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I'm thinking here largely of the context in which someone has an Elo rating model for a particular sport.

To calculate things such as how often the team makes the Finals series, or wins the Championship game, modellers might run Monte Carlo simulations of the entire season. Some modellers (e.g. FiveThirtyEight, at least with respect to their sports simulations as of 2015) run their simulations hot, in the sense that a team’s rating changes dynamically during the simulated season based on the results of the simulated games. Other modellers run their simulations cold, meaning that the rating at the commencement of the simulated season stays static irrespective of the results of the simulated games.

What's better, and why?

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    $\begingroup$ If you update your priors based on MC simulations simulating from your already pre-defined prior, does the mean of your estimators even change? It feels like all you do is add variance to the simulation. And running hot feels like trying to copy what you would do in the real world with real data when following Bayesian Inference, however you're kind of doing it erroneously since the data you're updating with is itself simulated. $\endgroup$
    – Dale C
    Commented Nov 16, 2020 at 4:16

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