information theory, find entropy given Markov chain

Question

There is an information source on the information source alphabet $A = \{a, b, c\}$ represented by the state transition diagram below:

a) The random variable representing the $i$-th output from this information source is represented by $X_i$. It is known that the user is now in state $S_1$. In this state, let $H (X_i|s_1)$ denote the entropy when observing the next symbol $X_i$, find the value of $H (X_i|s_1)$, entropy of this information source, Calculate $H (X_i|X_{i-1}) $ and $H (X_i)$ respectively. Assume $i$ is quite large

How can I find $H(X_i|s_1)?$ I know that $$H(X_i|s_1) = -\sum_{i,s_1} p\left(x_i, s_1\right)\cdot\log_b\!\left(p\left(x_i|s_1\right)\right) = -\sum_{i,j} p\left(x_i, s_1\right)\cdot\log_b\!\left(\frac{p\left(x_i, s_1\right)}{p\left(s_1\right)}\right)$$ but I don't know $p(s_1)$.

$$A=\begin{pmatrix}0.25 & 0.75 & 0\\0.5 & 0 & 0.5 \\0 & 0.7 & 0.3 \end{pmatrix}.$$

From matrix I can know that $p(s_1|s_1)=0.25$, etc.

But what is the probability of $s_1$? And how can I calculate $H (X_i|X_{i-1})$?

Answer 1

You can make the golloowing set of equations, where s1 - the probability of being in s1 state:

s1 = 0.25s1 + 0.5s2
...
s1+s2+s3=1

I don't know the theory, but think that entropy of each state can be computed as follows:

e1 = - 0.25*log(0.25) - 0.75*log(0.75)

with overall entropy given already computed probability and entropy of each state

e = e1*s1 + e3*s3 + e3*s3

I don't know what H(...) means.