 Hello everyone, this is Alice Gao. In this video, I will discuss the derivations for the smoothing formulas. The derivations for the smoothing formulas look even longer than those for the filtering formulas. Here's the first part and here's the second part. Don't worry about understanding all the formulas now. I will explain the derivation step-by-step. However, once we break it down and look at it step-by-step, you will see that each step can be justified by one of the five reasons that I discussed previously. For each step, please try to choose a justification yourself first. You will have five options as before. Base rule, rewriting the expression, the chain rule or the product rule, the Markov assumption and the sum rule. Let's look at the first part. In these steps, we wrote the probability as a normalized product of two messages, the first one from forward recursion and the second one from backward recursion. Step one, pause the video and choose an answer. Then keep watching. The correct answer is B, rewriting the expression. We split the sequence of observations into two parts. The second term contains the observations from the start to K to day K. The first term contains the observations from day K plus 1 to the end. If you imagine that you're in day K right now, then the two sequences correspond to past observations from the start up to yesterday, which was day K, and future observations from day K plus 1 to the end. Step two, pause the video and choose an answer. Then keep watching. The correct answer is A, base rule. It's easier to see this when you cross out the last variable in every term, O sub 0 to K, since it appears in all three terms. We switch the places of S sub K and O sub K plus 1 to T minus 1 using base rule. This is convenient since the observation sequence comes after the state on day K. It's natural to think about how the state on day K affects the observations in its future from day K plus 1 to the end. By the way, crossing out the last term doesn't mean that we can cancel it. I want you to disregard the last term so that it's easier to see the pattern in the remaining parts. This step is a version of the base rule where every term conditions on the same term. Here's a practice problem for you. Try proving the general equation yourself. You should be able to prove it with the basic rules of probability. Step three, pause the video and choose an answer. Then keep watching. The correct answer is D, the Markov assumption. This step removes O sub 0 to K from the second term. The Markov assumption is basically a conditional independence relationship. In this case, given S sub K, the state on day K, the future observations are independent of the past observations. That is, any observation up to day K is independent of any other observation from day K plus 1 onward, given the state S sub K. You can verify this relationship by applying D separation on the Bayesian network. Suppose that we're in day K right now. The path between any past observation and any future observation must go through S sub K. If we're considering O sub K, then the two red arrows are both pointing away from S sub K. If we're considering any past observation, the two orange arrows are pointing in the same direction. Either way, if we observe S sub K, it is as if we're cutting the chain at S sub K, making any past observation independent of any future observation. Let's look at the second part of the derivations. In these steps, we derive the recursive formula for backward recursion. We're going backward in time, given the probability of an observation sequence starting on day K plus 2. We can calculate the probability of an observation sequence starting on day K plus 1. Step 1, pause the video and choose an answer. Then keep watching. The correct answer is E, the sum rule. We use the sum rule to introduce the state on day K plus 1, S sub K plus 1. Introducing S sub K plus 1 is convenient, since it is a bridge between the state on day K or S sub K and the observations from day K plus 1 onward. Step 2, pause the video and choose an answer. Then keep watching. The correct answer is C, the chain rule or the product rule. This is easier to see if we cross out the last variable S sub K in every term. We use the product rule to write the probability as a product of two probabilities. Step 3, pause the video and choose an answer. Then keep watching. The correct answer is D, the Markov assumption. This step removes the value S sub K from the first term. This is another conditional independence relationship. Given the state on day K plus 1, S sub K plus 1, all the future observations are independent of the past state on day K, S sub K. You can verify this by applying the separation on the Bayesian network. The path between S sub K and any future observation has to go through S sub K plus 1. Whether we are considering the observation on day K plus 1 or any observation after that, the two arrows both point in the same direction. Therefore, if we observe S sub K plus 1, it is as if we cut the chain at S sub K plus 1, making the state on day K, S sub K, independent of any future observation. Step 4, pause the video and choose an answer. Then keep watching. The correct answer is B, rewriting the expression. We split the sequence of observations from day K plus 1 to the end into two parts. The first term contains the observation on day K plus 1 only. The second term contains the observations from day K plus 2 to the end. Step 5, this is our final step. Pause the video and choose an answer. Then keep watching. The correct answer is D, the Markov assumption. This is another conditional independence relationship. Given the state on day K plus 1, S sub K plus 1, the observation on day K plus 1, O sub K plus 1, is independent of all those future observations from day K plus 2 to the end. You can verify this by applying this operation on the Bayesian network. The path between O sub K plus 1 and any future observation has to go through S sub K plus 1. The two arrows are pointing away from S sub K plus 1. If we observe S sub K plus 1, it is as if we cut the chain at this state, making the observations on day K plus 1, O sub K plus 1, independent of any future observation. That's everything on the derivations of the smoothing formulas. Congratulations on making it through this video. If you will do one thing other than work or study this coming weekend, what would you do? Please post your plan in the week 8 post. That's everything for this video. Let me summarize. After watching this video, you should be able to describe the justification for every step of the derivation of the smoothing formulas. Thank you very much for watching. I will see you in the next video. Bye for now.