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Compute and estimate Markov transition probabilities

OtherData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This question evaluates understanding of discrete-time Markov chains, multi-step transition probabilities and empirical estimation of transition matrices, testing probabilistic reasoning and matrix-based computation in the Statistics & Math domain for a Data Scientist role; it is commonly asked because it probes both theoretical grasp of stochastic processes and practical data-driven parameter estimation. At an applied algorithmic abstraction level, it requires explicit mapping of state labels to matrix indices, computing k-step transitions from a given row-stochastic matrix, estimating transition probabilities from observed one-step transitions, and addressing zero-outgoing-observation cases and any chosen smoothing policy (or justification for none).

Part A (multi-step probabilities):

  • You are given a row-stochastic transition matrix P of shape (m x m), where P[i][j] = P(X_{t+1}=sj | X_t=si).
  • For several queries (start_state=a, end_state=b, steps=k), compute P(X_{t+k}=b | X_t=a).

Part B (estimate transition matrix from observations):

  • You are given a list of observed one-step transitions, e.g. [(s1,s2), (s2,s3), (s1,s3), (s1,s2), ...].
  • Estimate the transition probability matrix \hat{P} using empirical transition frequencies.

Requirements/notes:

  • Clearly define how you map state labels to matrix indices.
  • Handle states that may have zero outgoing observations in Part B.
  • State any smoothing you choose (or justify not using smoothing).