
Cauchy (mean & variance are undefined)
Used in particle physics

Another fun thing to do

Cumulative density function (CDF)
First integral of the PDF

Rather than P(X = x), P(X <= x)

Gaussian CDF:

They all look the same.

Bayesian methods

Treat all parameters as random variables
with associated probability distributions
rather than known quantities.

Distributions allow us to