Markov Chains: Building Blocks for MCMC
Markov chains are mathematical systems that move between states according to certain probabilities. The key feature of a Markov chain is the Markov property: the probability of moving to the next state depends only on the current state, not on the sequence of states that preceded it. This property makes Markov chains both memoryless and highly useful for modeling processes that evolve step by step.
To illustrate the Markov property, imagine you are playing a board game where your next move depends only on the square you are currently on — not on how you arrived there. For example, if you are on square 3, your next move is determined entirely by the rules for square 3, regardless of whether you came from square 2 or square 5. This "memoryless" property is what defines a Markov chain.
A Markov chain is defined by a set of states and a transition matrix that specifies the probability of moving from one state to another. Each row of the transition matrix corresponds to a current state, and each column represents the probability of transitioning to a possible next state.
A central concept in Markov chains is the stationary distribution. This is a probability distribution over states that does not change as the Markov chain evolves. If you start the chain in the stationary distribution, the probabilities of being in each state remain the same after each transition. Stationary distributions are important for sampling because, under certain conditions, the Markov chain will converge to this distribution regardless of the starting state. This means you can use a Markov chain to generate samples from complex distributions by running the chain long enough to reach stationarity.
12345678910111213141516171819import numpy as np # Define transition matrix for a simple 3-state Markov chain P = np.array([ [0.1, 0.6, 0.3], [0.4, 0.4, 0.2], [0.3, 0.3, 0.4] ]) # Start in state 0 current_state = 0 states = [current_state] # Simulate 10 steps of the Markov chain for _ in range(10): current_state = np.random.choice([0, 1, 2], p=P[current_state]) states.append(current_state) print("Sequence of states:", states)
Markov chains are useful for generating dependent samples — each new sample depends on the current one. This is different from independent sampling, where each sample is unrelated to the others. By designing Markov chains with the right transition probabilities and ensuring they have the desired stationary distribution, you can use them to sample from complex probability distributions that are otherwise difficult to access directly. This idea is the foundation for Markov Chain Monte Carlo (MCMC) algorithms, which are widely used in machine learning and statistics for approximate inference and estimation.
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Markov chains are mathematical systems that move between states according to certain probabilities. The key feature of a Markov chain is the Markov property: the probability of moving to the next state depends only on the current state, not on the sequence of states that preceded it. This property makes Markov chains both memoryless and highly useful for modeling processes that evolve step by step.
To illustrate the Markov property, imagine you are playing a board game where your next move depends only on the square you are currently on — not on how you arrived there. For example, if you are on square 3, your next move is determined entirely by the rules for square 3, regardless of whether you came from square 2 or square 5. This "memoryless" property is what defines a Markov chain.
A Markov chain is defined by a set of states and a transition matrix that specifies the probability of moving from one state to another. Each row of the transition matrix corresponds to a current state, and each column represents the probability of transitioning to a possible next state.
A central concept in Markov chains is the stationary distribution. This is a probability distribution over states that does not change as the Markov chain evolves. If you start the chain in the stationary distribution, the probabilities of being in each state remain the same after each transition. Stationary distributions are important for sampling because, under certain conditions, the Markov chain will converge to this distribution regardless of the starting state. This means you can use a Markov chain to generate samples from complex distributions by running the chain long enough to reach stationarity.
12345678910111213141516171819import numpy as np # Define transition matrix for a simple 3-state Markov chain P = np.array([ [0.1, 0.6, 0.3], [0.4, 0.4, 0.2], [0.3, 0.3, 0.4] ]) # Start in state 0 current_state = 0 states = [current_state] # Simulate 10 steps of the Markov chain for _ in range(10): current_state = np.random.choice([0, 1, 2], p=P[current_state]) states.append(current_state) print("Sequence of states:", states)
Markov chains are useful for generating dependent samples — each new sample depends on the current one. This is different from independent sampling, where each sample is unrelated to the others. By designing Markov chains with the right transition probabilities and ensuring they have the desired stationary distribution, you can use them to sample from complex probability distributions that are otherwise difficult to access directly. This idea is the foundation for Markov Chain Monte Carlo (MCMC) algorithms, which are widely used in machine learning and statistics for approximate inference and estimation.
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