What is the Markov assumption for the bigram model

The Markov assumption for the bigram model is a key assumption made in natural language processing (NLP) and probabilistic modeling. It states that the probability of observing a word in a sequence depends only on the preceding word, specifically the most recent word in the sequence.

Mathematically, the Markov assumption for the bigram model can be expressed as:

\[ P(w_n | w_1, w_2, \ldots, w_{n-1}) \approx P(w_n | w_{n-1}) \]

Where:
- \( w_n \) is the current word in the sequence.
- \( w_{n-1} \) is the preceding word.
- \( P(w_n | w_1, w_2, \ldots, w_{n-1}) \) is the conditional probability of observing word \( w_n \) given the entire history of preceding words.
- \( P(w_n | w_{n-1}) \) is the conditional probability of observing word \( w_n \) given only the preceding word \( w_{n-1} \).

In other words, according to the Markov assumption for the bigram model, the probability of the current word in a sequence depends only on the immediately preceding word, rather than the entire history of preceding words. This assumption simplifies the modeling process by reducing the amount of information needed to estimate the probabilities of word sequences, making it more computationally tractable.

The bigram model is a simple but widely used probabilistic model in NLP, particularly for tasks such as language modeling, part-of-speech tagging, and text generation. It assumes that each word in a sequence depends only on the preceding word, which allows for efficient estimation of probabilities and generation of text based on observed word frequencies.