daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Added Markov property example.

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I was getting a bit abstract describing Markov chains, so I gave a
concrete example of why it is a reasonable requirement.
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Roger Labbe 2018-02-10 08:47:21 -08:00
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07-Kalman-Filter-Math.ipynb
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@ -1565,13 +1565,14 @@
"\n",
"That is all well and good until the next measurement $\\mathbf z_{t+1}$ comes in, at which point we need to recompute the entire expression for the range $0:t+1$. \n",
"\n",
"\n",
"In practice this is intractable because we are trying to compute the posterior distribution $P(\\mathbf x_{0:t} \\mid \\mathbf z_{0:t})$ for the state over the full range of time steps. But do we really care about the probability distribution at the third step (say) when we just received the tenth measurement? Not usually. So we relax our requirements and only compute the distributions for the current time step.\n",
"In practice this is intractable because we are trying to compute the posterior distribution $P(\\mathbf x_{0:t} \\mid \\mathbf z_{0:t})$ for the state over the full range of time steps. But do we really care about the probability distribution at the third step (say) when we just received the tenth measurement? Not usually. So we relax our requirements and only compute the distributions for the current time step. \n",
"\n",
"The first simplification is we describe our process (e.g., the motion model for a moving object) as a *Markov chain*. That is, we say that the current state is solely dependent on the previous state and a transition probability $P(\\mathbf x_k \\mid \\mathbf x_{k-1})$, which is just the probability of going from the last state to the current one. We write:\n",
"\n",
"$$\\mathbf x_k \\sim P(\\mathbf x_k \\mid \\mathbf x_{k-1})$$\n",
"\n",
"In practice this is extremely reasonable, as many things have the *Markov property*. If you are driving in a parking lot, does your position in the next second depend on whether you pulled off the interstate or were creeping along on a dirt road one minute ago? No. Your position in the next second depends solely on your current position, speed, and control inputs, not on what happened a minute ago. Thus, cars have the Markov property, and we can make this simplification with no loss of precision or generality.\n",
"\n",
"The next simplification we make is do define the *measurement model* as depending on the current state $\\mathbf x_k$ with the conditional probability of the measurement given the current state: $P(\\mathbf z_t \\mid \\mathbf x_x)$. We write:\n",
"\n",
"$$\\mathbf z_k \\sim P(\\mathbf z_t \\mid \\mathbf x_x)$$\n",