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I'm doing a Bayesian analysis for a time series response and wonder whether it is possible to get the likelihood function without making distributional assumptions. I suppose my response is log-normal, but what if I do not want to make distributional assumptions?

My setting is a multi-armed bandit problem, so while my pay-off is log-normal, I could also define a discrete variable whether the pay-off increased from the last period to the current period. Then I would be in a binomial setting.

Hence, it is not obvious which likelihood to go for.

Is there a way to let the data decide what the likelihood is? So some form of non-parametric likelihood?

Sean Easter
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user13441344
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  • You can decide based on the marginal likelihood: Define as many different likelihood functions as you like; and select the most plausible ones based on their marginal likelihood (also known as evidence). – Summit Mar 02 '15 at 06:26
  • Thanks, Summit. Can you elaborate what you mean by choosing based on the marginal Likelihood? What properties am I evaluating? – user13441344 Mar 02 '15 at 07:39
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    Have a look at http://en.wikipedia.org/wiki/Marginal_likelihood#Bayesian_model_comparison and let me know if this did not answer your question. – Summit Mar 02 '15 at 11:15

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The choice as posed—namely, between a log-normal continuous variable and a discrete binomial one—seems distinct from that of whether to make distributional assumptions. In other words, whether the distribution is log-normal or binomial, it carries assumptions.

To the question of whether continuous or discrete, I would refer you to two posts by Andrew Gelman, arguing for modeling the continuous variable is more efficient, and more informative. A short snippet:

The key is that vote differential is available, and a simply performing a logit model for wins alone is implicitly taking this differential as latent or missing data, thus throwing away information.

In your example, by limiting your model to whether pay-off changed, you ignore how much it changed.

Choosing which distribution (and corresponding likelihood function) amounts to choosing a model, on which topic I'd direct you to the accepted answer to this question.

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Sean Easter
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