On Sat, Apr 19, 2003 at 05:59:55AM +0900, Daniel Carrera wrote:
> On Sat, Apr 19, 2003 at 03:03:44AM +0900, Chris Pine wrote:
> > ----- Original Message -----
> > From: "Chris Pine" <nemo / hellotree.com>
> > 
> > |  0,   0.5,  0.5  |
> > |  .67, 0,    .33  |
> > |  .67, .33,  0    |
> > 
> > 
> > |  0,   .25,  .75  |
> > |  1,   0,    0    |
> > |  .33, .67,  0    |
> > ----------------------------
> > 
> > Looking at this again, it seems that the latter matrix is flawed.  Not only
> > is it uglier (a reasonable complaint), but it isn't "locally random", so to
> > speak.  The long term probabilities are correct, but every 'b' is followed
> > by an 'a'.
> 
> My bigger concern is that those columns don't add up to 1.
> These aren't probability matrices.

The rows do. I guess the transposed is OK, then :-)
We're doing v' = vP where you expected u' = Pu. 

> > Perhaps a matrix is "naturally balanced" if the ratio of any two
> > probabilities in any row is the same as the ratio of the corresponding
> > probabilities in your initial vector (ignoring the zeros of the diagonal).
> 
> The idea I proposed was based on the objective of keeping the ratios the 
> same (except for the event that is not supposed to repeat).

But it does change the stationary distribution, doesn't it?

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