Algorithms Q&A - Recent questions in HMM

MLE for given HMM and Observation Sequence

Sat, 25 Apr 2026 15:40:44 +0000

We are given an MDP with 3 states: Red, Green, Blue (RGB).

Emission probabilities:

R --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

G --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

B --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

Transition Matrix

R --> R (p=1/3), G (p=1/3), B (p=1/3)

G --> G (p=1/2), B (p=1/2)

B --> B (p=1)

These are the observations: 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1

What is the most likely explanation for this observation sequence?
Initial state is R.

MLE for given HMM and Observation Sequence

Sat, 25 Apr 2026 13:26:54 +0000

We are given an MDP with 3 states: Red, Green, Blue (RGB).

Emission probabilities:

R --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

G --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

B --> 1 (p=1/3), 2 (p=1/3), 3 (p=1/3)

Transition Matrix

R --> R (p=1/3), G (p=1/3), B (p=1/3)

G --> R (p=1/3), G (p=1/3), B (p=1/3)

B --> R (p=1/3), G (p=1/3), B (p=1/3)

These are the observations: 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2

What is the most likely explanation for this observation sequence?
Initial state is not known and is equally likely from all states.

MLE for the given MDP and Observation Sequence (RGB Color Cycler)

Sat, 25 Apr 2026 01:09:37 +0000

We are given an MDP with 3 states: Red, Green, Blue (RGB).

Emission probabilities:

R --> 1 (p=0.4), 2, (p=0.5), 3 (p-0.1)

G --> 1 (p=0.3), 2 (p=0.3), 3 (p=0.4)

B --> 1 (p=0.4), 2 (p=0.1), 3 (p=0.1), 4 (p=0.4).

Transition Matrix: R -> G (p = 1.0), G -> B (p = 1.0), B -> R (p = 1.0).

These are the observations: 1, 2, 3, 4, 1, 3, 1, 2, 3

What is the most likely explanation for this observation sequence?
Initial state is not known and is equally likely from all states.

Heads - 2 in a row

Tue, 14 Apr 2026 21:06:07 +0000

I have two coins, Fair coin - gives H and T with 0.5 probability each, and a Biased coin, always gives H.

I pick one of the two coins randomly and toss it. It comes up as heads. I toss it again. What is the probability that it comes up as heads again?

Robot States and Sensors

Mon, 08 May 2023 21:05:20 +0000

Suppose we have a robot that can be in one of two states: state 1 or state 2. The robot moves between states according to the following rules:

If the robot is in state 1, it has a 70% chance of staying in state 1 and a 30% chance of transitioning to state 2.

If the robot is in state 2, it has a 60% chance of staying in state 2 and a 40% chance of transitioning to state 1.

The robot has two sensors: sensor 1 and sensor 2. The sensors are not perfect and sometimes give incorrect readings. The emission probabilities for each sensor are as follows:

If the robot is in state 1, sensor 1 gives the correct reading with a probability of 0.8 and an incorrect reading with a probability of 0.2. If the robot is in state 2, sensor 1 gives the correct reading with a probability of 0.4 and an incorrect reading with a probability of 0.6.

If the robot is in state 1, sensor 2 gives the correct reading with a probability of 0.3 and an incorrect reading with a probability of 0.7. If the robot is in state 2, sensor 2 gives the correct reading with a probability of 0.9 and an incorrect reading with a probability of 0.1.

Assume that the robot starts in state 1 with equal probability and generates the following sensor readings over time:

sensor 1: correct, correct, incorrect

sensor 2: incorrect, correct, correct

What is the most likely sequence of states that the robot went through to generate these observations?

Newsletter optimization using HMM

Sun, 02 May 2021 12:59:28 +0000

You are trying to sell a product by using an email newsletter (delivered weekly). A visitor can be interested in the product due to 2 different interests. If they are interested due to interest 1 (for example “soccer”), they may click a link (link 1) you have in the newsletter. If they are interested due to interest 2 (for example “tennis”), they may click a different link (link 2). Here is the probability table that shows the likelihood of viewer clicking these two links:

Interest	Link 1	Link 2
No Interest	0.01	0.05
Interest 1	0.3	0.6
Interest 2	0.4	0.8
Interest 1 and Interest 2	0.8	0.95

We do not know the initial state of the visitor. If the visitor has no interest in interest 1 or interest 2, she a 10% probability of developing either of those interests (independently) during the week. When she has one of the interests, there is a 20% probability of her developing the other interest as well. Once an interest is developed, that interest is never lost.

Draw and explain an HMM to model this problem.
You make the following observations in 8 weeks of newsletter delivers:

No Click, Link 1, Link 1, No Click, Link 1, Link 1 and Link 2, Link 1 and Link 2, Link 2, Link 1 and Link What is the best explanation for the interest level of the visitor during those 8 weeks?