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Based on my reading some course notes as well as the answers and comments to this SO post I started thinking about the general steps for creating a stationary distribution.

Assuming my problem were to model the chance of snow tomorrow based on weather conditions today I would need to be able to predict probabilites from each distribution:

  1. use Monte Carlo methods to generate a larger dataset based on the parameters for each variable I had observed in my data. know that 50% chance of sun tomorrow if sunny today. know that 10% chance of snow tomorrow if raining today. know that 60% chance of snow if temperature is below freezing.
  2. Feed that larger dataset into Use a convergence algorithm (like Metropolis-Hastings) to get a distribution for each variable. This distribution tells the likelihood of a particular change in that variable. (eg 10% chance of snow tomorrow if it is raining today for the snow distribution). or 50% chance of sun tomorrow if it is sunny today for the sun distribution.
  3. ??? Do magic???Create a single distribution that takes into account the likelihood of each variable (eg must be raining and cold to be snowing), so if raining today you have X percent chnace of it snowing tomorrow. This is the Probability Mass Function (PMF)
  4. Get a Probability Mass Function (PMF) aka stationary distribution that tells you the probability of a specific outcome given your starting state.

My question is, how do I do Step 3, taking the probability of one distribution (eg sunny or rainy from the SO example) and add its effect to another distribution (say temperature) to predict days snowing. How do I know how much effect rain has on snowing? Do I assume by default that it must be raining to be snowing (if I didnt know this based on logic)?

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  • $\begingroup$ As given, there is no connection between your question and MCMC algorithms. Please rephrase. $\endgroup$ – Xi'an Nov 23 '15 at 14:06
  • $\begingroup$ @Xi'an, I believe I have made my question more clear. If this still doesnt work, could you explain why? $\endgroup$ – Rilcon42 Nov 24 '15 at 19:06
  • $\begingroup$ I do not understand your step 1. It mixes dataset, data, parameters, and variable! $\endgroup$ – Xi'an Nov 24 '15 at 19:24
  • $\begingroup$ I do not understand your step 2. Again, there is a mix between simulated dataset, parameters as simulated by a Metropolis-Hastings algorithm, and variable. Since the Metropolis-Hastings algorithm aims at simulating a posterior, I do not understand how it relates to "the likelihood of a particular change in the variable". $\endgroup$ – Xi'an Nov 24 '15 at 19:26
  • $\begingroup$ @Xi'an, Hopefully I have made this clearer with a concrete example. How do I know how much effect rain has on snowing? Do I assume by default that it must be raining to be snowing? $\endgroup$ – Rilcon42 Nov 24 '15 at 21:18
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Diagonalize the stochastic transition matrix and then pick the Eigen vector with the Eigen value 1. That is what your data would converge to.

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