Hidden Markov Models

Summary

The video introduces the concept of hidden Markov models, where the underlying state is unobservable, but observable events provide clues. Using a weather example, it explains emission probabilities, illustrating how certain clothing items signal specific weather conditions. The focus shifts from predicting state sequences to analyzing the likelihood of observation sequences. By calculating probabilities based on various weather scenarios, the speaker highlights the importance of considering all potential weather patterns to determine the probability of observing specific items. This foundation sets the stage for deeper exploration of hidden Markov models in future discussions.

Takeaways

• 😀 A Hidden Markov Model (HMM) is used when the true state of a system cannot be observed directly.
• 🌤️ Observable elements, such as clothing, provide indirect evidence of the underlying state (like weather).
• 📈 Emission probabilities represent the likelihood of observing a specific symbol given a certain hidden state.
• 🧥 For example, the probability of seeing a bathing suit when it is sunny can be quantified using emission probabilities.
• 🔄 The model emphasizes analyzing sequences of observations rather than the hidden states themselves.
• 🔍 To calculate the likelihood of a sequence of observations, all possible state sequences must be considered.
• 💡 The formula combines the probabilities of each state sequence and the corresponding emission probabilities.
• 🌧️ Different weather conditions will lead to different probabilities for the observed clothing (coats, umbrellas, etc.).
• 📊 Understanding HMMs helps in modeling systems where direct observation is not possible, using related observable data.
• 🔗 HMMs are applicable in various fields, including weather forecasting, speech recognition, and bioinformatics.

Q & A

What is the primary difference between a standard Markov model and a hidden Markov model?

The primary difference is that in a hidden Markov model, the actual states are not observable. Instead, we can only observe certain evidence or outputs related to those states.

In the weather example provided, what are considered observable elements?

Observable elements in the weather example include items that people wear or carry, such as bathing suits, coats, and umbrellas, which provide clues about the weather state.

What are emission probabilities in the context of hidden Markov models?

Emission probabilities refer to the likelihood of observing a certain symbol or output given a specific hidden state. For instance, it quantifies the chance of seeing a bathing suit when the weather is sunny.

How is the likelihood of a sequence of observations computed in a hidden Markov model?

The likelihood of a sequence of observations is computed by considering all possible sequences of hidden states and their corresponding emission probabilities.

What does the notation 'B_sub_j of k' represent?

'B_sub_j of k' denotes the probability of observing symbol k given that the system is in a specific hidden state j.

Why does the order of observations not matter when calculating probabilities in a hidden Markov model?

The order does not matter because the probabilities are computed based on the current state alone, indicating that each observation is independent of others when the state is known.

How does one evaluate the probability of a specific sequence of observations?

To evaluate the probability of a specific sequence of observations, you need to account for all possible sequences of states and multiply the respective probabilities of the observations given those states.

What example is given for a specific sequence of weather states?

An example provided is a sequence where every day is sunny, with specific probabilities for observations such as coats and umbrellas in that scenario.

How can we mathematically represent the probability of observations given a sequence of states?

The probability can be represented as the product of the probability of the sequence of states and the emission probabilities for the corresponding observations.

What mathematical operation is used to express the likelihood of multiple observations from the same state?

The likelihood is expressed using multiplication, as shown in the example where the probability of observing multiple umbrellas given that it is raining is calculated.