And we can see this requirement concretely as well.
If you start in state j, the probability of being in the other states reachable from j will start increasing from 0, and the probability of remaining in state j will start decreasing from 1.
(Unl…
The requirement that the off-diagonal elements of the jth column be non-negative follows from the geometry of the simplex, as per the following.
Observe that \(\Gamma_j(0) = j\) = the corner of the simplex which is 1 at j, and 0 elsewhere.
The der…
The reason for this requirement stems from the master equation:
\[\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\]
Since \(\Gamma(t)\) must always remain in the simplex \(\Delta\) -- i.e. must always be a probability distribution -- a big constraint …
The second condition on \([H_{i,j}]\) is that it must be infinitesimal stochastic, which means this:
All the off-diagonal elements are non-negative
Each column sums to zero
But for the master equation to be valid, \([H_{i,j}]\) can't be any old infinite square matrix.
There are two criteria to be satisfied.
Look at the matrix multiplication involved in computing \(H(\sigma)\), for a general stochastic state \(\sigma\…
Recap: for each definite state \(j\), the jth column of \([H_{i,j}]\) equals the derivative of the stochastic state when the chain is started in \(j\).
Let's look at the master equation again:
\[\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\]
The linear operator \(H\) is just matrix multiplication by \([H_{i,j}]\).
Instantiating to t=0, we get:
\[\Gamma_{\sigma}'(0) = H(\Gamma_{\sigma}(0)) = H(\s…
This is exactly the same concept of flow rates that we used for the edge weights in the directed graph for the Markov chain.
The set of values \(H_{i,j}\) is an "infinite square matrix." The values can be picked off right from the graph: \(H_{i,j…
For definite states \(i, j \in D\), define:
\[H_{i,j} = (\Gamma_{i}'(0))(j)\]
This is the instantaneous rate of change of the probability of being in state \(j\), given that the chain began in state \(i\).
Loosely speaking, it is the rate at whic…
To make this all work smoothly, let's situate the stochastic states inside of a containing vector space.
Let D be the definite states -- these are the nodes in the transition graph.
The vector space that we will work within is the space of real-va…
The master equation for the Markov chain is a vector differential equation:
\[\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\]
where \(H\) is a linear operator to be constructed from the weighted graph for the Markov chain.
Once we have a graph labeled with transition rates, we can proceed to the master equation, which gives a deterministic law for how the stochastic state changes over time -- once an initial stochastic state is specified.
Let \(\Gamma_{\sigma}(t)\) b…
Made multiple updates.
For anyone who wants to lend a hand, there are many other references in our Corona discussion which deserve to be added to this syllabus / bibliography.
Dr. David Price, Weill Cornell Medical Center, Empowering and protecting your family during the COVID-19 pandemic (video, long), March 22, 2020. Lots of perspective from an ICU COVID-19 doctor, both on the disease and how to cope with it.
Just two more steps to wrap up the construction of the Markov chain graph for our Petri net.
First, form the union of the all of these per-transition graphs.
Second, wherever this union leads to multiple edges going from \(x\) to \(x'\), then sum …
Let's get more specific now.
Let \(t\) be a transition, and Stoich(\(t\)) be its stoichiometric vector.
For a species \(z\), let InpDegree(\(t,z\)) be the number of places that \(t\) inputs from \(z\), and OutDegree(\(t,z\)) be the number of place…
@WebHubTel
In comment #43 on this thread you cited a web page on how to make a DIY mask out of paper towel and rubber bands.
I did a quick Google search on making DIY masks, and found this survey.
Not sure how authoritative this is, but it's some…
@WebHubTel wrote:
Why was the wearing of face masks by doctors in the USA not encouraged? Every doctor interviewed by the media said they were not that effective. How can they not be effective? Even if they reduced transmission of droplets by 30…
First, let's tighten up the construction of a Markov chain for a given Petri net.
Here is a general recipe for constructing a CTMC. Form a labeled, directed graph which has Nodes = the definite states, and a directed edge from \(x\) to \(y\) when…
Here is the problem at hand, which the master equation will address.
Suppose that we know that at time 0, the net is in a stochastic state \(s_0\).
Then at time \(t\), what stochastic state will the net be in?
Say that our Petri net has species S.
Let a definite state to be a mapping from S into \(\mathbb{N}\), giving the size of all the populations.
So the set of all definite states is \(\mathbb{N}^S\).
Let a stochastic state to be a probability distr…
Master Equation
The 'master equation' determines the probabilistic structure of the process for a Petri net.
To do this, first we'll review how a Petri net is a kind of continuous time Markov chain (CTMC).
Then we can apply the general concept of…
Abstract:
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of ‘chemical reaction networks’, which describes the interaction of molecule…
Caveat: I'm only starting to work through this, so I can't speak will full confidence about it.
It's not about quantum mechanics per se, but rather the application of certain mathematical techniques from QM to at least a certain class of stochastic…
Above I wrote:
Specifically, the firing rate of the process is equal to its rate constant times the product of its input population counts.
And:
That the firing rates are proportional to the input population sizes is called the law of mass…