Design a site like this with WordPress.com

# Go Ahead, Make My Day. Excommunicate Me!

(Note: the Markov chain example starts about halfway through the post).

One of the issues that the Roman Catholic Church wrestles with on an ongoing basis concerns the status of those politicians which a) profess and call themselves Roman Catholics yet b) show that they do not ascribe to the teaching of the Church in the way they vote and the positions they take.  Since the Roman Catholic Church expects the faithful to follow its teachings without reservation, the question comes up: why doesn’t the Church excommunicate these people?

The answer to that question is like a lot of things these days: it’s complicated.  It ranges from the desire of the Church not to be unduly unpopular to not wanting the state to retaliate against it for such an action.  It also stems from the fact that neither the United States nor any other nation can be called a truly Catholic country, and thus the standard of expectation is not the same.

It’s fair to say that the Roman Catholic Church isn’t the only church that experiences this difficulty.  The tale that this blog piece deals with comes from Russia, a place where many strange things happen and many unexpected results come to reality.

In 1901 the Russian Orthodox Church excommunicated Lev Nikolayevitch Tolstoi, the famous Russian writer.  He had developed an idea of Christian anarchism and pacifism which (among other of his ideas) was unacceptable to the church.  The church wasn’t the only one unhappy with Tolstoi’s idea: in his last years his wife was increasingly disenchanted with his desire to renounce his wealth, as he came from an aristocratic background.

One of those in Russia who was likewise disenchanted with the state of things was the mathematician Andrei Andreyevitch Markov.  Markov is best known for his development of Markov Chains, an example of which can be found at the end of this piece.  Markov, far from basing his idea on Christianity, was an atheist.  Nevertheless, in protest of the Church’s excommunication of Tolstoi, he requested that the Church excommunicate him too.  The church made his day and did so, and he remained outside of its communion until his death after the Revolution.

It’s easily forgotten today, but Tolstoi was very influential in the development of non-violent resistance and action towards social change.  That influence was more felt outside of Russia through people such as Gandhi (India), Martin Luther King (United States) and in his later years Nelson Mandela (South Africa).  For that to be effective requires conditions which were present in the last century but which may be a thing of the past today.

In Russia, Markov’s fellow atheists the Bolsheviks took a more violent (and I might say a non-Markov Chain type of) course.  After the collapse of the Soviet Union, the same Russian Orthodox Church, almost driven to extinction under Stalin, made a comeback. After seventy years of atheism and all that went with it, Russians are chary of attacking an institution which stood against it, as Pussy Riot found out the hard way.  And they’re still hanging tough on Tolstoi’s excommunication.

For all of Russia’s strange and sometimes horrific history–a history that continues to play out in our time, with impact everywhere–one has to admire Markov when comparing him to the mealy-mouthed politicians who use their religious affiliation to garner votes which at the same time acting and voting in ways which go against its precepts.

Markov chains concern themselves with predicting the outcome of a sequence of events given the probability of an outcome at each step.  The following example comes from Marvin Marcus’ A Survey of Finite Mathematics (Boston: Houghton Mifflin Company, 1969).  We present another interesting example from Marcus’ book here.

In any case, consider a population of women which, like Gaul, is divided into three parts:

1. Those who are overweight at 40;
2. Those who are underweight at 40;
3. Those who are normal weight at 40.

(The example doesn’t define the break points for weight, which is always the tricky parts in studies like this).

The transference of this condition from mother to daughter runs as follows:

1. For mothers who are overweight at 40, 70% of their daughters are likewise, 20% are underweight and 10% are normal weight.
2. For mothers who are underweight at 40, 30% of their daughters are overweight, 50% are likewise, and 20% are normal weight.
3. For mothers who are normal weight at 40, 15% of their daughters are overweight, 60% are underweight, and 25% are likewise.

We arrange these results in what we call a transition matrix, shown below.  Each category of mother represents a column in the matrix and each category of daughter represents a row in the matrix.

P=\left [\begin {array}{ccc} {\frac {7}{10}}&3/10&{\frac {3}{20}} \\\noalign{\medskip}1/5&1/2&3/5\\\noalign{\medskip}1/10&1/5&1/4 \end {array}\right ]

We now want to diagonalise the matrix.  We do this first by finding the eigenvalues and eigenvectors for the matrix.  These are reproduced below, with the following notation: [eigenvalue, number of occurrences, {[eigenvector]}]

$[[{\frac {9}{40}}+1/40\,\sqrt {73},1,\left \{[-7/4-1/4\,\sqrt {73},3/4 +1/4\,\sqrt {73},1]\right \}],[{\frac {9}{40}}-1/40\,\sqrt {73},1, \left \{[-7/4+1/4\,\sqrt {73},3/4-1/4\,\sqrt {73},1]\right \}],[1,1, \left \{[{\frac {17}{14}},1,3/7]\right \}]]$

In floating point form, the eigenvalues are .4386000936, .0113999064 and 1.

We now construct a matrix of the eigenvectors and its inverse, as follows:

Q=\left [\begin {array}{ccc} {\frac {17}{6}}&-{\frac {13}{16}}-1/16\, \sqrt {73}&-{\frac {13}{16}}+1/16\,\sqrt {73}\\\noalign{\medskip}7/3&1 &1\\\noalign{\medskip}1&-3/16+1/16\,\sqrt {73}&-3/16-1/16\,\sqrt {73} \end {array}\right ]

Q^{-1}=\left [\begin {array}{ccc} {\frac {6}{37}}&{\frac {6}{37}}&{\frac {6}{ 37}}\\\noalign{\medskip}-{\frac {1}{2701}}\,\left (69+7\,\sqrt {73} \right )\sqrt {73}&{\frac {1}{5402}}\,\left (-27+23\,\sqrt {73}\right )\sqrt {73}&-{\frac {1}{2701}}\,\left (-227+7\,\sqrt {73}\right ) \sqrt {73}\\\noalign{\medskip}-{\frac {1}{2701}}\,\left (-69+7\,\sqrt {73}\right )\sqrt {73}&{\frac {1}{5402}}\,\left (27+23\,\sqrt {73} \right )\sqrt {73}&-{\frac {1}{2701}}\,\left (227+7\,\sqrt {73}\right )\sqrt {73}\end {array}\right ]

Careful observers will note that there is a scalar multiple between the original eigenvectors and these arrays.  This is an artefact of a struggle with Maple I didn’t quite win, and will cancel itself out in the diagonalisation process.

That being the case, we multiply them to obtain

$D=Q^{-1}PQ=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & {\frac{9}{40}}+1/40\,\sqrt{73} & 0\\ 0 & 0 & {\frac{9}{40}}-1/40\,\sqrt{73} \end{array}\right]$

We obtain, as we would expect, a matrix with the eigenvalues along the diagonal.

We then use the diagonalising matrices again by multiplying to obtain the distribution of results after an “infinite” number of generations, thus

$A=QDQ^{-1}=\left[\begin{array}{ccc} \frac{17}{37} & \frac{17}{37} & \frac{17}{37}\\ \frac{14}{37} & \frac{14}{37} & \frac{14}{37}\\ \frac{6}{37} & \frac{6}{37} & \frac{6}{37} \end{array}\right]$

The result is as we want: the three columns are identical.  That result is a good check on whether your result is correct; I found it very easy to make mistakes in the entries of the transition matrix, which will show up if either those entries are invalid (or unsuitable for a Markov chain) or an incorrect value is entered.

The three rows represent the final outcomes of the chain.  Thus, in this case, the top row represents the women who will be overweight at age 40, the second row those who will be underweight, and the bottom those of normal weight.  We thus see that the result is that at age 40, 46% of the women will be overweight, 38% be underweight, and 16% be of normal weight.

Notes:

1. The example above leaves out a great deal of the theory of how the diagonalisation process is used to analyse the Markov chain.  Marcus goes into this in some detail, but in the example he actually uses another method to get his result.
2. Although some will find this example objectionable, linear algebra is full of examples like this.  When I took advanced linear algebra, I fell ill during Spring Break, and ended up in a Catholic hospital, where I saw the election of Pope Francis.  (I told them during the process, “You better pay attention, you’re getting a new boss…”)  I came back from this experience only to be presented with an example involving people dying in the hospital!

## 3 Replies to “Go Ahead, Make My Day. Excommunicate Me!”

1. Don,

Thanks for the note about my Twitter being hacked. I had another one from Microsoft Security with a scary list of the people who were hitting on several of my accounts. (the known ones, i.e. that they had identified.) Five or six, repeatedly, in Eastern Russia, Turkmenistan, and China!

I’m still paying the price of being hacked about four years ago, so I’m taking this seriously. Fortunately the people at PayPal and my bank, HSBC, are pros.

There are some difficulties. E.g., I’ve been on the Internet Engineering Task Force since 1984, and was active in its development when I was a consultant to MIT, in 1971~72. But this means I have to negotiate my own security, which is sometimes difficult — because I think I’m being careful when I don’t write things down. Duh! 🙂

Anyway, I should have all me accounts in order in another week or so.

Again, thanks,

-dlj.

Like

1. me says:

Good to hear from you. It’s never easy to get that kind of thing wiped from your system once it gets in there.

The truth is that, even though there are safeguards out there, just about everyone has to negotiate their own security.

Like