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Compute multi-step Markov probabilities and estimate transitions

Quick Overview

This question evaluates understanding of discrete-time Markov chains, computation of multi-step transition probabilities from a one-step transition matrix, and empirical estimation of transition matrices from observed state-pair data including handling states with zero outgoing transitions.

Problem (Markov Chain)

Assume a discrete-time Markov chain over a finite set of states $S = \{s_1, \dots, s_n\}$ .

Part A — Multi-step transition probabilities

You are given a 1-step transition probability matrix $P \in \mathbb{R}^{n \times n}$ , where $P_{ij} = \Pr(X_{t+1}=s_j \mid X_t=s_i)$ .

Answer queries of the form:

$\Pr(X_{t+k}=s_j \mid X_t=s_i)$ for various $i, j, k$ .

You may use matrix multiplication (e.g., matmul) in your solution.

Part B — Estimate the transition matrix from observed transitions

You are given a list of observed consecutive transitions (state pairs), e.g.:

[(s1, s2), (s2, s3), (s1, s3), (s1, s2), ...]

Construct an estimated transition matrix $\hat P$ using transition frequencies:

$\hat P_{ij} = \frac{\#(s_i \to s_j)}{\sum_{j'} \#(s_i \to s_{j'})}$

Requirements / Edge cases to address

How to handle states that appear in the state list but have zero outgoing transitions in the data.
Ensure each row of $\hat P$ is a valid probability distribution (sums to 1 where defined).

Quick Overview

Problem (Markov Chain)

Assume a discrete-time Markov chain over a finite set of states

S = \{s_1, \dots, s_n\}

Part A — Multi-step transition probabilities

You are given a 1-step transition probability matrix

P \in \mathbb{R}^{n \times n}

, where

P_{ij} = \Pr(X_{t+1}=s_j \mid X_t=s_i)

Answer queries of the form:

\Pr(X_{t+k}=s_j \mid X_t=s_i)

for various

i, j, k

You may use matrix multiplication (e.g., matmul) in your solution.

Part B — Estimate the transition matrix from observed transitions

You are given a list of observed consecutive transitions (state pairs), e.g.:

[(s1, s2), (s2, s3), (s1, s3), (s1, s2), ...]

Construct an estimated transition matrix

\hat P

using transition frequencies:

\hat P_{ij} = \frac{\#(s_i \to s_j)}{\sum_{j'} \#(s_i \to s_{j'})}

Requirements / Edge cases to address

How to handle states that appear in the state list but have zero outgoing transitions in the data.

Ensure each row of

\hat P

is a valid probability distribution (sums to 1 where defined).

Compute multi-step Markov probabilities and estimate transitions

Quick Overview

Problem (Markov Chain)

Part A — Multi-step transition probabilities

Part B — Estimate the transition matrix from observed transitions

Requirements / Edge cases to address

Solution

Comments (0)

Compute multi-step Markov probabilities and estimate transitions

Quick Overview

Problem (Markov Chain)

Part A — Multi-step transition probabilities

Part B — Estimate the transition matrix from observed transitions

Requirements / Edge cases to address

Solution

Comments (0)