Viterbi once again!

• Model generates numbers– 312453666641

1:1/62:1/63:1/64:1/65:1/66:1/6Fair

1:1/102:1/103:1/104:1/105:1/106:1/2Loaded

0.95

0.10

0.05

0.9

The unfair casino: Loaded dice p(6) = 0.5; switch fair to load:0.05; switch load to fair: 0.1

Model decoding (Viterby)

• Example: 566. What was the series of dice used to generate this output?

1:1/62:1/63:1/64:1/65:1/66:1/6Fair

1:1/102:1/103:1/104:1/105:1/106:1/2Loaded

0.95

0.10

0.05

0.9

FFF = 0.167*0.95*0.167*0.95*0.167 = 0.0042

FFL = 0.167*0.95*0.167*0.05*0.5 = 0.00666

FLF = 0.167*0.05*0.5*0.1*0.167 = 0.000070

FLL = 0.167*0.05*0.5*0.9*0.5 = 0.0019

LFF

LFL

LLF

LLL

Model decoding (Viterby)

• Example: I now add one more say 5!• After my first 3 throws I have ended up

having either a loaded or a fair dice in my hand

• The most likely “path” to a loaded dice is

FLL = 0.167*0.05*0.5*0.9*0.5 = 0.0019

• The most likely path to a fair dice isFFF = 0.167*0.95*0.167*0.95*0.167 = 0.0042

• Since the Markov model has no memory, the most likely path to having a fair dice after the 4th through is

1:1/62:1/63:1/64:1/65:1/66:1/6Fair

1:1/102:1/103:1/104:1/105:1/106:1/2Loaded

0.95

0.10

0.05

0.9

€

Max(FLLF,FFFF) = 0.167 ⋅max(FFF ⋅0.95,FLL ⋅0.1)

€

Max(FLLF,FFFF) = 0.167 ⋅max(0.0042 ⋅0.95,0.0019 ⋅0.1)

€

Max(FLLF,FFFF) = 0.167 ⋅max(0.004,0.0002)

€

Max(FLLF,FFFF) = 0.00067

Model decoding (Viterby)

1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78

Fair

1:-12:-13:-14:-15:-16:-0.3

Loaded

-0.02

-1

-1.3

-0.05Log model

1 2 4 5 6 6 6

F -0.78 -1.58

L Null -3.08 -3.88

€

log(PL (4)) = −1−0.05 − 3.08 = −4.13 or

log(PL (4)) = −1−1.3−1.58 = −3.88

Model decoding (Viterby)

1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78

Fair

1:-12:-13:-14:-15:-16:-0.3Loaded

-0.02

-1

-1.3

-0.05Log model

1 2 4 5 6 6 6

F -0.78 -1.58

L Null -3.08 -3.88

Identify what series of dice was used to generate this output?

Model decoding (Viterby)

1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78

Fair

1:-12:-13:-14:-15:-16:-0.3Loaded

-0.02

-1

-1.3

-0.05Log model

1 2 4 5 6 6 6

F -0.78 -1.58 -2.38 -3.18 -3.98 -4.78 -5.58

L Null -3.08 -3.88 -4.68 -4.78 -5.13 -5.48

Series of dice is FFFFLLL