ABSTRACT
In the previous chapter, we saw that particle filtering corresponds to approximating the conditional distribution of the state https://www.w3.org/1998/Math/MathML"> X t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11233.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> at time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11234.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> given the observations, https://www.w3.org/1998/Math/MathML"> Y 0 : t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11235.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , up to time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11236.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . In this chapter, we extend the idea to obtaining smoothers. Particle smoothing corresponds to approximating the conditional distribution of the state https://www.w3.org/1998/Math/MathML"> X t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11237.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> at time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11238.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (or a subsequence of states https://www.w3.org/1998/Math/MathML"> X s : t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11239.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for https://www.w3.org/1998/Math/MathML"> s ≤ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11240.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) given all of the available observations, https://www.w3.org/1998/Math/MathML"> Y 0 : n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11241.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , up to time https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11242.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> t ≤ n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11243.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The smoothing distribution may be thought of as a correction or an update to the filter distribution that is enhanced by the use of additional observations from time https://www.w3.org/1998/Math/MathML"> t + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11244.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq11245.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . In general, smoothing distributions are an integral part of inference for state space models. For example, we saw that in the linear Gaussian case, the smoothers were essential for MLE via the EM algorithm (see Section 2.3.2) and, of course, in fitting smoothing splines (see Section 2.4).
