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Markov assumption is a fundamental concept in probability theory and machine learning, particularly in Markov models, which is named after the Russian mathematician Andrey Markov. This assumption is based on the idea that the future state of a system depends only on its current state and not on the sequence of events that preceded it. In other words, the Markov assumption posits that the future is conditionally independent of the past given the present state. 

Mathematically, the Markov assumption can be expressed as: 

         P(X_t+1∣X_t, X_t−1, …, X₁) = P(X_t+1∣X_t) ------------------------------------ [3611a]

where, 

         X_t represents the state of the system at time t, and X_t+1 represents the state at the next time step. The assumption implies that the probability distribution over future states only depends on the current state and is not influenced by the entire history of states. 

Markov assumptions are commonly employed in various machine learning models, including Markov chains, Hidden Markov Models (HMMs), and Markov Decision Processes (MDPs). These models are particularly useful in scenarios where the evolution of a system can be characterized by a sequence of states, and the Markov assumption simplifies the modeling process by focusing on the immediate dependencies. 

In many practical situations, it is not realistic or computationally feasible to consider an infinite history of past states to predict the current state. Instead, the Markov assumption simplifies the modeling process by asserting that the current state depends only on a finite and fixed number of previous states. 

In the example of weather, this would mean that the weather today depends on the weather of the past few days but not necessarily on the weather from a year ago. The specific number of past states considered depends on the order of the Markov model being used. For example, a first-order Markov model (Markov chain) considers only the immediate previous state, while a higher-order model might consider more past states. This assumption makes it more practical to model and analyze complex systems by focusing on a limited history of states, leading to more tractable and efficient algorithms for prediction and decision-making. 

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