Sir R----- my fine fellow! Come in from the cold and join me at my table for a tumbler of restorative spirits!

Might I also tempt you with a wager?

Good man!

I propose a game that was popular amongst the notoriously unsuccessful lunar prospectors of '29. Spurred on by rumours of gold nuggets scattered upon the ground simply for the taking, they arrived en-masse during winter woefully unprepared for the inclement weather. By the time that I arrived on a diplomatic mission to the king of the moon people they were in a frightful state, desperately short of provisions and futilely trying to work the frost bitten land to grow more.

Some time ago we saw how Newton's method used the derivative of a univariate scalar valued function to guide the search for an argument at which it took a specific value. A related problem is finding a vector at which a multivariate vector valued function takes one, or at least comes as close as possible to it. In particular, we should often like to fit an arbitrary parametrically defined scalar valued functional form to a set of points with possibly noisy values, much as we did using linear regression to find the best fitting weighted sum of a given set of functions, and in this post we shall see how we can generalise Newton's method to solve such problems.

The Baron and Sir R-----'s latest wager comprised of first placing a draught piece upon the fifth lowest of a column of twelve squares and subsequently moving it up or down by one space depending upon the outcome of a coin toss until such time as it should escape, either by moving above the topmost or below the bottommost square. In the former outcome the Baron should have had a prize of three coins and in the latter Sir R----- should have had two.

Last time we took a look at linear regression which finds the linear function that minimises the differences between its results and values at a set of points that are presumed, possibly after applying some specified transformation, to be random deviations from a straight line or, in multiple dimensions, a flat plane. The purpose was to reveal the underlying relationship between the independent variable represented by the points and the dependent variable represented by the values at them.
This time we shall see how we can approximate the function that defines the relationship between them without actually revealing what it is.

Over the course of the year my fellow students and I have been utilising our free time to explore the behaviour of cellular automata, which are mechanistic processes that crudely approximate the lives and deaths of unicellular creatures such as amoebas. Specifically, they are comprised of unending lines of boxes, some of which contain cells that are destined to live, dive and reproduce according to the occupancy of their neighbours.
Most recently we have seen how we can categorise automata by the manner in which their populations evolve from a primordial state of each box having equal chances of containing or not containing a cell, be they uniform, constant, cyclical, migratory, random or strange. It is the latter of these, which contain arrangements of cells that interact with each other in complicated fashions, that has lately consumed our attention and I shall now report upon our findings.

Several months ago we saw how we could use basis functions to interpolate between points upon arbitrary curves or surfaces to approximate the values between them. Related to that is linear regression which fits a straight line, or a flat plane, though points that have values that are assumed to be the results of a linear function with independent random errors, having means of zero and equal standard deviations, in order to reveal the underlying relationship between them. Specifically, we want to find the linear function that minimises the differences between its results and the values at those points.

Season's greetings Sir R-----! Come take a glass of mulled wine to warm your spirits on this chill winter's night!

Will you also accept a wager to warm your blood?

It gladdens my heart to hear so sir!

I propose a game that oft puts me in mind of the banquet held in the great hall upon Mount Olympus to which I was invited as the guest of honour by Zeus himself!

Over the last few months we have been looking at Bernoulli processes which are sequences of Bernoulli trails, being observations of a Bernoulli distributed random variable with a success probability of p. We have seen that the number of failures before the first success follows the geometric distribution and the number of failures before the r^th success follows the negative binomial distribution, which are the discrete analogues of the exponential and gamma distributions respectively.
This time we shall take a look at the binomial distribution which governs the number of successes out of n trials and is the discrete version of the Poisson distribution.

Recall that the Baron's most recent game involved advancing kings from the first and last ranks of a three by three chequerboard in a pawn-like manner until either he or Sir R----- reached the opposing rank or blocked all of the other's kings from moving, having the game in either eventuality.

Lately we have been looking at Bernoulli processes which are sequences of independent experiments, known as Bernoulli trials, whose successes or failures are given by observations of a Bernoulli distributed random variable. Last time we saw that the number of failures before the first success was governed by the geometric distribution which is the discrete analogue of the exponential distribution and, like it, is a memoryless waiting time distribution in the sense that the distribution for the number of failures before the next success is identical no matter how many failures have already occurred whilst we've been waiting.
This time we shall take a look at the distribution of the number of failures before a given number of successes, which is a discrete version of the gamma distribution which defines the probabilities of how long we must wait for multiple exponentially distributed events to occur.