# Calibrating a Markov Chain with little data

I am trying to calibrate a Markov Chain. Usually you would have the amount that "moves" from one state to the other and then the resulting value, with data at any given time with the following arrangement:

$$\begin{matrix} F_{a,a} & F_{a,b} & F_{a,c}\\ F_{b,a} & F_{b,b} & F_{b,c}\\ F_{c,a} & F_{c,b} & F_{c,c}\\ \end{matrix}$$

The problem is that I have data in this arrangement for the complete set:

$F_i = F_{i,a}+F_{i,b}+F_{i,c}$ for $i = a,b,c$

I was wondering if there was any way I could obtain the $F$ parameters for all the combinations of $a,b$ and $c$; some sort of proxy of them; or even a way of getting the resulting parameters of the calibrated model with this data.

• Little data just means your trained parameters may be inaccurate, not that you can't train. What prevents you from using the usual training algorithms? I don't understand what you are doing with your $F_i$; surely, $F_i = 1$ always? – Raphael Jul 13 '15 at 15:08
• I meant the $F$s as the flux that goes from one state to the other, so in this case its just amounts of money that go from one investment to the other. Its the resulting transition matrix (Where the rows do sum 1) that I'm having trouble getting to. – eduardo0 Jul 13 '15 at 19:01