prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How many novels did Margaret Way write?

Read the following text and answer briefly.

Context 1: Margaret Way (b  Brisbane d  Cleveland, Queensland, Australia ) was an Australian writer of romance novels and women's fiction  A prolific author, Way wrote more than 120 novels since 1970, many through Mills & Boon, a romance imprint of British publisher Harlequin UK Ltd , owned by Harlequin Enterprises 

Biography
Before her marriage, she was a well-known pianist, teacher, vocal coach and accompanist  She began writing when her son, Laurence Way, was born, a friend took a pile of Mills & Boon books to her, she read all and decided that she also could write these types of novels  She began to write and promote her country with her stories set in Australia  She sold her first novels in 1970  Margaret Way lives with her family in her native Brisbane  Beginning in 2013, Margaret began to self-publish, releasing her first "e-book" mid-July 

Margaret died on the 10th of August 2022 in Cleveland, Queensland 

Bibliography

Single Novels
King Country (1970)
Blaze of Silk (1970)
The Time of the Jacaranda (1970)
Bauhinia Junction (1971)
Man from Bahl Bahla (1971)
Summer Magic (1971)
Return to Belle Amber (1971)
Ring of Jade (1972)
Copper Moon (1972)
Rainbow Bird (1972)
Man Like Daintree (1972)
Noonfire (1972)
Storm Over Mandargi (1973)
Wind River (1973)
Love Theme (1974)
McCabe's Kingdom (1974)
Sweet Sundown (1974)
Reeds of Honey (1975)
Storm Flower (1975)
Lesson in Loving (1975)
Flight into Yesterday (1976)
Red Cliffs of Malpara (1976)
Man on Half-moon (1976)
Swan's Reach (1976)
Mutiny in Paradise (1977)
One Way Ticket (1977)
Portrait of Jaime (1977)
Black Ingo (1977)
Awakening Flame (1978)
Wild Swan (1978)
Ring of Fire (1978)
Wake the Sleeping Tiger (1978)
Valley of the Moon (1979)
White Magnolia (1979)
Winds of Heaven (1979)
Blue Lotus (1979)
Butterfly and the Baron (1979)
Golden Puma (1980)
Temple of Fire (1980)
Lord of the High Valley (1980)
Flamingo Park (1980)
North of Capricorn (1981)
Season for Change (1981)
Shadow Dance (1981)
McIvor Affair (1981)
Home to Morning Star (1981)
Broken Rhapsody (1982)
The Silver Veil (1982)
Spellbound (1982)
Hunter's Moon (1982)
Girl at Cobalt Creek (1983)
No Alternative (1983)
House of Memories (1983)
Almost a Stranger (1984)
A place called Rambulara (1984)
Fallen Idol (1984)
Hunt the Sun (1985)
Eagle's Ridge (1985)
The Tiger's Cage (1986)
Innocent in Eden (1986)
Diamond Valley (1986)
Morning Glory (1988)
Devil Moon (1988)
Mowana Magic (1988)
Hungry Heart (1988)
Rise of an Eagle (1988)
One Fateful Summer (1993)
The Carradine Brand (1994)
Holding on to Alex (1997)
The Australian Heiress (1997)
Claiming His Child (1999)
The Cattleman's Bride (2000)
The Cattle Baron (2001)
The Husbands of the Outback (2001)
Secrets of the Outback (2002)
With This Ring (2003)
Innocent Mistress (2004)
Cattle Rancher, Convenient Wife (2007)
Outback Marriages (2007)
Promoted: Nanny to Wife (2007)
Cattle Rancher, Secret Son (2007)
Genni's Dilemma (2008)
Bride At Briar Ridge (2009)
Outback Heiress, Surprise Proposal (2009)
Cattle Baron, Nanny Needed (2009)

Legends of the Outback Series
Mail Order Marriage (1999)
The Bridesmaid's Wedding (2000)
The English Bride (2000)
A Wife at Kimbara (2000)

Koomera Crossing Series
Sarah's Baby (2003)
Runaway Wife (2003)
Outback Bridegroom (2003)
Outback Surrender (2003)
Home to Eden (2004)

McIvor Sisters Series
The Outback Engagement (2005)
Marriage at Murraree (2005)

Men Of The Outback Series
The Cattleman (2006)
The Cattle Baron's Bride (2006)
Her Outback Protector (2006)
The Horseman (2006)

Outback Marriages Series
Outback Man Seeks Wife (2007)
Cattle Rancher, Convenient Wife (2007)

Barons of the Outback Series Multi-Author
Wedding At Wangaree Valley (2008)
Bride At Briar's Ridge (2008)

Family Ties Multi-Author
Once Burned (1995)

Hitched! Multi-Author
A Faulkner Possession (1996)

Simply the Best Multi-Author
Georgia and the Tycoon (1997)

The Big Event Multi-Author
Beresford's Bride (1998)

Guardian Angels Multi-Author
Gabriel's Mission (1998)

Australians Series Multi-Author
7  Her Outback Man (1998)
17  Master of Maramba (2001)
19  Outback Fire (2001)
22  Mistaken Mistress (2002)
24  Outback Angel (2002)
33  The Australian Tycoon's Proposal (2004)
35  His Heiress Wife (2004)

Marrying the Boss Series Multi-Author
Boardroom Proposal (1999)

Contract Brides Series Multi-Author
Strategy for Marriage (2002)

Everlasting Love Series Multi-Author
Hidden Legacy (2008)

Diamond Brides Series Multi-Author
The Australian's Society Bride (2008)

Collections
Summer Magic / Ring of Jade / Noonfire (1981)
Wife at Kimbara / Bridesmaid's Wedding (2005)

Omnibus in Collaboration
Pretty Witch / Without Any Amazement / Storm Over Mandargi (1977) (with Lucy Gillen and Margaret Malcolm)
Dear Caliban / Heart of the Eagle / Swans' Reach (1978) (with Jane Donnelly and Elizabeth Graham)
The Bonds of Matrimony / Dragon Island / Reeds of Honey (1979) (with Elizabeth Hunter and Henrietta Reid)
The Man Outside / Castles in Spain / McCabe's Kingdom (1979) (with Jane Donnelly and Rebecca Stratton)
Winds From The Sea / Island of Darkness / Wind River (1979) (with Margaret Pargeter and Rebecca Stratton)
Moorland Magic / Tree of Idleness / Sweet Sundown (1980) (with Elizabeth Ashton and Elizabeth Hunter)
The Shifting Sands / Portrait of Jaime / Touched by Fire (1982) (with Jane Donnelly and Kay Thorpe)
Head of Chancery / Wild Heart / One-Way Ticket (1986) (with Betty Beaty and Doris Smith)
Heart of the Scorpion / The Winds of Heaven / Sweet Compulsion (1987) (with Janice Gray and Victoria Woolf)
One Brief Sweet Hour / Once More With Feeling / Blue Lotus (1990) (with Jane Arbor and Natalie Sparks)
Marry Me Cowboy (1995) (with Janet Dailey, Susan Fox and Anne McAllister)
Husbands on Horseback (1996) (with Diana Palmer)
Wedlocked (1999) (with Day Leclaire and Anne McAllister)
Mistletoe Magic (1999) (with Betty Neels and Rebecca Winters)
The Australians (2000) (with Helen Bianchin and Miranda Lee)
Weddings Down Under (2001) (with Helen Bianchin and Jessica Hart)
Outback Husbands (2002) (with Marion Lennox)
The Mother's Day Collection (2002) (with Helen Dickson and Kate Hoffmann)
Australian Nights (2003) (with Miranda Lee)
Outback Weddings (2003) (with Barbara Hannay)
Australian Playboys (2003) (with Helen Bianchin and Marion Lennox)
Australian Tycoons (2004) (with Emma Darcy and Marion Lennox)
A Mother's Day Gift (2004) (with Anne Ashley and Lucy Monroe)
White Wedding (2004) (with Judy Christenberry and Jessica Steele)
A Christmas Engagement (2004) (with Sara Craven and Jessica Matthews)
A Very Special Mother's Day (2005) (with Anne Herries)
All I Want for Christmas    (2005) (with Betty Neels and Jessica Steele)
The Mills and Boon Collection (2006) (with Caroline Anderson and Penny Jordan)
Outback Desire (2006) (with Emma Darcy and Carol Marinelli)
To Mum, with Love (2006) (with Rebecca Winters)
Australian Heroes (2007) (with Marion Lennox and Fiona McArthur)
Tall, Dark and Sexy (2008) (with Caroline Anderson and Helen Bianchin)
The Boss's Proposal (2008) (with Jessica Steele and Patricia Thayer)
Island Heat / Outback Man Seeks Wife / Prince's Forbidden Virgin / One Night Before Marriage / Their Lost-and-found Family / Single Dad's Marriage Wish (2008) (with Robyn Donald, Marion Lennox, Carol Marinelli, Sarah Mayberry and Anne Oliver)
Australian Billionaires (2009) (with Jennie Adams and Amy Andrews)
Cattle Baron : Nanny Needed / Bachelor Dad on Her Doorstep (2009) (with Michelle Douglas)

External links
Margaret Way at Harlequin Enterprises Ltd

Australian romantic fiction writers
Australian women novelists
Living people
Year of birth missing (living people)
Women romantic fiction writers
Context 2: 2015-05-14 Assigned to ROVI GUIDES, INC  reassignment ROVI GUIDES, INC  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: TV GUIDE, INC 
2015-05-14 Assigned to UV CORP  reassignment UV CORP  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: UNITED VIDEO PROPERTIES, INC 
2015-05-14 Assigned to TV GUIDE, INC  reassignment TV GUIDE, INC  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: UV CORP 
Methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  In some embodiments, a media application may use a content-recognition module to determine the context of an event and distribute itemized tasks to multiple entities in order to generate the supplemental information about the event 
While viewing media assets (e g , a television program), users may wish to learn more information about an event (e g , a statement made by a person appearing in the media asset, the validity of a claim in an advertisement, etc ) occurring in the media asset  While some media assets allow a user to select additional options or added features (e g , pop-up biographies about the cast and crew), when the added features appear and what topic the added features concern are determined by the content producer and not the user  Furthermore, as the added feature is derived from the content producer, the added feature may be biased or may present limited viewpoints about an event  Therefore, added features provided by a content producer may not provide the added information about an event that a user desires 
In order to gain the added information that a user desires, the user may use additional devices (e g , a laptop computer) to search (e g , using an Internet search engine) for more information about the event  However, without knowing the proper context (e g , who said the statement, what was the tone of the statement, when was the statement said, etc ) of the event or what search terms to use to describe the context of the event (e g , how to describe the tone of the statement), a user may not be able to determine (even using a search engine) more information about the event  Moreover, the use of general search terms may not provide the accuracy or precision needed by the user  Furthermore, even if a user may eventually determine the information, the effort and time required may distract the user from the media asset 
Accordingly, methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  In some embodiments, a media application may use a content-recognition module to determine the context of an event in a media asset and distribute itemized tasks to multiple users in order to generate the supplemental information about the event  The context-recognition module prevents the user from being distracted from the media asset (e g , while the user attempts to describe the context of the event or search for information about the event)  In addition, by distributing tasks to multiple entities (e g , crowd-sourcing), the media application may collect large amounts of information in relatively short periods of time (or in real-time) and aggregate and/or filter the information to generate the supplemental information about the event based on multiple viewpoints and/or sources  By using multiple viewpoints and/or sources, the media application enhances the completeness (e g , by providing unbiased information) and accuracy of the supplemental information 
For example, when a statement or action is made by a character or person appearing on a media asset (e g , a television program), a user may request supplemental information about the statement or action  In response, the media application may determine the context of the statement (e g , who said the statement and to what the statement was referring) or action (e g , what was the reason for the action)  After determining the context of the statement or action, the media application may itemize into tasks the additional information it requires in order to generate the supplemental information  The media application may then transmit requests including the tasks to a plurality of other users  Based on the responses from the plurality of other users, the media application may generate the supplemental information for display to the user 
In some embodiments, a media application may use multiple types of content-recognition modules and/or algorithms to determine the context of an event  For example, the media application may process data associated with the event in order to determine the context of an event  In some embodiments, processing the various types of data may include cross-referencing the data in a database indicating the different contexts the event may have 
In some embodiments, a media application may generate supplemental information about an event in a media asset in response to a user request  In order to generate the supplemental information, the media application may transmit, to multiple users, a request for additional information regarding a context of an event shown in a media asset  Upon receiving messages from the plurality of users that include the requested additional information, the media application may generate the supplemental information associated with the context of the event based on the messages 
It should be noted, the systems and/or methods described above may be applied to, or used in accordance with, other systems, methods and/or apparatuses 
FIG  9 is a flowchart of illustrative steps for generating supplemental information based on additional information provided by a plurality of users in accordance with some embodiments of the disclosure 
Accordingly, methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  The methods and systems described herein alleviate the need for a user to determine the proper context (e g , who said a statement, what was the tone of the statement, when was the statement said, etc ) of an event in a media asset, or the search terms to use to describe the event (e g , the proper search terms to describe the tone of the statement), in order to determine more information about the event  In addition, the methods and systems increase the completeness and accuracy of the information compared to information gathered using traditional searching methods (e g , an Internet search engine), without distracting the user from the media asset 
In some embodiments, a media application may receive a user input from a user device for supplemental information about the context of an event shown in a media asset  The media application may determine additional information required to generate the supplemental information about the context of the event shown in a media asset, and transmit requests for the additional information to one or more users  The media application may receive one or more messages, which include the requested additional information, from the one or more users and generate the supplemental information based on the one or more message  The media application may then instruct the user device to display the supplemental information 
As used herein, “supplemental information” refers to any information related to or associated with an event in a media asset  For example, supplemental information may include, but is not limited to, the verification of a statement or claim in a media asset, further descriptions and/or information about objects or entities shown and/or described in a media asset, and/or any other information, including, but not limited to, a video or audio segment, that may interest a user about an event in a media asset  In some embodiments, the media application may generate supplemental information based on one or more pieces of additional information 
As used herein, “additional information” refers to any information used to generate supplemental information  For example, in an embodiment in which supplement information is the verification of a statement made by a person displayed in a media asset, and a request for the additional information from the media application includes a request for a fact needed to verify the factual basis of the statement, the additional information may be the fact used to verify the statement  For example, if an advertisement claims to have the best product on the market, the media application may use additional information such as the name of the product in question, a list of all other products in the market, and the results of a comparison study of the product in question to all other products to determine whether or not the product is actually the “best” product on the market  Additionally or alternatively, the media application may request industry and/or user reviews related to the event (e g , reviews indicating the quality of the product)  The media application may then use the information in the reviews to generate the supplemental information 
As used herein, an “event” is any action (e g , a verbal statement, opinion and/or physical movement), segment (e g , a portion of a news broadcast featuring a particular topic), or other occurrence during a media asset that may be of particular interest to a user  For example, in some embodiments an event may be a statement or gesture made by a character or person in a media asset affirming or denying a claim 
As referred to herein, the terms “media asset” and “content” should be understood to mean an electronically consumable user asset, such as television programming, as well as pay-per-view programs, on-demand programs (as in video-on-demand (VOD) systems), Internet content (e g , streaming content, downloadable content, Webcasts, etc ), video clips, audio, content information, pictures, rotating images, documents, playlists, websites, articles, books, electronic books, blogs, advertisements, chat sessions, social media, applications, games, and/or any other media or multimedia and/or combination of the same  Media applications also allow users to navigate among and locate content  As referred to herein, the term “multimedia” should be understood to mean content that utilizes at least two different content forms described above, for example, text, audio, images, video, or interactivity content forms  Content may be recorded, played, displayed or accessed by user equipment devices, but can also be part of a live performance 
As referred to herein, the phrase “user equipment device,” “user equipment,” “user device,” “electronic device,” “electronic equipment,” “media equipment device,” or “media device” should be understood to mean any device for accessing the content described above, such as a television, a Smart TV, a set-top box, an integrated receiver decoder (IRD) for handling satellite television, a digital storage device, a digital media receiver (DMR), a digital media adapter (DMA), a streaming media device, a DVD player, a DVD recorder, a connected DVD, a local media server, a BLU-RAY player, a BLU-RAY recorder, a personal computer (PC), a laptop computer, a tablet computer, a WebTV box, a personal computer television (PC/TV), a PC media server, a PC media center, a hand-held computer, a stationary telephone, a personal digital assistant (PDA), a mobile telephone, a portable video player, a portable music player, a portable gaming machine, a smart phone, or any other television equipment, computing equipment, or wireless device, and/or combination of the same 
In some embodiments, the user equipment device may have a front facing screen and a rear facing screen, multiple front screens, or multiple angled screens  In some embodiments, the user equipment device may have a front facing camera and/or a rear facing camera  On these user equipment devices, users may be able to navigate among and locate the same content available through a television  Consequently, media may be available on these devices, as well  The media provided may be for content available only through a television, for content available only through one or more of other types of user equipment devices, or for content available both through a television and one or more of the other types of user equipment devices  The media applications may be provided as on-line applications (i e , provided on a website), or as stand-alone applications or clients on user equipment devices  Various devices and platforms that may implement media applications are described in more detail below 
In some embodiments, a media application may transmit, to a plurality of users, a request for additional information regarding a context of an event shown in a media asset  As used herein, a “plurality of users” may include, but is not limited to any device, entity, or source of information that may process a request for additional information  For example, the plurality of users may include a person operating a user equipment device  In some embodiments, the person may receive (e g , via e-mail, Internet posting, advertisement, or any other applicable information delivery method) the request from the media application for additional information, and in response generate a message (e g , via a return e-mail, an answer to the Internet posting, a user input in the advertisement, or any other applicable method of transmitting information) that includes the additional information  It should be noted that in some embodiments, transmitting a request to a plurality of users may also include querying one or more databases (e g , an Internet search engine or any other storage device, including, but not limited to, databases containing previously generated supplemental information and/or additional information) or consulting one or more data gathering services (e g , a intelligent personal assistant application) for the additional information 
In some embodiments, a media application may use a content-recognition module or algorithm to determine the context of an event and distribute itemized tasks to multiple users in order to generate the supplemental information about the event  The content-recognition module may use object recognition techniques such as edge detection, pattern recognition, including, but not limited to, self-learning systems (e g , neural networks), optical character recognition, on-line character recognition (including but not limited to, dynamic character recognition, real-time character recognition, intelligent character recognition), and/or any other suitable technique or method to determine the objects and/or characteristics in media assets  For example, the media application may receive media assets in the form of a video  The video may include a series of frames  For each frame of the video, the media application may use a content-recognition module or algorithm to determine the context (e g , the person that is speaking or a facial gesture affirming or denying a statement) of an event occurring during the frame or series of frames 
In some embodiments, the content-recognition module or algorithm may also include speech recognition techniques, including but not limited to Hidden Markov Models, dynamic time warping, and/or neural networks (as described above) to translate spoken words into text  The content-recognition module may also use other techniques for processing audio and/or visual data  For example, the media application may monitor the volume of a statement in a media asset to determine the tone of the statement (e g , a high volume may indicate an angry tone) 
In addition, the media application may use multiple types of optical character recognition and/or fuzzy logic, for example, when determining the context of a keyword(s) retrieved from data (e g , media data, translated audio data, subtitle data, user-generated data, etc ) associated with the media asset (or when cross-referencing various types of data with databases indicating the different contexts of events as described below)  For example, the particular data field may be a textual data field  Using fuzzy logic, the system may determine two fields and/or values to be identical even though the substance of the data field or value (e g , two different spellings) is not identical  In some embodiments, the system may analyze particular data fields of a data structure or media asset frame for particular values or text  The data fields could be associated with characteristics, additional information, and/or any other data required for the function of the embodiments described herein  Furthermore, the data fields could contain values (e g , the data fields could be expressed in binary or any other suitable code or programming language) 
As used herein, the “context” of an event refers to the set of circumstances or facts that surround a particular event that influence or affect the meaning of the event  For example, when determining the context of a written and/or spoken statement, the media application may determine who or what authored/stated the statement, the written and/or spoken words and/or other statements that preceded and/or followed the statement, the tone of the statement, and/or any other conditions that may alter the connotation of the statement 
FIG  1 shows an illustrative example of a media application that may be used to display supplemental information in accordance with some embodiments of the disclosure  Display 100 illustrates a display on a user device displaying a media asset  Display 108 illustrates a display featuring supplemental information as described and/or generated in FIGS  6-9  It should be noted that display 100 and display 108 may be presented on any of the devices shown in FIGS  3-4  For example, in some embodiments, display 100 and display 108 may be displayed on user equipment 402, 404, and/or 406 (FIG  4) 
In FIG  1, display 100 represents a display of a media asset (e g , a streaming television program) on a user device (e g , user equipment 402, 404, and/or 406 (FIG  4))  Display 100 includes entity 102 and entity 104  In display 100, entity 104 is currently speaking as indicated by event 106  As shown in FIG  1, event 106 is a statement (e g , “We export a lot of coal”) by a person in the media asset 
In some embodiments, display 108 represents the continued display of the media asset on a user device, after a user has requested supplemental information about event 106  For example, a media application may have received a user input (e g , via user input interface 310 (FIG  3)) while entity 104 was speaking  Using the systems and methods described herein (e g , FIGS  6-9), the media application generated supplemental information 110  Supplemental information 110 represents more information about event 106 
For example, the media application (e g , media application 206 (FIG  2)) may have determined the context of event 106  Specifically, the media application may determine via a content-recognition module or algorithm the words spoken and/or actions by the person during the event  Additionally or alternatively, the media application may analyze the words and/or action during a predetermined amount of time (e g , ten seconds) before and/or after the event (e g , in order to better understand the context of the event)  Furthermore, by cross-referencing the words and/or other information obtained by the content-recognition module (e g , as discussed below in relation to FIG  5) with a database, the content-recognition module determines that the term “we,” the person in the media asset refers to an organization or body
Context 3: \section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}  Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation 

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary  This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency} 

Although these two approaches share some underlying relations, there are very few connections in the literature  The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i e  Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al } \cite{haykin1997adaptive}  The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model  This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes 

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation  The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step  However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm 

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS)  Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution  We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS  Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}  Here, we show the usefulness of this probabilistic approach for adaptive filtering 

The probabilistic perspective we adopt throughout this work presents two main advantages  Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments  The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w r t  these algorithms and standard LMS  Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications 

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works  However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering \\


\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors 


In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process  In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:


\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix  In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}) \nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$ 
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner  The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i e  the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k  
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive} 

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$  Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful  In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution  We also show that this approximation leads to an LMS-like estimation 
 


\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution 

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ) 
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i e , 

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}  \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A



ground_truth: Margaret Way wrote more than 120 novels.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does a media application determine the context of an event?

Read the following text and answer briefly.

Context 2:  The content-recognition module or algorithm may also determine that the term “export” refers to shipping goods out of a country  The content-recognition module or algorithm may also determine that the term “a lot” refers to a particular numerical amount  Finally, the content-recognition module or algorithm may also determine that the term “coal” refers to a mineral of fossilized carbon 
The content-recognition module or algorithm may also determine the relationships between words and/or other information obtained by the content-recognition module  For example, by processing the relationship between the words, the media application determines that event 106 is a statement regarding a particular amount of a particular substance shipped out of a particular country  Therefore, the media application determines that the request for supplemental information is likely a request to determine the validity of the statement  The media application then generates the supplemental information 
The media application may also have stored supplemental information generated by previous requests (e g , supplemental information generated in response to the same or different user viewing the media asset at an earlier date), and display the supplemental information again during the event (either in response to a user input requesting supplemental information or automatically without a user requesting supplemental information) 
FIG  2 shows an illustrative example of a system that may be used to generate supplemental information (e g , supplemental information 110 (FIG  1)) based on additional information provided by a plurality of users in accordance with some embodiments of the disclosure  For example, in some embodiments, system 200 may be used to generate supplemental information (e g , supplemental information 110 (FIG  1)) on a display (e g , display 108 (FIG  1)) of a user device (e g , user equipment 402, 404, and/or 406 (FIG  4))  It should be noted that in some embodiments, the devices shown in FIG  2 may correspond to one or more devices in FIGS  3-4 
FIG  2 shows system 200  In system 200, a user is currently accessing a media asset on display 202  In some embodiments, display 202 may correspond to display 100 (FIG  1))  During an event (e g , event 106 (FIG  1)) a user may have requested supplemental information about an event (e g , event 106 (FIG  1)) in display 202 using user device 204  Media application 206, which in some embodiments, may be implemented on user device 204 or at a remote location (e g , supplemental information source 418 (FIG  4)), receives the request for supplemental information 
Media application 206 determines the context of the event (e g , who said the statement making up the event and to what the statement was referring)  After determining the context of the statement, the media application may itemize into one or more tasks, additional information (e g , facts) it requires in order to generate the supplemental information (e g , a verification or correction of the factual basis of the statement)  For example, if the event is a statement about the amount of coal that is exported from the United States (e g , as described in relation to FIG  1 above), media application 206 may determine the fact required to generate the supplemental information is the exact numerical amount of coal that is exported from the United States  The media application may then transmit requests for the additional information (e g , a request for the exact numerical amount of coal that is exported from the United States) to a plurality of other users 
In FIG  2, users operating user device 208, user device 210, and user device 212 represent a plurality of users  Having determined the additional information it requires in order to generate the supplemental information, media application 206 requests the additional information from the plurality of users  In system 200, media application 206 has transmitted the same task (e g , the same question) to each of the plurality of users  In some embodiments, one or more of the users may receive different tasks  For example, by breaking the additional information into small, independent tasks, media application 206 may increase the speed (e g , multiple users may work concurrently to solve different parts of a problem) and accuracy (e g , reducing the tasks to smaller, less complex problems reduces the chance of human error) of the additional information returned by the plurality of users 
In addition, by breaking the additional information into small, independent tasks, the plurality of users may not know to what they are contributing (enhancing the privacy of the user that requested the supplemental information), however, the plurality of users can still be effective in their individual tasks  In addition, by breaking the additional information into small, independent tasks, the media application may more easily outsource the requests for additional information  For example, one or more of the tasks used to generate the additional information may be the same as one or more of the tasks used to generate other additional information (e g , additional information used to generate different supplemental information in response to a request for supplemental information about the same or a different event issued by the same or a different user)  The response to each of the request and/or the additional information may be stored (e g , on any of the devices accessible by communications network 414 (FIG  4)) for subsequent retrieval 
Based on the responses, transmitted as messages including the additional information, from the plurality of other users, media application 206 may generate the supplemental information (e g , supplemental information 110 (FIG  1)) for display to the user on the user device 204  For example, media application may aggregate, append, and/or compare the additional information in each of the messages received from the plurality of users  The supplemental information may then be generated based on the aggregated, appended, and/or compared additional information (e g , as described in FIG  9 below) 
In some embodiments, the plurality of users may receive summary information about the event with the request for additional information  (e g , a video clip of a portion or segment of the media asset, a textual description, etc ), which may help the plurality of users provide additional information  For example, in some embodiments, the media application may instead of (or in addition to) determining the context of an event, determine a particular portion of the event that would be needed for the plurality of users to provide additional information about the event 
For example, the media application may use progress information associated with the progress of the media asset (e g , line 506 (FIG  5)) to determine at what point during the progression of the media asset the event occurred, and in response, transmit a portion of the media asset beginning ten second before that point and ending ten seconds after that point  For example, if the event is a statement made by a character or person in a media asset, the media application may determine when the statement began (e g , the point of progress of the media asset in which the statement began) and ended  The media application may then include the portion containing the entire statement (and the event) in the request for additional information sent to the plurality of users 
The selected portion may include any amount of summary information that the media application determines is necessary for the user or any one of the plurality of users to understand the main action sequence  This summary information (e g , a portion of the media asset) may be included with the request for additional information (e g , in a file transmitted with the request), or may be included with the generated supplemental information as a reference for the user  For example, the media application may select a segment of the play length of the media asset or a particular scene of the media asset, which includes the event, for to display to the plurality of users along with the request for additional information 
For example, if an event (e g , a statement) was in response to a question, the media application may also determine when the question began and ended, and send the entire question (or the play length of the media asset corresponding to the question) to the plurality of users as well  After determining the portion to provide to the plurality of users (e g , a segment including the ten seconds before and the ten seconds after the event), the media application may provide the summary information of the event and any other material needed by the plurality of users to understand the event and/or request for supplemental information from the user 
In some embodiments, a portion of the media asset containing the event, as selected by the media application, may also include any amount of the play length of the media asset, or any amount of scenes or segments from the media asset  In some embodiments, the portion may include segments of the play length of the media asset or scenes from the media asset that are not adjacent during the normal playback of the media asset  For example, in some embodiments, a portion of the media asset may include one or more sequences or scenes of interest to the plurality of users, even though the particular sequences or scenes are featured at different points in the play length of the media asset  The media application may determine the segments or scenes to include based on a content recognition file (e g , data structure 500 (FIG  5)) describing the media asset  For example, if a plot point or other information, which may be relevant to an event is displayed earlier in the media asset, the summary information may include a portion of the media asset displaying the plot point 
In some embodiments, the length of a portion may be determined based on the genre of the media asset  In some embodiments, the length of the portion may depend on a user profile for the user or for anyone of the plurality of users  For example, a user profile and/or a content recognition file (e g , data structure 500 (FIG  5)) may indicate that a particular user may require more or less additional content  For example, the user may be aware of particular characters or plot points in the media asset and, therefore, may not require the additional content to introduce those aspects 
In some embodiments, the plurality of users may receive a particular user interface, which organizes the data about the event (e g , a clip of the actual event, summary information about the event, information about the request for supplemental information issued by the user, etc )  The interface may also include an automatic submission form, which may be used to generate a message, which is sent to the media application 
In some embodiments, the media application may also receive user input from the user requesting the supplemental information that further affects the generation of supplemental information by the media application  For example, the user may request the supplemental information includes particular information (e g , the factual basis of a statement), may request a multimedia format of the supplemental information (e g , textual description, a video clip, etc ), may request a form of the supplemental information (e g , a short description about the event, an Internet link to other sources of information on the event, or a true or false designation about the event) by entering user inputs (e g , via user input interface 310 (FIG  3)) 
It should be noted that any information or process referred to in this disclosure that is referred to as being in response to a user input may alternatively and/or additionally be performed automatically by the media application (e g , via control circuitry 304 (FIG  3))  For example, in some embodiments, a user may request a true or false designation (e g , an on-screen pop-up box indicating whether an event was true or false)  Additionally and/or alternatively, in some embodiments, the true or false designation may appear automatically based on predetermined settings indicating to the media application to display a true or false designation in response to detecting an event 
In some embodiments, an indicator that supplemental information has previously been generated or is currently ready to generate (e g , a plurality of users are available) may be displayed to a user (e g , on display 100 (FIG  1) during the event)  The indicator may also indicate the particular information, the multimedia format, and/or the form of supplemental information that is available  An indicator may also appear with the supplemental information (e g , supplemental information 110 (FIG  1)), which allows the user to request additional supplemental information or provide feedback/responses (e g , rating the quality of the supplemental information) to the media application and/or plurality of users 
In some embodiments, a user may also access (e g , via selection of an indicator and/or automatically upon the supplemental information being generated) summary information about the event  For example, in some embodiments (e g , when the supplemental information is not generated in real-time), the media asset may have progressed to a different point by the time the supplemental information is ready for display  Therefore, the media application may need to provide a video clip of the event or other summary information, so that the user remembers about what or why the supplemental information was requested 
FIG  3 is a block diagram of an illustrative user equipment device in accordance with some embodiments of the disclosure  It should be noted that the components shown in FIG  3 may be used to store, receive, transmit, and/or display the media assets, additional information, and/or supplemental information as described herein  For example, media application 206 (FIG  2) may be implemented on user equipment device 300, and may issue instructions (e g , displaying supplemental information 110 (FIG  1)) via control circuitry 304 
Users may access media assets and the media application (and its display screens described above and below) from one or more of their user equipment devices  FIG  3 shows a generalized embodiment of illustrative user equipment device 300  More specific implementations of user equipment devices are discussed below in connection with FIG  4  User equipment device 300 may receive content and data via input/output (hereinafter “I/O”) path 302  I/O path 302 may provide content (e g , broadcast programming, on-demand programming, Internet content, content available over a local area network (LAN) or wide area network (WAN), and/or other content) and data to control circuitry 304, which includes processing circuitry 306 and storage 308  Control circuitry 304 may be used to send and receive commands, requests, and other suitable data using I/O path 302  I/O path 302 may connect control circuitry 304 (and specifically processing circuitry 306) to one or more communications paths (described below)  I/O functions may be provided by one or more of these communications paths, but are shown as a single path in FIG  3 to avoid overcomplicating the drawing 
Control circuitry 304 may be based on any suitable processing circuitry such as processing circuitry 306  As referred to herein, processing circuitry should be understood to mean circuitry based on one or more microprocessors, microcontrollers, digital signal processors, programmable logic devices, field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc , and may include a multi-core processor (e g , dual-core, quad-core, hexa-core, or any suitable number of cores) or supercomputer  
Context 3:  In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M} 
\label{eq:sigma_hat}
\end{equation}


We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule  First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$  Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}), 
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber 
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch  4]{murphy2012machine})  Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where 
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k   
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq ~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size 


Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2) 
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k, 	
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} \nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$ 

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}  In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$  

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}  More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases  We will comment more about this in the next section  
\end{itemize}



\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments  In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$  The entries of the vector are independently and uniformly chosen in the range $[-1,1]$  Then, the vector is normalized so that $\|{\bf w}^o\|=1$  Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix  The additive noise variance is such that the SNR is $20$ dB  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1



ground_truth: It uses a content-recognition module or algorithm.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the proposed approach in this research paper?

Read the following text and answer briefly.

Context 3: 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}




ground_truth: This research paper proposed an approach based on approximating the posterior distribution with an isotropic Gaussian distribution.