The fundamental difference between a Hidden Markov Model (HMM) and a Maximum Entropy Markov Model (MEMM) lies primarily in their approach to probability modeling: HMMs are generative models that predict the joint probability of observations and states, while MEMMs are discriminative models that predict the conditional probability of states given observations. Both are directed graphical models commonly used for sequence labeling tasks.
Here's a detailed breakdown of their distinctions:
Understanding HMMs and MEMMs
Hidden Markov Model (HMM)
An HMM is a statistical Markov model where the system being modeled is assumed to be a Markov process with unobserved (hidden) states. It directly models the transition probability between hidden states and the emission (or phenotype) probability of observing a specific output given a hidden state. By modeling these directly, HMMs calculate the probability of co-occurrence of a sequence of observations and a sequence of hidden states.
- Generative Approach: HMMs model the joint probability $P(\text{Observations}, \text{States})$. This means they can be used to generate sequences of observations.
- Independence Assumption: HMMs assume that each observation is conditionally independent of other observations given the hidden state that produced it. The probability of transitioning to a new state depends only on the previous state.
- Parameters: Consists of:
- Initial state probabilities ($P(S_1)$)
- Transition probabilities ($P(St | S{t-1})$)
- Emission probabilities ($P(O_t | S_t)$)
Maximum Entropy Markov Model (MEMM)
An MEMM is a discriminative model that predicts the probability of the next state given the previous state and the current observation. Unlike HMMs, MEMMs establish the probability of co-occurrence of states and observations by directly modeling the conditional probability of a state sequence given an observation sequence. They achieve this by using maximum entropy classifiers to define the transition probabilities, which allows them to incorporate rich, overlapping features from the input observations.
- Discriminative Approach: MEMMs model the conditional probability $P(\text{States} | \text{Observations})$. They are designed to discriminate between possible state sequences for a given observation sequence, not to generate data.
- Flexibility: MEMMs can incorporate a wider range of features from the observation sequence into their transition functions, allowing for more complex relationships between observations and states. The transition probability to a new state is conditioned on both the previous state and the current observation (and potentially surrounding observations).
- "Local Normalization": A critical aspect of MEMMs is that their probability distributions for transitions are normalized locally at each state. This leads to a significant issue known as the "label bias problem."
Key Differences Between HMM and MEMM
| Feature | Hidden Markov Model (HMM) | Maximum Entropy Markov Model (MEMM) |
|---|---|---|
| Model Type | Generative | Discriminative |
| Graphical Model | Directed Graph | Directed Graph |
| Probability Modeled | Joint probability $P(\text{Observations}, \text{States})$ | Conditional probability $P(\text{States} |
| Independence Assumption | Observations conditionally independent given hidden state. | Transition probability depends on previous state and current observation. |
| Transition Probability | $P(S_t | S_{t-1})$ |
| Emission Probability | $P(O_t | S_t)$ (modeled explicitly) |
| Feature Usage | Limited to simple, independent features. | Can use rich, arbitrary, overlapping features from observations. |
| "Label Bias Problem" | No | Yes, a significant issue due to local normalization. |
| Primary Applications | Speech recognition, Bioinformatics, Time-series analysis. | Named Entity Recognition (NER), Part-of-Speech (POS) tagging (often superseded by CRFs). |
The Label Bias Problem in MEMMs
The "label bias problem" is a major limitation of MEMMs. It arises because the transition probabilities in an MEMM are normalized locally, meaning the sum of probabilities of exiting a particular state equals one.
- How it Occurs: States with fewer outgoing transitions tend to "steal" probability mass from states with many outgoing transitions. In essence, an MEMM might favor paths through states with fewer possible next states, regardless of how strong the evidence is for other paths.
- Example: Imagine a state in an MEMM that has only one possible next state. The probability of transitioning to that next state will always be 1 (after normalization) from that state, regardless of the observation. If another state has 10 possible next states, the probability of any single transition will be much lower, even if the evidence for it is strong. This bias can lead to incorrect predictions, as the model might prefer simpler paths even when the data suggests otherwise.
Moving Beyond HMMs and MEMMs: Conditional Random Fields (CRFs)
The label bias problem in MEMMs led to the development of Conditional Random Fields (CRFs). CRFs are also discriminative models, but unlike MEMMs, they perform global normalization over the entire sequence of labels, rather than local normalization at each state. This global perspective allows CRFs to consider the probabilities of all possible output sequences for a given input sequence simultaneously, effectively overcoming the label bias problem. It's also noteworthy that while HMMs and MEMMs are directed graphs, CRFs are often modeled as undirected graphs.
Practical Insights
- HMMs: Simpler to understand and implement. Good for tasks where the independence assumptions hold reasonably well and features are not overly complex. They require less data for training compared to discriminative models.
- MEMMs: Offer more flexibility in incorporating complex features, making them potentially more powerful for tasks like natural language processing. However, the label bias problem can make them less reliable in scenarios with varying numbers of outgoing transitions.
- CRFs: Generally preferred over MEMMs for sequence labeling tasks due to their ability to mitigate the label bias problem. They combine the advantages of discriminative modeling and rich feature sets without the local normalization issue.
In summary, while both HMMs and MEMMs are directed graphical models for sequence analysis, their core difference lies in their probability modeling paradigm—generative vs. discriminative—which in turn dictates their independence assumptions, feature handling capabilities, and susceptibility to issues like the label bias problem.
