Investigate this more

[MichalisPanayides]

Be confident that we understand why this is happening:

The red lines (convex) indicate when μ is higher than λ The blue lines (concave) indicate when μ is less than λ

[geraintpalmer]

In investigate_concavity_convexity.ipynb in #5 I wrote:

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We can think about this function as the 'acceleration' towards infinity:

• If the closer we get to the boundary (a larger 𝑠) the rate at which the probability of reaching the boundary increases, then the MC 'accelerates' towards the boundary. That means the system 'accelerates' towards infinity, and shoots off. This is shown with a concave function.

• If the closer we get to the boundary (a larger 𝑠) the rate at which the probability of reaching the boundary decreases, then the MC 'decelerates' towards the boundary. That means the system 'decelerates' towards infinity, and so doesn't shoots off, it stabilises. This is shown with a convex function.

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I propose also:

• say this only works for MCs with a certain structure, that is, as you go 'close' to the bound the 'further away' you are from the abosrbing state.
• defining $d(s)$, the distance state $s$ is from the absorbing state. (How? maybe considering the structure as a graph, looking at shortest paths to the absorbing state? not sure yet).

[geraintpalmer]

After a chat with @drvinceknight and @11michalis11:

• Can we user $h_{sb}$ itself as a measure of distance between the $s$ and the bound $b$?

Advantage: much simpler, all we have to do is order the states by $h_{sb}$. Disadvantage: It would be neater to have a separate definition of $d(s)$ is order to help with #4 and defining a 'reasonable region'