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I'm slightly abusing the Azimuth Blog tag to talk about some stuff that will feed in to the Math Horizons article I'm helping John to write.

One of the things that will probably be helpful is having a simple example of using various parameter's hindcasts' fit to the observed data as part of the estimate of that parameter's probability. In particular, if the parameter space is 2-D then we can actually plot both the full likelihood surface and the values the MCMC gives and show how well they agree where they share points.

But I haven't immediately managed to come up with a good system to do this on. I'd been trying

$$x_t = \lambda * x_{t-1} + noise$$ for lambda in $(-0.75,-0.25)$ and various noise distributions (normal, uniform, "pyramid shaped", etc) and using the appropriate likelihood model on random simulations of the system for given parameters, either the likelihhood is very, very highly peaked over the true value or the noise dominates and tehre's no way to get an useful posterior distribution. John has pointed out that if $|\lambda| \gt 1$ then eventually exponential behaviour will kick in while for $1 \gt |\lambda|$ it's likely the noise will swamp things too much.

One vague thought is to see if a discrete analogue of the simple harmonic motion equation is any better for getting a "non-trivial but not too spread out" posterior probability density out, but there might be some better ideas?

## Comments

I'd guess that making the noise proportional to $x_t$ would help in your example.

`I'd guess that making the noise proportional to $x_t$ would help in your example.`

Thanks: i'll try that this evening and report back.

`Thanks: i'll try that this evening and report back.`

You could try the 0-dimensional linear energy balance model for temperature, $c d T/d t = F(t) - \lambda T$, $y = T+\xi$, where $F(t)$ is a forcing function, $\xi$ is a noise process (e.g. first-order autoregressive), and $y$ is the observed time series. (Or you could put the stochastic forcing right into the differential equation, but then it's a bit harder to tweak the behavior of the observed process.) The problem is to estimate $\lambda$ and $c$, which are confounded.

`You could try the 0-dimensional linear energy balance model for temperature, $c d T/d t = F(t) - \lambda T$, $y = T+\xi$, where $F(t)$ is a forcing function, $\xi$ is a noise process (e.g. first-order autoregressive), and $y$ is the observed time series. (Or you could put the stochastic forcing right into the differential equation, but then it's a bit harder to tweak the behavior of the observed process.) The problem is to estimate $\lambda$ and $c$, which are confounded.`

If you wanted to do a single-parameter example, you could try the energy balance model in Eq. 1b of Kelly and Kolstad (1999), which (in the linear-Gaussian setting) has an analytic solution for the Bayesian Kalman update (Eq. 7).

`If you wanted to do a single-parameter example, you could try the energy balance model in Eq. 1b of [Kelly and Kolstad (1999)](http://are.berkeley.edu/courses/ARE263/fall2008/paper/Learning/KellyJEDC.pdf), which (in the linear-Gaussian setting) has an analytic solution for the Bayesian Kalman update (Eq. 7).`

Hi Nathan, thanks for the suggestions. In case I wasn't clear, it's not necessary for the example to (necessarily) have anything to do with AMOC, or climate; the goal is to come up with something very simple to demonstrate the MCMC hindcast technique, altnough if it happens to connect with climate that's better. I'll experiment with a discretised variation of the 0-dimensional balance you mention in post 4 (so that there's no need to mention about the DE solution step and issues arising from that) and see how that goes.

`Hi Nathan, thanks for the suggestions. In case I wasn't clear, it's not necessary for the example to (necessarily) have anything to do with AMOC, or climate; the goal is to come up with something very simple to demonstrate the MCMC hindcast technique, altnough if it happens to connect with climate that's better. I'll experiment with a discretised variation of the 0-dimensional balance you mention in post 4 (so that there's no need to mention about the DE solution step and issues arising from that) and see how that goes.`

Great! While not necessary, it'll be pedagogically very nice if the model has something to do with climate physics. It makes for a better segue: we can briefly say how the equations are inspired by climate physics, and then use them to illustrate something.

`Great! While not necessary, it'll be pedagogically very nice if the model has something to do with climate physics. It makes for a better segue: we can briefly say how the equations are inspired by climate physics, and then use them to illustrate something.`