Topics 3 Probable

There many perspectives on statistics and probability stemming from the two predominate schools, the frequentest and the Bayesian’s. It’s not a choice between the two schools, one needs to adopt their own perspective or mental model of to master this body of knowledge.

The perspective that dominates these notes, can be labelled, if it must, as ‘rational quantification’. It’s a perspective that is can be characterised as:

  • Identification & measurement of known unknown by deduction.

  • Seeking to identify the unknown unknowns collectivity or individually by inference.

  • Focusing on growing a collection of models that can survive the onslaught of limited, noise laden, messy data rather than depending upon transformed and well wrangled data.

  • Practical incorporation of empirical knowledge into Priors.

  • Preference for arriving at outcomes via simulation then analytical calculations.

  • Preference for model that can explain themselves over black box or mathematical devices.

  • Incorporation time vectors in all, but the simplest of models.

Adoption of the above opinionated perspective focuses the descriptions and examples in these notes, where shortcoming are evident they are called out.

3.1 Counting

At the core of probability is ability to count (Combinatorics) the number of way a given value/outcome could arise.

  • A ‘Conjectures’ is the label given to a possible outcome.
  • The plausibility of conjecture is in portion to the number of way it could produce the observed outcome.
  • Not all conjectures start with the same plausibility.
  • Assuming all conjectures are initially equally likely is a form of ignorance, as it excludes scientific context.

When looking a set of objects, asking yourself: ‘how many … ?’ your in the land of combinatorics.

  • Combinations
  • Permutations
  • Subsets

3.1.1 Factorial

These are direct combinations of the members within a set, where each member can combine with any other member. The number of possible ways they could combine is given be the Factorial of number of members in the set.

Where a set has:

1 Member = 1! = 1,

2 Members = 2! = 1 x 2 = 2,

3 Members = 3! = 1 x 2 x 3 = 6,

4 Members = 4! = 1 x 2 x 3 x 4 = 24,

5 Members = 5! = 1 x 2 x 3 x 4 x 5 = 120,

6 Members = 61 = 1 X 2 X 3 X 4 X 5 X 6 = 720,

Does not take long before there is combination explosion, 10! = 3,628,800 & 20! = 2.43 x 10^18.

R factorial function

factorial(0)
[1] 1
factorial(1)
[1] 1
factorial(3)
[1] 6
factorial(6)
[1] 720
factorial(12)
[1] 479001600
factorial(24)
[1] 6.204484e+23

Dart Lib factorial function

var fact = 6.factorial;
return fact;
-> 720