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I have a "process-based" water quality model, which is essentially a black-box full of differential equations describing various chemical and hydrological processes. The model is deterministic and takes a set of input time series (weather and chemical variables etc.) and produces simulations of stream flow and water chemistry with daily resolution.

The model has various adjustable parameters corresponding to constants in the underlying differential equations. Some of these can be fixed based on system knowledge, but others need to be calibrated empirically by comparing the model output to observations. In general, it's common to have much more calibration data for some variables than for others e.g. stream flow might be measured daily, whereas water chemistry is only measured monthly.

I am using Bayesian MCMC to explore parametric uncertainty in my model. For a single output variable (e.g. stream flow), I can write down a log-likelihood function that compares simulated to observed values, but I am not sure how to write down an appropriate likelihood for my multi-variable output, given the different amounts of observed data for each variable.

My log-likelihood for each variable is just the sum of log-probabilities for each observed data point. Initially, I just added together the log-likelihoods for each variable:

$$LL_{Total} = LL_{Flow} + LL_{Chem}$$

However, because $LL_{Flow}$ is based on >1000 data points and $LL_{Chem}$ <50, this feels somehow "biased" towards fitting flow at the expense of water chemistry.

I guess this situation must be common in many areas of statistics, but I'm lacking the background to be able to research effectively. I'm hoping you can point me in the right direction. My questions:

  1. My current approach gives equal "weight" to each observation, which I guess is reasonable. If I'm more interested (from a practical perspective) in water chemistry than flow, would it also be reasonable to "re-weight" the likelihoods e.g. normalised by the number of observations? I suppose ultimately I can define any "skill metric" I think is relevant, but I suspect there must be principled ways of doing this? What are the implications?

  2. What about a "two-stage" approach, such as calibrating the stream flow parameters first and the water chemistry parameters second (while fixing the stream flow parameters at the MAP from the first stage)? This only considers a subset of the overall parameter space and therefore seems less satisfying, but it would help to have a better intuition regarding the implications.

  3. Other options, methods or topics that I should research?

Thank you!

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