= Solution
For each <match state>, count the nongap <amino acids> and divide by the number of nongap observations in that column. A gap is not one of the twenty emitted <amino acids>; including it in the denominator would make the <emission probabilities> sum to less than one:
``
def emission_counts(column):
counts = Counter(str(x) for x in column if x != ".")
n = sum(counts.values())
if n == 0:
raise ValueError("an all-gap column has no empirical emission distribution")
return counts, n
match_emissions = {}
for state, j in enumerate(match_columns, 1):
counts, n = emission_counts(A[:, j])
match_emissions[state] = {aa: count/n for aa, count in counts.items()}
print(state, j+1, match_emissions[state])
``
The table gives an exact compact representation of \b[every <emission probability>]: divide each displayed count by the denominator $n$ in its row; all unlisted <amino acids> have probability zero. The state number follows the order of <match states>, while the column refers to the original <multiple sequence alignment>.
|| State
|| Alignment column
|| $n$
|| Nongap counts
| 1
| 1
| 9
| I:1, K:1, R:1, S:2, V:3, Y:1
| 2
| 2
| 9
| D:1, E:2, G:1, I:1, L:1, Q:2, S:1
| 3
| 3
| 9
| E:2, I:1, K:1, Q:1, R:3, T:1
| 4
| 4
| 9
| L:8, M:1
| 5
| 5
| 9
| E:3, G:1, L:2, N:1, Q:1, S:1
| 6
| 6
| 9
| A:1, D:2, G:1, K:1, N:1, S:2, T:1
| 7
| 7
| 9
| A:1, E:3, I:2, L:1, M:1, V:1
| 8
| 8
| 9
| I:2, L:7
| 9
| 9
| 9
| D:3, E:1, K:1, L:2, N:1, Q:1
| 10
| 10
| 9
| D:1, E:3, G:1, N:1, P:1, S:2
| 11
| 11
| 9
| A:5, D:1, G:1, P:1, T:1
| 12
| 12
| 9
| I:1, L:7, Y:1
| 13
| 13
| 9
| E:1, F:1, G:1, H:2, M:1, Q:2, T:1
| 14
| 14
| 9
| F:9
| 15
| 15
| 9
| F:1, L:8
| 16
| 16
| 9
| E:1, I:1, L:1, N:3, Q:1, R:1, Y:1
| 17
| 17
| 9
| D:6, E:2, S:1
| 18
| 18
| 9
| C:7, F:1, S:1
| 19
| 19
| 9
| K:1, N:6, R:2
| 20
| 20
| 9
| E:1, F:1, I:5, L:1, V:1
| 21
| 21
| 9
| K:3, P:1, Q:1, S:4
| 22
| 29
| 7
| K:4, P:1, R:1, S:1
| 23
| 30
| 7
| E:2, F:1, G:1, I:1, K:1, V:1
| 24
| 31
| 7
| D:3, E:1, G:1, K:1, S:1
| 25
| 32
| 8
| D:1, E:1, G:1, K:1, N:1, R:3
| 26
| 33
| 8
| D:7, N:1
| 27
| 34
| 8
| A:1, P:6, S:1
| 28
| 35
| 9
| C:1, H:1, I:1, K:1, M:1, R:1, T:3
| 29
| 36
| 9
| E:1, F:1, G:1, L:1, Q:3, W:1, Y:1
| 30
| 37
| 9
| I:5, L:1, T:2, Y:1
| 31
| 38
| 9
| A:1, G:1, P:3, S:4
| 32
| 39
| 9
| I:1, K:6, L:1, R:1
| 33
| 40
| 9
| F:1, N:1, Q:4, T:2, V:1
| 34
| 41
| 9
| A:1, F:1, I:1, V:6
| 35
| 42
| 9
| I:1, L:7, V:1
| 36
| 43
| 9
| A:1, C:1, E:1, N:1, R:1, S:3, V:1
| 37
| 44
| 9
| D:3, E:5, Q:1
| 38
| 45
| 9
| A:1, C:7, I:1
| 39
| 46
| 9
| K:2, L:3, R:4
| 40
| 47
| 9
| A:3, E:1, L:1, N:3, T:1
| 41
| 48
| 9
| I:4, L:1, T:1, V:3
| 42
| 49
| 9
| L:9
| 43
| 50
| 9
| D:1, E:1, H:1, Q:1, S:1, T:1, V:1, Y:2
| 44
| 51
| 9
| E:1, K:1, Q:2, R:1, V:4
| 45
| 52
| 9
| L:9
| 46
| 53
| 9
| G:9
| 47
| 54
| 9
| P:8, T:1
| 48
| 55
| 9
| W:9
| 49
| 56
| 9
| A:2, C:7
| 50
| 57
| 9
| A:6, C:1, L:1, T:1
| 51
| 58
| 9
| A:1, D:3, F:1, S:2, V:1, Y:1
| 52
| 59
| 9
| K:4, R:2, S:3
| 53
| 60
| 9
| A:2, F:1, I:2, L:2, V:2
| 54
| 61
| 9
| A:7, G:1, Y:1
| 55
| 62
| 9
| E:3, G:3, Q:3
| 56
| 63
| 9
| I:3, M:3, T:1, Y:2
| 57
| 64
| 9
| F:3, L:2, M:3, N:1
| 58
| 65
| 9
| H:1, I:3, L:1, V:4
| 59
| 66
| 9
| K:1, M:1, P:1, R:4, S:1, T:1
| 60
| 67
| 9
| D:1, E:3, Q:4, Y:1
| 61
| 68
| 9
| I:1, L:7, M:1
| 62
| 69
| 9
| D:1, E:4, Q:4
| 63
| 70
| 9
| K:7, R:2
| 64
| 71
| 9
| A:1, I:2, M:1, N:1, R:1, Y:3
| 65
| 72
| 9
| A:1, C:1, D:1, E:2, I:2, V:2
| 66
| 73
| 9
| K:9
| 67
| 74
| 9
| H:4, L:1, Q:2, S:1, V:1
| 68
| 75
| 9
| E:8, Q:1
| 69
| 76
| 9
| Q:3, R:1, S:2, T:3
| 70
| 77
| 9
| C:2, E:3, G:1, R:1, S:1, V:1
| 71
| 78
| 9
| A:1, D:2, E:3, I:1, K:1, Y:1
| 72
| 79
| 9
| I:2, L:3, M:1, N:1, Q:1, V:1
These are unregularized <maximum-likelihood estimates> of the <emission probabilities>. No additional artificial observations are added. Although only emissions are requested here, a full <profile hidden Markov model> would also require transitions and deletion mechanisms to represent the gaps that remain in some match columns.
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