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There are a few different ways to answer this question. The answer isn’t as simple as it should be.The first known case in Wuhan was identified in late December. Based on this case they were able to identify other cases that they diagnosed as incorrectly earlier in December and because of the incubation time the person must have caught it from someone/something it in mid-November.So key dates:First Offical case: 24th of December (from memory)First case with symptoms after going back through records: 1st of DecemberBest guess of when Covid-19 was in Wuhan: middle of November 2019.So what are the dates for America:First Offical case: 2nd of January. (Updated)First case reported to a hospital after going back through records: The CDC has said this has happened BUT has not released any information or dates[1] (this was in early March so they could be talking about February or January but nothing has been released)Best Guess: The government either hasn’t tried or hasn’t officially released any information. All we have to go on is rumours:Mayor Michael Melham thinks he got Covid-19 in November but he may have just gotten asymptomatic Covid-19.There was a large number of cases in Washington and it is possible for the government to go back and test more Flu samples to see if it was Covid-19 BUT they don’t want to. [3] Update: Tested! No Covid-19 found in Seatle. [5]So there is one government that is not wanting to be open and provide information but it isn’t the one you would expect. Now China could easily be lying so it may not be a good comparison. That being said mid-November also matches with when France had it’s first confirmed case which is in line with the first case in Wuhan (ignore the headline) [4]It is worth being aware that Dr Chu or the CDC could easily go back through these Flu samples and find out if more of them are Covid-19 (They did back in April but I missed the story). They can do this at any time to find the earliest case but chose not to. (They would need to get permission which would take time but it isn’t impossible or illegal. It was approved after a few weeks but since nothing was found it wasn’t a big story).So the answer to the question at the moment is:No, at the moment the first case that actually has a positive Covid-19 anti-body is mid-November around the same time as Wuhan & France but not before Wuhan. Even that case is a low probabilty. The CDC has stated that it knows cases of Flu were Covid-19 but has not released the data it has (which may be from January & February). The Flu samples from 2019 were tested for Covid-19 and the earliest case was February 21st. The model they created predicts that the virus entered Seatle mid-January which is slightly earlier than the first official case on the 20th of January but not at the start of November.Also, I highly recommend you check out the ongoing research form Seatle tracing the virus. It really is impressive: Narrative: August 2020 update of COVID-19 genomicIt is worth being aware that the majority of the cases DON’T trace back to the Wuhan Covid-19 version.Red = WuHanBlue = First seen in Europe and Australia.This DOESN’T mean that the virus didn’t come from Wuhan and came from America. What it means is that there is a parent strain and we never got a sample of it (and probably never will). That strain existed/happened before the samples collected at in Wuhan in December. The question is where did that parent strain come from?[1] User Clip: Diagnosed as Flu, but actually COVID 19[2] Mayor Michael Melham thinks he got Covid-19 in November:[3] Seattle lab only uncovered extent of Washington coronavirus outbreak after breaking government rules[4] French patients were sick with Covid-19 in mid-November and before China - researchers[5] Seattle researcher debunks theory COVID-19 spread in Calif. in NovemberUpdate: Thanks Keith RobisonI don’t know how I missed the story. I assume it was buried in all the other noise. Seatle Flu samples tested. No Covid-19 before January 20.

(This is the fourth answer in a 7 part series on Probabilistic Graphical Models ('PGMs').)So far, our running definition of inference has been:The task of using a given graphical model of a system, complete with fitted parameters, to answer certain questions regarding that system.Here, we'll discuss a general class designed to approximate these answers with simulations. That is, we'll get samples drawn from a distribution which approximates the distribution we're asking about.Due to one reason only, I enjoy these techniques the most. It's their generality. Exact inference algorithms demand the graphs are sufficiently simple and small. Variable Inference, a clever approximate approach, demands defining approximate distribution spaces and a means to search them effectively. Monte Carlo methods, however, demand no qualifying inspection of these graph - all graphs are fair game. This provides us with a much wider class of models to fit our reality.This isn't to say these methods are a cure-all. Yes, there are circumstances in which we fail to get answers. But the responsible reason, outside of the vague 'lack of convergence', is not well understood. So we accept these problem specific battles as the cost of supreme generality.But before we dive in, a short review will help.Refresher (Skip this if you’ve read answers 1 and 2!)In the first answer, we discovered why PGMs are useful for representing complex system. We defined a complex system as a set, [math]\mathcal{X}[/math], of [math]n[/math] random variables (‘RVs’) with a relationship we'd like to understand. We assume there exists some true but unknown joint distribution, [math]P[/math], which govern these RVs. We take it that a 'good understanding' means we can answer two types of questions regarding this [math]P[/math]:Probability Queries: Compute the probabilities [math]P(\mathbf{Y}|\mathbf{e})[/math]. This means: what is the distribution of [math]\mathbf{Y}[/math] given we have some observation of [math]\mathbf{E}[/math]?MAP Queries: Determine [math]\textrm{argmax}_\mathbf{Y}P(\mathbf{Y}|\mathbf{e})[/math]. That is, determine the most likely values of some RVs given an assignment of other RVs.(Where [math]\mathbf{E}[/math] and [math]\mathbf{Y}[/math] are two arbitrary subsets of [math]\mathcal{X}[/math]. If this notation is unfamiliar, see the 'Notation Guide' from the first answer).The idea behind PGMs is to estimate [math]P[/math] using two things:A graph: a set of nodes, one for each RV in [math]\mathcal{X}[/math], and a set of edges between them.Parameters: objects that, when paired with a graph and a certain rule, allow us to calculate probabilities of assignments of [math]\mathcal{X}[/math].Depending on these two, PGMs come in two main flavors: Bayesian Networks ('BNs') and Markov Networks ('MNs').A Bayesian Network involves a graph, denoted as [math]\mathcal{G}[/math], with directed edges and no directed cycles. The parameters are Conditional Probability Tables ('CPDs' or 'CPTs'), which are, as the naming suggests, select conditional probabilities from the BN. They give us the right hand side of the Chain Rule, which dictates we calculate probabilities this way:where [math]\textrm{Pa}_{X_i}^\mathcal{G}[/math] is the set of parent nodes of [math]X_i[/math] in the graph.A Markov Network's graph, denoted as [math]\mathcal{H}[/math], is different in that its edges are undirected and it may have cycles. The parameters are a set of functions (called ‘factors’) which map assignments of subsets of [math]\mathcal{X}[/math] to nonnegative numbers. Those subsets, which we'll call [math]\mathbf{D}_i[/math]'s, correspond to complete subgraphs of [math]\mathcal{H}[/math]. We can refer to this set as [math]\Phi=\{\phi_i(\cdots)\}_{i=1}^m[/math]. With that, we say that the Gibbs Rule for calculation probabilities is:where [math]Z[/math] is defined such that our probabilities sum to 1.To crystallize this idea, it's helpful to imagine the 'Gibbs Table', which lists that above product (without [math]Z[/math]) for each assignment. In the second answer, we pictured an example where [math]\mathcal{X}=\{C,D,I,G,S\}[/math] as:Lastly, we recall that the Gibbs Rule may express the Chain Rule. That is, we can always recreate the probabilities produced by a BN's Chain Rule with an another invented MN and its Gibbs Rule. Essentially, we define factors as those that reproduce looking up in a BN's CPDs. This equivalence allows us to reason solely in terms of the Gibbs Rule, while assured that whatever we discover will also hold for BNs. In other words, with regards to inference, if something works for [math]P_M[/math], then it works for [math]P_B[/math].Great, now… what's our starting point?We are handed a MN (might be the converted form of a BN). That is, we get a graph [math]\mathcal{H}[/math] and a set of factors [math]\Phi[/math]. We're interested in the distribution of a subset of RVs, [math]\mathbf{Y} \subset \mathcal{X}[/math], conditional on an observation of other RVs ([math]\mathbf{E}=\mathbf{e}[/math]). We'll have our answer presumably (to both queries) if we can generate samples, [math]\mathbf{y}[/math]'s, that come (approximately) from this distribution.The first step is to address conditioning. To do so, let's steal one idea from the second answer. That is, inference in a MN conditional on [math]\mathbf{E}=\mathbf{e}[/math] gives the same answer as unconditional inference in a specially defined MN. Let’s call its graph [math]\mathcal{H}_{|\mathbf{e}}[/math] and its factors [math]\Phi_{|\mathbf{e}}[/math]. It turns out [math]\mathcal{H}_{|\mathbf{e}}[/math] is [math]\mathcal{H}[/math], but with all [math]\mathbf{E}[/math] nodes and any edges involving them deleted. [math]\Phi_{|\mathbf{e}}[/math] is the set of factors [math]\Phi[/math], but with [math]\mathbf{E}=\mathbf{e}[/math] fixed as an input assignment. The point is if we can do unconditional inference, we can do conditional inference, granted we performed this change. For the sake of cleanliness, I'll drop the '[math]|\mathbf{e}[/math]' and assume we've already done this conditioning conversion.Finally, we're ready for the big idea.Markov Chain Monte Carlo (MCMC)In a nutshell, MCMC finds a way to sequentially sample [math]\mathbf{y}[/math]'s such that, eventually, these [math]\mathbf{y}[/math]'s are distributed as [math]P_M(\mathbf{Y}|\mathbf{e})[/math].To see this, we must first defined a Markov Chain. All this is is a set of states and transition probabilities defined between such states. These probabilities are the chances we transition to any other state given the current state. For our purposes, the set of states is [math]Val(\mathbf{X})[/math] where [math]\mathbf{X}=\mathcal{X}[/math] - all possible joint assignments of all variables. For any two states [math]\mathbf{x},\mathbf{x}' \in Val(\mathbf{X})[/math], we write the transition probability as [math]\mathcal{T}(\mathbf{x} \rightarrow \mathbf{x}')[/math]. We may refer to all such probabilities or the whole Markov Chain as [math]\mathcal{T}[/math].As a simple example, suppose our system was one RV, [math]X[/math], that could take 3 possible values (so [math]Val(X)=[x^1,x^2,x^3][/math]). Then we might have this Markov Chain:Thinking generally again, to 'sample' a Markov Chain means we sample a starting [math]\mathbf{x}^{(0)}\in Val(\mathbf{X})[/math] according to some starting distribution [math]P_\mathcal{T}^{(0)}[/math]. Then, we use our [math]\mathcal{T}(\mathbf{x} \rightarrow \mathbf{x}')[/math] probabilities to determine the next state, giving us [math]\mathbf{x}^{(1)}[/math]. Then we repeat, giving us a long series of [math]\mathbf{x}^{(t)}[/math]'s. If we were to restart the sampling procedure many times and select out the [math]t[/math]-th sample, we'd observe a distribution that we'll call [math]P_\mathcal{T}^{(t)}[/math].So, a sample of our toy examples might be: [math]x^1[/math] (33% chance) [math]\rightarrow x^3[/math] (75% chance) [math]\rightarrow x^2[/math] (50%) [math]\rightarrow x^2[/math] (70%). The first has a 33% chance because we assume [math]P_\mathcal{T}^{(0)}[/math] is uniform. Simple enough, right?By the nature of this procedure, we can figure this relation:This is to say: the probability of the next state one step from now is equal to the sum of the probabilities of being in another state and transitioning from there to that next state.Now, for a large [math]t[/math], it's reasonable to expect [math]P_\mathcal{T}^{(t)}[/math] to be very similar to [math]P_\mathcal{T}^{(t+1)}[/math]. Under some conditions, that's correct intuition. Whatever that common distribution is, we call it the stationary distribution of [math]\mathcal{T}[/math], and it's written as [math]\pi_\mathcal{T}[/math]. It is the single distribution that works for both [math]P_\mathcal{T}^{(t+1)}[/math] and [math]P_\mathcal{T}^{(t)}[/math] in that above relation. That is, it solves:In effect, [math]\pi_\mathcal{T}[/math] is the distribution that [math]P_\mathcal{T}^{(t)}[/math] converges too.With that, we're ready for the big insight:We may choose our Markov Chain, [math]\mathcal{T}[/math], such that [math]\pi_\mathcal{T} = P_M[/math]Now, conceivably, we could make our choice of [math]\mathcal{T}[/math] with [math]P_M[/math] in mind and solve for the stationary distribution to get our answer. However, in general, this isn't possible. Hence, we need our next 'MC'. That is, we'll use Monte Carlo simulations to solve our problem. Instead of trying to solve for [math]\pi_\mathcal{T}[/math], we execute the sampling procedure to produce a series of [math]\mathbf{x}^{(t)}[/math]'s, and then we observe an empirical approximation to [math]\pi_\mathcal{T}[/math] (and hence [math]P_M[/math]) after a number of iterations. If we are concerned with a subset [math]\mathbf{Y}[/math] from [math]\mathcal{X}[/math], we simply select out the [math]\mathbf{Y}[/math]-elements from our series of [math]\mathbf{x}^{(t)}[/math]'s and use those.OK, but...How do we choose [math]\mathcal{T}[/math]?This is where things get hairy. Fortunately, the algorithms make this decision for us, but to understand their relative advantages, we need to understand their common aim. None of them entirely nail that aim.In a nutshell, that aim is:We'd like a [math]\mathcal{T}[/math] for which sampling will converge quickly to a single [math]\pi_\mathcal{T}[/math], equal to [math]P_M[/math], from any starting [math]P^{(0)}_\mathcal{T}[/math].This is hard. Here are the major ways it may crash and burn:Imagine a [math]\mathcal{T}[/math] with two states where if you're on one, you always transition to the other. This makes for a heavy dependence on [math]P^{(0)}_\mathcal{T}[/math] and no stationary distribution [math]\pi_\mathcal{T}[/math]. Cyclic behavior like this means the Markov chain is periodic - we hate periodic chains.A [math]\mathcal{T}[/math] with a low conductance is one in which there are regions of the state space which are very hard to go between. This means that if you start in one, it'll take you a long time before you explore the other. Therefore, we have a near-dependency on [math]P^{(0)}_\mathcal{T}[/math]. Also, convergence will require traversing that narrow bridge, so it certainly won't be quick. If dotted lines imply small transition probabilities, this is an example of a low conductance [math]\mathcal{T}[/math]:If there exist two states such that if you're on one, you can never reach the other, that [math]\mathcal{T}[/math] is called reducible and the consequence is that there may be more than one stationary distribution.The protection against this unruly behavior are some theoretical properties you may demand of a [math]\mathcal{T}[/math]. You may demand it's aperiodic for example.Probably the most helpful one is called detailed balance. A [math]\mathcal{T}[/math] with this property has a [math]\pi_\mathcal{T}[/math] such that:for any pair of [math]\mathbf{x}, \mathbf{x}' \in Val(\mathbf{X})[/math].Compare this to the equation that defined the stationary distribution. The right side of that has many more terms, and as such, it has many more degrees of freedom for [math]\pi_\mathcal{T}(\mathbf{x})[/math] to sit within. Intuitively, detailed balance means the stationary distribution follows from single step transitions, and not from large cycles of many transitions. It's a kind of 'well connectedness'. If [math]\mathcal{T}[/math] implies any two states are reachable from each other, is aperoidic and has detailed balance, than we'll converge to the unique stationary distribution from any starting distribution.Enough with the theoretics! Let's see an algorithm.Gibbs Sampling - a type of MCMCIn this method, our first step is to uniformly sample a [math]\mathbf{x}^{(0)}[/math] from [math]Val(\mathcal{X})[/math]. Here, our transition probabilities will be such that we are forced to change this vector one element at a time to produce our samples. Specifically:Pick out [math]X_1[/math] from [math]\mathbf{X}[/math] and list out all values of [math]Val(X_1)[/math]. For example, [math]Val(X_1)=[x_1^1,x_1^2,x_1^3,x_1^4][/math].For each [math]x_1^i \in Val(X_1)[/math], create a new vector by subbing that [math]x_1^i[/math] into the [math]X_1[/math]-position of [math]\mathbf{x}^{(0)}[/math] (call it [math]\mathbf{x}^{(0)}_{i-subbed}[/math]).Plug each [math]\mathbf{x}^{(0)}_{i-subbed}[/math] into our Gibbs Rule[1], giving us 4 positive numbers and normalize them into probabilities.Use this size-4 probability vector to sample one of these [math]\mathbf{x}^{(0)}_{i-subbed}[/math]‘s. Use that as [math]\mathbf{x}^{(1)}[/math].Go back to step 1, but use [math]X_2[/math] this time to produce [math]\mathbf{x}^{(2)}[/math].And we keep cycling through these steps for as long as we'd like. In effect, steps 2-4 are sampling from a specially defined set of transition probabilities. As a result, if we pick a [math]t[/math] large enough, [math]\mathbf{x}^{(t)}[/math] will come from [math]P_M(\mathbf{X})[/math][2].But, you may have noticed an issue. [math]\mathbf{x}^{(t)}[/math] is only different from [math]\mathbf{x}^{(t-1)}[/math] by one element, so these sequences are seriously correlated over short distances. This series isn't a set of independent samples!To address this, we keep track of the effective sample size of our simulations. Imagine our sampling produced a series that is perfectly correlated - they are all the same. Clearly, this is effectively a single independent sample. At the other end, imagine there is no correlation - we have our independence and we have as many independent samples as samples (beyond that 'large enough' [math]t[/math]). So given the case where we have some correlation, the effective sample size is a number between these two extremes. There exists some heuristics to estimate this figure from lagged correlations. It's an important figure to keep handy, as it gives you a clue as to how good our approximated inference is.But correlation isn't the only issue. Since we update one [math]X_i[/math] at a time, it's also fairly slow.We need to get more general.The Metropolis-Hastings AlgorithmThe issue in Gibbs Sampling is that we move through the states very slowly. It would help if we could control how fast we explore this space. To guarantee that the resulting [math]\pi_\mathcal{T}[/math] equals our [math]P_M[/math], we'd have to apply a correction to compensate for that control. Roughly, this is the idea behind the Metropolis-Hasting algorithm.More specifically, we invent another Markov Chain, which we'll call [math]\mathcal{T}^Q[/math], which is our proposal Markov Chain. That is, it's defined over the same space ([math]Val(\mathbf{X})[/math]) and is responsible for proposing the next state, given whatever current state. This is our opportunity to invoke large leaps. Our correction will be to sample a yes-no event according to a certain acceptance probability. If we draw a yes, we transition to the proposed state. If we draw a no, we remain at the current state. This acceptance probability is specially design to ensure detailed balance. It is:This acceptance probability ensures that we will converge to [math]P_M[/math] from any starting distribution, given [math]\mathcal{T}^Q[/math] isn't especially badly behaved.For the intuiters, let's decompose [math]\mathcal{A}[/math]. It's effectively made up of [math]\frac{\tilde{P}_M(\mathbf{x}')}{\tilde{P}_M(\mathbf{x})}[/math] and [math]\frac{\mathcal{T}^Q(\mathbf{x}'\rightarrow\mathbf{x})}{\mathcal{T}^Q(\mathbf{x}\rightarrow\mathbf{x}')}[/math].[math]\frac{\tilde{P}_M(\mathbf{x}')}{\tilde{P}_M(\mathbf{x})}[/math] makes us more likely to reject states that are unfavorable according to [math]P_M[/math]. Notice it uses the unnormalized probabilities - no intractable [math]Z[/math] involved! Also, since it's a ratio involving the Gibbs Rule, if [math]\mathbf{x}'[/math] and [math]\mathbf{x}[/math] share some identical terms (maybe by design of our [math]\mathcal{T}^Q[/math]), it's possible some factors cancel. So we could save time by avoiding computation of the full Gibbs Rule.[math]\frac{\mathcal{T}^Q(\mathbf{x}'\rightarrow\mathbf{x})}{\mathcal{T}^Q(\mathbf{x}\rightarrow\mathbf{x}')}[/math] makes us unlikely to accept states that are easy to transition to (according to our proposal) but difficult to return from. Such behavior is at odds with detailed balance, so it's curtailing it for that sake of this property.Now, there are some practical considerations to keep in mind. First, [math]\mathcal{T}^Q[/math] should be able to propose everything in [math]Val(\mathbf{X})[/math], otherwise we'll give a zero probability to something [math]P_M[/math] might favor. Second, a rejected proposed sample is a waste of your computer’s time, so in this sense, we like high [math]\mathcal{A}[/math]'s. However, very high [math]\mathcal{A}[/math]'s might mean you aren't exploring the space quickly. Together, this means that [math]\mathcal{T}^Q[/math] must be tuned right to explore the space efficiently.Let's see it!I can't think of a better visual than the one I've seen in Kevin Murphy's text (Chapter 24, see source [2]). So, I’ll use that. First, let's pretend our 'intractable' [math]P_M[/math] is a mixture of two Gaussian distributions, like this:We'd like a set of samples which spent time along the horizontal axis in proportion to the height of this graph. To do so, we'll use a normal distribution centered on our current state: [math]\mathcal{T}^Q(\mathbf{x}\rightarrow\mathbf{x}') = \mathcal{N}(\mathbf{x}'|\mathbf{x},v)[/math] where [math]v[/math] is the variance (which we'll play with). We'll assume we can evaluate the probability ratio of two samples according to the intractable distribution. With that, we know how to generate proposals, accept/reject them and produce samples. From the book, this process can be represented as:As pointed out, there is a happy middle ground. There is a particular proposal variance that strikes the right balance of exploring the space and being accepted frequently.And that just about does it.MCMC is our most general tool for sampling from any given PGM. We need skill and intuition to shepherd these methods away from their failure points, but with that, we can perform inference on some rather exotic graphs. That shepherding becomes more difficult with the size and complexity of the PGM, so understanding these difficulties beforehand is the key to knowing when to use MCMC.What's next?At this point, especially if you've read answers 2 and 3 as well, you may have had your fill of inference. You might be curious as to how we actually learn the parameters of a BN or a MN. Well, you might enjoy:5. How are the parameters of a Bayesian Network learned?6. How are the parameters of a Markov Network learned?Footnotes[1] Actually, you don't need to compute the full Gibbs Rule product - you only need to consider the factors for which [math]X_1[/math] appears in. Just think about it - for all the factors for which [math]X_1[/math] doesn't appear, their product remains constant as you plug in different assignments of [math]X_1[/math]. When we normalize, this constant will be divided out, so we don't need to consider it. For large MNs, this is a huge efficiency gain![2] Well, [math]P_M(\mathbf{X}|\mathbf{e})[/math] in fact.Sources[1] Koller, Daphne; Friedman, Nir. Probabilistic Graphical Models: Principles and Techniques (Adaptive Computation and Machine Learning series). The MIT Press. The Markov Chain visuals are from this book, with Daphne’s permission.[2] Murphy, Kevin. Machine Learning: A Probabilistic Perspective. The MIT Press.

On of more hip schools[1] in Warsaw is named after Digvijaysinhji Ranjitsinhji Jadeja[2], Maharaja Jam Sahib of Navanagar.That story is an important part of the school’s tradition. Below, a sample kid’s project done in that school.I mentioned that WWII story in one of my posts, How were the Polish Forces in the West reinforced (in terms of manpower) during WW2?BTW, not hundreds of orphans, but thousands. Much was written about this, a fairly recent documentary film is below.In short, in 1939/41 Soviet Union deported estimated 700000-1500000 of people (mainly but not exclusively Poles, gentile and Jewish) from their homes on territories conquered September 1939 together with Nazi Germany. In many cases the parents were either shot outright, or killed in Gulag camps (roughly estimated 200000). Kids were sometimes shipped to more survivable locations.Gulag orphanage. “I was only ten years old. I could not understand, how I could threaten the great Soviet Union”[3].When Nazis turned around and bit the Soviets (June 1941), UK and Polish governments negotiated “amnesty” for those prisoners and deportees, and Soviets allowed to form an army out of them. Processions of starved ghosts from across all kinds of Soviet bushes and deserts started to walk or travel as they could to designated muster points, often thousands of kilometres.As the Soviet permission (and food rations) were only “to form an army”, uniforms were issued to everybody arriving, including kids. Eventually, those kids were evacuated to India, where Digvijaysinhji Ranjitsinhji Jadeja organized, across Indian society, permanent care for them, including himself funding and building a large care center in his state. Estimated number of kids saved this way is at least 5000.One of witnesses reports the message Digvijaysinhji Ranjitsinhji Jadeja send to the children arriving in India, when he learned of their fate: “Please tell them they are no longer orphans. I am becoming their father”.Footnotes[1] RASZ[2] Digvijaysinhji Ranjitsinhji - Wikipedia[3] Przedszkolaki w pasiakach? Stalin zsyłał do łagrów nawet kilkuletnie dzieci