I have been trying to understand hidden Markov models with observational probability but I often find myself confused. I have discussed with my tutor for further help however, he is often rude and does not help and so I have decided to turn to the community.
I am trying to determine the probability of observing the following sequences: AABGA, ABBGT.
From what I understand, you would first need to find probabilities of the sequence occurring without observations and then begin looking for probabilities of the sequence with observational probability. I think I am often confused on questions like this because I do not know how I could check my answer to ensure I have obtained all the possible probabilities of the sequence. Would appreciate on insight into whether my approach was correct and if I have correctly obtained the probability of observing the sequences. Thanks in advance.
States with their initial probabilities
$A(0.3) $
$A(0.4) $
$B(0.2) $
$G(0.0) $
$T(0.1) $
Markov model - Updated based on suggestion:
AABGA
$A,A,A,G,G = 0.3 * 0.7 * 0.1 * 0.5 * 0.2 = 0.21%$
$A,A,A,A,A = 0.3 * 0.7 * 0.1 * 0.3 * 0.5 = 0.315%$
$A,A,A,A,A = 0.3 * 0.7 * 0.1 * 0.3 * 0.1 = 0.063%$
$A,A,A,G,G = 0.4 * 0.5 * 0.1 * 0.5 * 0.2 = 0.2%$
$A,A,A,G,G = 0.4 * 0.1 * 0.1 * 0.5 * 0.2 = 0.04%$
$A,A,A,A,A = 0.4 * 0.1 * 0.1 * 0.3 * 0.5 = 0.06%$
$A,A,A,A,A = 0.4 * 0.1 * 0.1 * 0.3 * 0.1 = 0.012%$
$P(AABGA)$ = 0.21% + 0.315% + 0.063% + 0.2% + 0.04% + 0.06% + 0.012% = $0.9%$
ABBGT
$A,A,A,G,G = 0.4 * 0.1 * 0.1 * 0.5 * 0.2 = 0.04%$
$A,A,A,A,T = 0.4 * 0.1 * 0.1 * 0.3 * 0.5 = 0.06%$
$P(ABBGT)$ = 0.04% + 0.06% = $0.1%$