Input: What did Mary tell the disciples?

Context:
A Homily from Easter Sunday, 2017.
Early on the first day of the week, while it was still dark, Mary Magdalene came to the tomb and saw that the stone had been removed, indicating that the tomb was empty. But Mary stood weeping outside the tomb. As she wept, she bent over to look[a] into the tomb; and she saw two angels in white, sitting where the body of Jesus had been lying, one at the head and the other at the feet. They said to her, “Woman, why are you weeping?” She said to them, “They have taken away my Lord, and I do not know where they have laid him.” When she had said this, she turned around and saw Jesus standing there, but she did not know that it was Jesus. Jesus said to her, “Woman, why are you weeping? Whom are you looking for?” Supposing him to be the gardener, she said to him, “Sir, if you have carried him away, tell me where you have laid him, and I will take him away.” Jesus said to her, “Mary!” She turned and said to him in Hebrew,[b] “Rabbouni!” (which means Teacher). Jesus said to her, “Do not hold on to me, because I have not yet ascended to the Father. But go to my brothers and say to them, ‘The tomb is empty and he is no longer there.’” Mary Magdalene went and announced to the disciples, “The tomb is empty”; and she told them that he had said these things to her.
Early in the morning, while it was still dark, Mary wept in the throws of grief. Early in the morning, while it was still dark, Mary dragged herself out of bed after a sleepless night and walked to the tomb in a kind of trance. Early in the morning, while it was still dark, Mary cried—scared, confused, alone. Early in the morning, while it was still dark, Mary thought that the powers of death had the last Word. Early in the morning, while it was still dark, Mary heard a voice in the darkness calling her name—Mary.
Throughout this Lenten season, we’ve examined the ways that the powers and principalities hold us captive—how they push us towards securing our own survival, dominating others, using God for our own agenda. We’ve seen how in Jesus’ ministry, he’s constantly in resistance mode—exposing the powers for what they really are and envisioning an alternative way of living in the world. He describes this way as “the kingdom of God,” the living water we drink so we never thirst again, the light of the world. Jesus invites those who follow him into similar acts of resistance—to free us from the power money has on us by giving it away, to choose to see ourselves as Jesus sees us, resisting the shame that says I’m not enough, to practice Sabbath that contradicts productivity, to untie the grave clothes of someone who’s hands and feet are still tied in the trappings of death.
But all Jesus’ acts of resistance had a cost. All of the times he just wouldn’t shut up, all of the crowds he attracted because he actually noticed those who were normally ignored, the powers finally said enough is enough and put an end to his resistance the only way they could guarantee silence and division—by nailing him to a tree.
Just then, she turned and saw a man the shadow of a man behind her; a man she assumed was the gardener, his face unfamiliar in the darkness. He repeated the question—“Woman, why are you crying?” Thinking that perhaps he knew what happened or worse, that he was a culprit, she begged, “Sir if you have carried him away, tell me where you have put him and I will get him.” But Jesus interrupted her pleading, interrupted her desperation, and called her by name from the darkness, Mary.
Mary. He calls her name. Her name. The name that captures the particularity of her life. To the gardener, she would just be the crying woman. At other points in her life, she was the possessed woman, the woman who wasn’t enough, the woman on the outside of the group. Never nameless—but still unnamed. Never not Mary, but still, not known.
Early in the morning, while it was still dark, God defeated the powers and principalities in the ultimate act of resistance—resurrection. The grave could not contain the Lord. Even death wasn’t enough.
In the resurrection, God defeats the powers of death and shows that it’s God who has the final Word. Nothing, not even death, can keep us from being fully known by God. The powers try to have the final say on our names, our identities, the markers by which we measure ourselves, the systems that hold people captive or keep people in oppression. But Jesus calls us out of the darkness by name.
On this Easter Sunday, we hear our Risen Lord calling our names from the darkness—Jesus, the resurrected one, the name above all names, the great I am, the Prince of Peace, the alpha and omega, the light of the world. The risen Lord has spoken.
This is the name unto which you were baptized. As you come forward and mark the sign of the cross on your forehead today, hear Jesus speaking your name from the darkness and drawing you into the light.
From our worship service on the fifth Sunday of Lent, April 2, 2017.
“Is the Lord really with us or not?” “Is the Lord really with us or not?” Why did you bring us all the way from Egypt to let us die of thirst in this desert? At least in Egypt, we had water. At least in Egypt, we weren’t so thirsty. At least in Egypt, we knew what tomorrow would hold. At least in Egypt, we weren’t so thirsty.
But no, that’s not the story they give us. They are hard on their ancestors. They tell how it is. The elders who sat and wrote down these stories understood something about our bodies, who we are and how we work. After all the generations these stories passed through, they tell the truth about how quickly we forget, about how quickly we complain, about how quickly we grow thirsty, about how much we need water.
It doesn’t take long, does it. By the end of this sermon, I will no doubt feel thirsty, not from walking on hard dusty ground in the heat of the day, but just from speaking with you. Most of us wake up in the morning needing a drink. Our bodies depend on water. We cannot live without it. Thirst, then, doesn’t happen only one time. When the Israelites panicked that they had no water, they weren’t only thinking of the present moment. They knew what was coming! We need water to live! Without water, we will die! Even if we have water for today, we will need water again tomorrow! We can drink until we are satisfied, only to know that we will eventually be thirsty for more.
The gospel of John tells a story about a woman who gave up on this question all together. She moved beyond wondering if the Lord was really with her, so confident God had forgotten her that she gave up wondering at all. Born a Samaritan into a world that valued other bodies as better than her body: male bodies, Jewish bodies, even married bodies. Even after encountering Jesus, she still leaves their conversation without a name, numbered as one of many, simply called, “Samaritan woman.” She too, was thirsty. Most believe that her shame led her to drink water in the heat of the day, when no one else would be at the rocky well, when she could get a drink alone, without experiencing the stigma and stares of others. When she came to get a drink, Jesus was also at the well, thirsty himself and in need of rest and water from the long journey through Samaria.
The Israelites complaint for water sends Moses to the only one who can satisfy, the only one who can meet this need. Moses turns to God, “What should I do with these people? How can I satisfy their thirst? I’ve looked around, I’ve checked far and wide, turned the house upsidedown, looked under the seats of the car, at the bottle of every bottle, I’ve even looked for dew on the ground and under the lids of jars and there is no water to be found. Where do we go for water? Is the Lord really with us or not?
The Israelites who wrote down this story and allowed the ancestors to look like desperate complainers who doubted God and tested God, they were onto something. They knew that we are thirsty people. Jesus knew also. All who are thirsty, come! All who believe in me, drink this living water! We are desperate to feel God’s presence, to be bathed in the water of the Spirit, to know that this is not all there is, to feel a sense of belonging to the One who is greater than I. We can only make it so long in the desert, so long wandering from one trial to the next, without a drink.
And yet, Jesus also says, “Blessed are those who hunger and thirst for righteousness, for they will be filled. Notice with me: not blessed are those who are righteous, but those who thirst for righteousness. Not blessed are those who are righteous, but those who thirst for righteousness. Blessed are those who thirst for relationship with God, to know God, to see God.
I wonder, “Does Jesus want us to keep wanting?” Does Jesus want us to keep thirsting? Many faithful followers of Jesus throughout history have never claimed their thirst was quenched, never fully satisfied. You know that moment when you quench your thirst, when you sigh with relief when your throat is at ease once again, that’s the opposite of how many of God’s children have described the life of faith. They describe wanting more, being satisfied at times, while knowing they will be thirsty again.
What will be waiting for you at the rock? Will the water gush out, bursting forth, covering you from head to toe with God’s presence, drenching you in hope, cleansing you from the dust that’s caked to your feet and renewing you for a new day, a new hour, a new moment basking in the presence of God?
What will be waiting for you at the rock? Will the water drip slowly, quenching your thirst for but a moment, giving you just a glimpse of God’s spirit? Will it be so hard to get the water from the rock, that you’ll have to bend down, get underneath that dripping water to try and catch a drop? Will it be just enough for you to know, if only for a moment, that God is really with you? Will it be just enough to satisfy you for this hour, but keep you coming back for more?
What will be waiting for you at the rock? What if it seems like the water has run out, like there isn’t a drop left, the way that Mother Teresa described? What then? We follow her example. She still goes to the rock, over and over again, not to get water quench her own thirst, but to relieve the thirst of God’s other children.
This post was adapted from our sermon series on Interpreting Exodus. Pastor Megan preached this sermon at Butner Federal Prison complex on August 30, 2015.
On Father’s Day 2015, we gathered for worship at the labyrinth in front of UNC hospital, having devoted the month of June to exploring the question, “What happens after we die?” Many have watched their father’s die in this place or other similar spaces. We shared in a time of both remembrance and prayer/meditation, participating in the ancient spiritual practice of walking the labyrinth. A labyrinth is a kind of maze, laid out in a circle.
Tony graciously shared the following reflections from his experience at the labyrinth on the hot June day.
It’s smaller than I expected, stark and hard‐surfaced, with no landscaping for ornamentation or shade. I don’t know what to expect from it… or from myself. But that’s part of the appeal. I stand at the entrance, hesitating, trying to clear my mind. This doesn’t work very well, so I just start walking.
Almost immediately, the path presents itself as a linear and chronological symbol of my life’s journey. Like my physical lifetime, it has a beginning and an end, with an as‐yet undetermined amount between. This could be interesting. I like it so far… although I’m insecure about my style… and unsure about proper protocol. Is someone staring at me? Do I have to meditate? How slowly should I walk? Is it better to focus my thoughts… or to simply let them come? Will I control this thing, or allow it to control me?
I begin to see each step as an increment of elapsed time, an irretrievable expenditure of life energy. I equate my initial discomfort to the natural immaturity of my childhood years. I gradually move beyond it, into metaphorical adulthood. This is much better.
Most of the path is a series of gentle arcs. These are fairly easy to maneuver, like my comfortable life. But these segments are connected by intermittent sharp turns, mostly 180‐degree switchbacks. I see these as representing significant life changes or challenges, requiring more concentration and skill to negotiate. I notice that I am executing some of these turns mechanically, and some more gracefully. I begin to anticipate upcoming turns, and try to maintain good form around each one.
I can’t see much of the path ahead, nor the end. I spend a significant amount of mental energy dealing with this uncertainty, constantly wanting to know my real‐time ratio of “distance walked” to “distance remaining”. This is a recurring distraction.
Today is Father’s Day, and my Dad is on my mind. He recently completed his well‐walked journey, and is now watching me… even if as mere metaphor… or only as an element of my own (self‐) consciousness. I feel his presence embedded in his absence. I’m aware that it’s not only my turn to walk… it’s my only turn to walk.
I think about my children, grandson, soon‐to‐arrive granddaughter, and their descendants. The familiar succession of life, death and new life seems magical, divinely‐derived, and strangely better than living forever. My role is limited, but critical. I love the part, and embrace it.
I am acutely aware that others are journeying all around me. These are friends of mine. We meet, almost brushing, as we walk. The path seems purposefully narrow, perhaps perfectly so. I suddenly understand that it is impossible to walk this close to others without being affected by them. I affect them too… seen as small adjustments in their position or posture. As we meet, I try not to encroach too much, but making sure not to pull away. I put creative energy into maintaining the perfect degree of separation between our bodies. This feels like more art than science… each friend deserving a customized approach. This closeness seems good to me.
There is a much younger walker behind me, getting ever closer. I’m clearly holding her back. Maybe this means that the younger generation wants me to hurry up and get out of their way. I remind myself not to stretch the symbolism too far… as I pick up my pace.
I now see the end of the path ahead. I have been expecting this part to be emotionally complicated, but it is not. The final section is round… large and unrestrictive… a qualitative change from the narrow linear pathway. The circle opens up to welcome me. It is easy to step into, a perfectly natural thing to do at the end of my walk. Inside the circle, I am centered… comfortable… peaceful… thankful.
16 As Jesus passed alongside the Galilee Sea, he saw two brothers, Simon and Andrew, throwing fishing nets into the sea, for they were fishermen. 17 “Come, follow me,” he said, “and I’ll show you how to fish for people.” 18 Right away, they left their nets and followed him. 19 After going a little farther, he saw James and John, Zebedee’s sons, in their boat repairing the fishing nets. 20 At that very moment he called them. They followed him, leaving their father Zebedee in the boat with the hired workers.
This is a story about 4 fishermen, Simon, Andrew, James and John. It’s a normal morning at the docks. Each one of them is going about business as usual. They arrived at dawn, bundled up in the cool morning air and started work without much conversation. Simon and Andrew are working on one fishing boat and see the Teacher approaching. “Hey. There he is,” says Andrew. Jesus from Nazareth. You can’t go anywhere without hearing about him lately. What’s he doing down here?” They paddle back to shore, not wanting to miss any trouble this Jesus fellow might stir up. Simon and Andrew get the beach and Jesus comes over to talk to them. It’s like he had come there that morning just to find these two guys. Jesus didn’t say much, “Come and follow me.” Jesus invited these 2 fishermen to be his disciples, to follow after him, to walk behind him, tracing his every step.
Further down the beach, the same scene repeats. This time, Jesus walks directly up to James and John who are focused on repairing their fishing net. Jesus says the same thing to them and now all four fishermen walk behind their rabbi with no idea of what’s ahead of them.
It’s a big deal! The four normal guys, working a normal job, on a normal morning, decide to follow Jesus. Maybe you’ve wondered like I have, how is it that Simon, Andrew, James and John do it? How do they drop everything to follow Jesus? What were they thinking? How did they feel?
It’s interesting. The story doesn’t tell us. There’s nothing about how they felt. It doesn’t say they were excited, or moved, or scared, or joyful or resistant. This story about four fisherman gives us only verbs. Jesus passed alongside the Galilee Sea. He saw two brothers. He said, Come, follow. Then, Simon and Andrew left and followed. Jesus saw James and John. Jesus called them. They followed him.
This is a story about four fisherman who decided to follow Jesus.
This is also a story about fishing. I’ve been fishing been fishing three or four times. Once I realized that fishing was primarily a crack of dawn activity, I knew it wasn’t really for me. Jesus uses a kind of puzzling image about fishing. He says, “Come, follow me, and I’ll show you how to fish for people.” I don’t know about you, but this I find this to be very strange. I realized this week why his image is so confusing to me. What do you imagine when someone talks about fishing? What I imagine when I hear the word “fish” or “fishing” is a fishing pole, the rod, reel, bait, tackle box, worms, that kind of fishing. So I’ve always interpreted what Jesus said this way.
I will make you fishers of men, fishers of men, fishers of men. I will make you fishers of men, if you follow me.
The road Jesus invites these four fishermen to follow him on will mean casting a net of love and welcome to people that they do not anticipate. Jesus will cast his net into the sea of a broken world, filled with sinners, people who have messed up, people who are outsiders, who don’t belong. Jesus will stay in the homes of poor, be guilty of associating prostitutes and touching the hands of people with communicable diseases. Jesus will throw his net into the sea and invite everyone in. Jesus will eventually be arrested and executed because those in power decided his fishing net included a few too many of the wrong people. This is a story about fishing.
This isn’t only a story about four fisherman, or only a story about fishing. It’s also, and perhaps, most importantly, a story about God.
If this is only a story about four fisherman who decide to follow Jesus, the pressures on you and me! After all, aren’t we too called to follow Jesus? Called to be his disciples? Wasn’t that the invitation you first heard when you first heard about Jesus? God has called us and we must decide. Jesus wants us all to follow him, to be like him, to walk in his footsteps, to do what he does. Of course this story is about that! And they do it, don’t they? Simon, Andrew, James, John, they do it! They decide and they do follow Jesus, imperfectly at that. Still, it’s a lot of pressure, a lot of responsibility. If life becomes all about what we do for Jesus, something is missing.
If this is only a story about fishing, have some of us failed? Is it too late for us? Some of us might not be the best at fishing, not all the great about casting Jesus’ loving net to our brothers and sisters. His net is sometimes, or maybe more than sometimes, a bit more expansive than we might be comfortable with. He calls us to be like him and fish for people, and yet, sometimes we can barely get the net into the water. Perhaps for others, we aren’t even convinced that Jesus would include us in the net at all, no matter how deep into the water he goes. He can really mean me? Would his net really reach me? There’s still more to the story.
This is a story about God, who God is, how God acts, what God does. Before Andrew, Simon, James and John follow Jesus, Jesus finds them. Before they follow Jesus, Jesus comes to them! They don’t have to go searching, they have been found. Jesus saw. Jesus spoke. Jesus called. Jesus said, “Come.“ We don’t follow Jesus in order to find him, to prove our worthiness with what we do, or even by showing Jesus how big our nets are. We follow Jesus because he first came to us. He came down to the beach to meet these four fishermen. He came specifically for Simon and for Andrew, for James and for John, for you and me.
This blog post was adapted from Pastor Megan’s sermon at Butner Federal Prison on January 25, 2015.
The life of faith consists of seasons. One scholar suggests that we can categorize these seasons of life as seasons of being securely oriented, painfully disoriented, and surprisingly reoriented. These generalizations could apply to our self-acceptance, our relations to significant others, and our participation in public or private life. We might think about these seasons as passages of life, stages of growth, or even identity crises. Acknowledging where we find ourselves in a particular season can allow us to be honest about where we are at in our lives and where we are in relation to God.
The Psalms, a collection of prayers, songs, and poems addressed to God, correspond to these seasons of orientation, disorientation, and reorientation. As we read through the book, we find Psalms where the writer is full of thanksgiving to God, securely oriented in life. We also find Psalms that demonstrate disorientation, perhaps categorized by loss, transition, grief, suffering, or even anger. Finally, some Psalms are written from a perspective of reorientation, wherein the Psalmist transitions from a period of being disoriented to being reoriented in relation to God and others.
The Psalms can become our partner in prayer. Giving us words when we have none, we pray the Psalms joining with all those who have prayed them before us and all who will pray them after we are gone. As we pray the Psalms, we find permission to be utterly honest with God about our feelings and situation, free to speak openly and deeply to God about what we are experiencing. Praying the Psalms also helps us to envision God’s future when we can’t see it ourselves. Lastly, the Psalms guard us against religion or merely thinking about God. Using their words in prayer brings us into direct conversation with the living God, in language we may never have imagined would come from our lips.
-Pray the assigned Psalm from the daily lectionary, with set Scriptures to read each day. Click here to see today’s readings, subscribe to the daily readings by email, or download the app.
-Pray the Psalms using the practice of praying in color. Click here for an excerpt from Sybil MacBeth’s book that gives instructions for praying in color. I have the book available if anyone would like to borrow it. You can read more about praying in color on her website.
-Pray a Psalm, followed by journal writing. Consider these prompts: Where do I find myself in this Psalm? Where do I find my community? How am I being oriented to God in this prayer? What images or metaphors do I find striking? Explore the image more deeply.
-Pray through a list of Psalms, one per day or the same one each day for a week.
-Pray them as a family or with housemates at mealtime or bedtime.
-Pray abbreviated Psalms as breath prayers. A breath Ppayer rhythm is simple: Breathe in slow and deep as you whisper or think on a phrase… Hold your breath… Then exhale.
I will sing to my God as long as I am.
Psalm 8: Lord, our master, how great is your name in all the earth.
Psalm 104: Seek the Lord and his power; seek his face forever. Remember the wonders he has done.
-Pray the Psalms using lectio divina. For instructions on praying lectio divina individually or in groups, click here. There are also instructions for doing lectio divina in color from Sybil MacBeth’s book.
-Pray a Psalm from the category of life within which you find yourself—orientation, disorientation, or reorientation.
Psalms of Orientation: These Psalms reflect a confident belief that the world is well ordered, reliable, and life-giving to the person of faith.
Psalms of Disorientation: These Psalms reflect the brokenness of life, when it is no longer orderly but savage. Spoken out of the depths, they are still bold acts of faith.
Psalms of New Orientation: The pit is not the end of life; there is more. New orientation Psalms reflect the surprise of new possibilities that are experienced as pure gift from God. They are full of thanks.
Citations: The Message of the Psalms and Praying the Psalms by Walter Brueggemann and Getting Involved with God, by Ellen Davis.
Joseph’s story opens in Genesis 37 and it’s a long one. Joseph was one of 11 kids, the youngest son. In Genesis 37, the story says, “Now Jacob (Joseph’s dad) loved Joseph more than any of his other sons because he was born when Jacob was old. Jacob had made for him a long robe. When his brothers saw that their father loved him more than any of his brothers, they hated him and couldn’t even talk nicely to him.” Sibling rivalry, jealousy, family drama—maybe a little too familiar for some of us.
Just when you want to feel bad for Joseph, show him sympathy, “Poor kid—he can’t help that he’s the favorite,” Joseph makes himself quickly unlikeable. When Joseph’s head hits the pillow at night, he has vivid dreams about the future, dreams where he rules over his brothers. In one of these dreams, he’s in the field working with his brothers. They each tie a bundle of grain together…I imagine it like a hay bail. His bail rises up, towering and floating in the air above the others, while each of his ten older brother’s bails of hay, bows down to his bail, as if he’s ruling over them like a king. What’s worse—he didn’t keep his mouth shut about his dreams. Nope. He went ahead and announced them at the dinner table. When I imagine this scene, I’m reminded of the importance of friends. He seriously needed a friend to say, “Dude, listen, you have some dreams where you’re awesome and your brothers treat you like a king. They hate you, man. Keep your dreams to yourself.” Joseph lacked such a friend, so he bragged about his dreams—that combined with his fancy North Face jacket that Daddy bought for him only and the favoritism their dad showed him, brought his brothers to plot about how they might rid themselves of this pesky brat forever.
Joseph’s brothers considered killing Joseph, but they settled on kidnapping him and selling him into slavery instead. That way, they wouldn’t have his death on their foreheads, without having to put up with him anymore. They took Joseph’s fancy coat and destroyed it, making it look like a wild animal killed Joseph. This they showed to their father, so that he would assume that Joseph was dead; their dad would never suspect they had any part in his disappearance.
Meanwhile, Joseph was taken off to Egypt where he worked as a slave. Though he did well there and followed all the rules, he became a victim for a second time, when his master’s wife accused him of a crime he didn’t commit. Over a period of 13 years, Joseph worked as a slave and spent years locked up in prison. After a series of unlikely events, some terrible and some remarkable, Joseph rose to power and became the king’s right hand man, his adviser.
With the king’s blessing and support, Joseph led his country in preparing for a famine, putting food away on reserve during seven years of plenty. When a famine struck the land, Egypt was in a good position, able to lean on the reserved food that Joseph had put away. The surrounding lands, including Joseph’s homeland, had to lean on Egypt for food or else they would starve.
Joseph shows his brother’s enormous generosity. He has them go home, pack up and move their entire family, including their elderly father Jacob to Egypt to be near Joseph. Not long after making the trip to Egypt and being reunited with his father, their father, an elderly man at this point, Jacob dies.
And the final chapter of the story opens. Jacob is dead. Their father is gone. Now what?
Realizing that their father was dead, Joseph’s brothers said, “What if Joseph still bears a grudge against us and pays us back in full for all the wrong that we did to him?” 16 So they approached[b] Joseph, saying, “Your father gave this instruction before he died, 17 ‘Say to Joseph: I beg you, forgive the crime of your brothers and the wrong they did in harming you.’ Now therefore please forgive the crime of the servants of the God of your father.” Joseph wept when they spoke to him. 18 Then his brothers also wept,[c] fell down before him, and said, “We are here as your slaves.” 19 But Joseph said to them, “Do not be afraid! Am I in the place of God? 20 Even though you intended to do harm to me, God intended it for good, in order to preserve a numerous people, as he is doing today. 21 So have no fear; I myself will provide for you and your little ones.” In this way he reassured them, speaking kindly to them.
Fear is a powerful force. Fear motivates and fear paralyzes.
It’s a little funny how they phrase the words of their father. The brothers put a great deal of distance between themselves and Joseph. Instead of saying, “Our father told us to tell you…” they say, “Your father to us to tell you…” They distance themselves from Joseph and from the message that their dad supposedly gave them to pass along.
And then, they do it again. “Please forgive the crime of the servants of the God of your father,” his brothers say. They refer to themselves in third person—“the servants of the God of your father.” It’s not “our crime” that “we committed,” but the crime of these others.
Fear not only keeps them from confession, fear also keeps them from receiving forgiveness. They are scared for their lives the moment their father breathes his last, but haven’t they already been through this conversation with Joseph? At the dinner table, when Joseph revealed his identity to them, he tells them not to worry. “It’s ok. Yeah, it was awful, but look where I am! Look at how God has used me to help save those who would be starving now. I’m even saving you!” Joseph has already offered them forgiveness, but they haven’t fully received it. They haven’t believed what he’s said. Perhaps their views of themselves were so low that they didn’t see themselves worthy of forgiveness. Maybe they’ve carried the guilt for so long about what they’ve done, they fear what life will be like without it. It’s become so much an engrained part of their identity, they don’t know who they are apart from the guilt of what they’ve done. They fear receiving Joseph’s forgiveness. They fear forgiving themselves.
To the plea of the 10 brothers, to this made-up, manipulative, last cry for safety, Joseph has two responses. First, he weeps. His weeping—his display of vulnerability and emotion—causes his brothers to begin to weep also. There they are, 11 grown brothers, weeping on the floor of the house. Why did Joseph begin to weep? The story doesn’t say. Let’s notice, brothers and sisters—the road to releasing fear and offering and receiving forgiveness may not come without weeping.
Fear is a powerful force. Fear is an excellent motivator—moving us to do particular things and act in particular ways. But fear not only motivates, it can also paralyze, cause us to freeze right where we’re at, accept things for how they are. This final chapter begins with the brothers saying to one another, “What if…?” What if Joseph still bears a grudge against us…? Fear finishes the sentence, beginning in the words, “What if…?” Fear finishes the sentence. What if…he still bears a grudge against us? What if…we confess our evil to Joseph and he says that’s the end of us? What if…we ask for forgiveness and he denies it—if I say, will you forgive me, and he says, “no”?
“What if’s” sneak into our minds and hearts.
What if…I never get out of here?
What if I fail as a parent?
What if I don’t belong?
What if no one notices I’m gone?
What if I stand up for what I believe is right and it costs me my reputation?
What if I make a mistake at work and lose my job?
What if I risk opening myself up to someone and get hurt or betrayed again?
What if my body fails me?
What if I can never accept that the past can’t change?
What if I’m not worthy of God’s forgiveness or the forgiveness of those I’ve wronged?
What if I can never forgive myself?
We do not have to live in fear. We do not have to be motivated or paralyzed by it. Look at the God that we serve! Joseph explains how God has been with him. He says to his brothers, “Even though you intended to do harm to me, God intended it for good, in order to preserve a numerous people, as he is doing today.” Does this mean that God wanted or desired Joseph’s brothers to kidnap him, throw him into slavery and ruin his life to avenge their jealousy? No. God doesn’t desire that jealousy and revenge rule our lives. God doesn’t will for us to do evil or to harm other people. Rather, God is able to overcome evil and transform it. God can overcome evil! When Jesus was captured, tried as a criminal and sentenced to death, God overcame death, raising Jesus from the death.
This post was adapted from my sermon preached at Butner Federal Prison on September 14, 2014.
We were gathered at the plaza, right between the giant bull statue and the unattractive fences of a construction site. Luminary bags weighted with rice and lit candles marked the sacred space surrounding 30 of us, one to represent each person who died as a result of domestic violence the previous year in our state. The vigil began as planned, simple, but meaningful, to remember victims of this tragedy and raise awareness about the suffering that takes place behind closed doors. About halfway through the simple service, a woman stumbled into the vigil, interrupting the solemn mood without realizing that a group was gathered and someone was speaking. She stood silent for a few moments, listening to the speaker. When she realized that the speaker was talking about domestic violence, she began to interrupt, asking questions to the speaker, sharing details from her own experience with abuse. “What would you do…what would you do if…?” she cried. Then, as unexpectedly as she joined us and as abruptly as her interruption, she began to weep, uncontrollably crying for the rest of the vigil. A couple of women gathered around her and held her as she wept. Before long, it was my turn to pray. I barely got the words out…I could hardly project my shaking voice over her loud sobs.
“Blessed are those who mourn, for they will be comforted,” Jesus proclaims in the second line of the beatitudes. Blessed are those who mourn. How is this weeping woman, this victim of abuse, blessed? She mourns the injustices she’s experienced, her suffering, the ways her life has been shaped by pain and her inability to free herself from her oppression. Jesus says that this woman and all her sisters and brothers that mourn with her are blessed.
The Jewish culture that Jesus was born into has a rich history of mourning or practicing lament, stretching back hundreds of years before he was born. The prophets and the Psalms include poems, songs, and speeches, recounting the words of people gathered together for public mourning. This mourning wasn’t a kind of crying about having a bad day or because of a frustration at home or work. The mourning Jesus is referencing is the kind of mourning that is a response to injustice and oppression, those who mourn the impact of the powers, both material and spiritual, on the lives of the most vulnerable.
Blessed are those who mourn. Another beatitude and another paradox. Once again, Jesus’ words are outlandish and nonsensical. How is it that those who mourn are blessed? Aren’t those who are happy and fulfilled, aren’t they the ones that are blessed? Yet, in this beatitude, in this paradox, Jesus once again exposes the powers and envisions an alternative. Jesus exposes the powers that cause people to mourn in the first place, those who experience unjust suffering and loss, the same injustices that cause people to be poor in spirit. It’s these people, the mourners, that are blessed, Jesus says. These are the people that Jesus came for. In God’s empire, mourners are not written off or ignored as uncivilized, uneducated, or badly behaved. Instead, in God’s empire, they are the ones who receive God’s comfort and consolation; God’s hears their cries.
Our culture tends to restrict mourning or public displays of emotion to something appropriate for home life or private time. Further, spending time in mourning may be quickly relegated to a waste of time or an inactive posture. The expression, “Don’t just cry about it, do something,” illustrates this clearly. But mourning is not a useless waste of time or an inactive practice. Mourn is a verb. In fact, mourning elicits action and engagement. Mourning exposes the powers, shows their true colors. Seeing people in mourning is disorienting. It interrupts the lives we lead that are detached from suffering and injustice, forcing us to take another look, to pause, to listen, and to join.
The woman who interrupted our solemn vigil for victims of domestic violence exposed the powers with her loud wailing. She made me feel uncomfortable, like I wanted to look away and get away from her as quickly as possible. And yet, her cries made it impossible for me to forget her. The sound of her weeping echoed in my ears for weeks following and if I try, I can still hear them now, over nine months later. Her mourning moves me to engage in seeking justice for others who have suffered like she has.
1. James Howell, The Beatitudes for Today. Louisville: Westminster John Knox Press, 2005., 45.
2. James Howell, The Beatitudes for Today, 46.

Answer: The tomb is empty.


Input: What is the future direction mentioned in the conclusion?

Context:
Paper Info

Title: Is In-hospital Meta-information Useful for Abstractive Discharge Summary Generation?
Publish Date: 10 Mar 2023
Author List: Mamoru Komachi (from Tokyo Metropolitan University), Takashi Okumura (from Kitami Institute of Technology), Hiromasa Horiguchi (from National Hospital Organization), Yuji Matsumoto

Figure

Fig. 1.Example of part of a discharge summary which is a dummy we created.
Fig. 2. Overview of our proposed method.A new feature embedding layer encoding hospital, physician, disease, and length of stay is added to the standard transformer architecture.The figure shows an example of hospital embedding.
Statistics of our data for experiment.
of summarization models with different meta-information.The best results are highlighted in bold.Each score is the average of three models with different seeds.The BS and BR indicate BERTScore and BLEURT, respectively.
Statistics on the number of cases handled by physicians.C/P denotes Cases/Physician, which indicates how many cases an individual physician has.Method of Grouping Physician IDs A most naive method of mapping physician IDs to features is without any grouping process.The data contains 4,846 physicians, so |M | was set to 4,846.However it caused our model's training to be unstable.This might be due to the many physician IDs appearing for the first time in the test time.Table

abstract

During the patient's hospitalization, the physician must record daily observations of the patient and summarize them into a brief document called "discharge summary" when the patient is discharged. Automated generation of discharge summary can greatly relieve the physicians' burden, and has been addressed recently in the research community.
Most previous studies of discharge summary generation using the sequenceto-sequence architecture focus on only inpatient notes for input. However, electric health records (EHR) also have rich structured metadata (e.g., hospital, physician, disease, length of stay, etc.) that might be useful. This paper investigates the effectiveness of medical meta-information for summarization tasks.
We obtain four types of meta-information from the EHR systems and encode each meta-information into a sequence-to-sequence model. Using Japanese EHRs, meta-information encoded models increased ROUGE-1 by up to 4.45 points and BERTScore by 3.77 points over the vanilla Longformer. Also, we found that the encoded meta-information improves the precisions of its related terms in the outputs.
Our results showed the benefit of the use of medical meta-information.

INTRODUCTION

Clinical notes are written daily by physicians from their consults and are used for their own decision-making or coordination of treatment. They contain a large amount of important data for machine learning, such as conditions, laboratory tests, diagnoses, procedures, and treatments. While invaluable to physicians and researchers, the paperwork is burdensome for physicians , .
Discharge summaries, a subset of these, also play a crucial role in patient care, and are used to share information between hospitals and physicians (see an example in Figure ). It is created by the physician as a summary of notes during hospitalization at the time of the patient's discharge, which is known to be very time-consuming.
Researchers have begun to apply automatic summarization techniques to address this problem - . Previous studies used extractive or abstractive summarization methods, but most of them focused on only progress notes for inputs. Properly summarizing an admission of a patient is a quite complex task, and requires various meta-information such as the patient's age, gender, vital signs, laboratory values and background to specific diseases.
Therefore, discharge summary generation needs more medical meta-information, than similar but narrower tasks such as radiology report generation. However, what kind of meta-information is important for summarization has not been investigated, even though it is critical not only for future research on medical summarization but also for the policy of data collection infrastructure.
In this paper, we first reveal the effects of meta-information on neural abstractive summarization on admissions. Our model is based on an encoder-decoder transformer with an additional feature embedding layer in the encoder (Figure ). Hospital, physician, disease, and length of stay are used as meta-information, and each feature is embedded in the vector space.
For experiments, we collect progress notes, discharge summaries and coded information from the electronic health record system, which are managed by a largest multi-hospital organization in Japan. Our main contributions are as follows: • We found that a transformer encoding meta-information generates higher quality summaries than the vanilla one, and clarified the benefit of using meta-information for medical summarization tasks.
• We found that a model encoding disease information can produce proper disease and symptom words following the source. In addition, we found that the model using physician and hospital information can generate symbols that are commonly written in the summary. • We are the first to apply the abstractive summarization method to generate Japanese discharge summaries.
In the studies of summarization of medical documents, it is common to retrieve key information such as disease, examination result, or medication from EHRs - . Other researchs more similar to our study targeted to help physicians get the point of medical documents quickly by generating a few key sentences - .
Studies generating contextualized summaries can be categorized by the type of model inputs and architectures. Some studies produced a whole discharge summary using structured data for input - The sensitivity of the gram stain for bacterial meningitis is about 60%, and the sensitivity of the culture is not high either.
Also, the glucose in the cerebrospinal fluid would have been slightly lower. Although no definitive diagnosis could be made, bacterial meningitis was the most suspicious disease. The causative organism was assumed to be MRSA, and vancomycin and meropenem (meningitis dose) were used to cover a wide range of enteric bacteria.
a whole discharge summary from free-form inpatient records - . The free-form data is more challenging since it is noisier than structured data. In inputting of the free-form data, extractive summarization methods, which extract sentences from the source, are commonly used , - . On the other hands, an encoder-decoder model was used for abstractive summarization , , with a limited number of studies.
The various issues in the abstractive generation of discharge summary would be studied in the future. Studies using medical meta-information have long been conducted on a lot of tasks - . In abstractive summarization on discharge summary,  developed a model incorporating similarity of progress notes and information of the record author.
They presented an idea of integrating meta-information into the abstractive summarization model on medical documents, but did not reveal how meta-information would affect the quality of the summaries. Our method is based on the encoder-decoder transformer model. The transformer model is known for its high performance and has been widely used in recent studies, thus it is suitable for our purpose.
As shown in Figure , the standard input to a transformer's encoder is created by a token sequence T = [t 0 , t 1 , ..., t i ] and position sequence P = [p 0 , p 1 , ..., p i ], where i is the maximum input length. The token and position sequences are converted into token embeddings E T and positional embeddings E P by looking up the vocabulary tables.
The sum of E T and E P is input into the model. In this paper, we attempt to encode meta-information to feature embeddings. We follow the segment embeddings of BERT and the language embeddings of XLM , which provide additional information to the model. It is not a new idea but is suitable for our validation.
Our method is formulated as follows: Let M be feature type, M ∈ {Vanilla, Hospital, Physician, Disease, Length of stay}, since we set five types of features. Feature embeddings E M is created by looking up the feature table where m j is featue value (e.g., pysician ID, disease code, etc.) and |M | is the maximum number of differences in a feature.
In our study, |M | is set to four different values depending on features. Specifically, they are as follows. a) Hospital: As shown in Table , the data includes five hospital records. They were obtained mechanically from the EHR system. b) Physician: Physicians are also managed by IDs in the EHR systems. We hashed the physician IDs into 485 groups containing 10 people each.
Specifically, as a naive strategy, we shuffled and listed the cases within each hospital, and hashed them into groups in the order of appearance of the physician IDs. So each group has the information about the relevance of the hospitals. The reason for employing a grouping strategy is described in Appendix A.
c) Disease: Two types of disease information exist in our EHRs: disease names and disease codes called ICD-10 . We did not use any disease names in the inputs for our experiment. Instead, we encoded diseases with the first three letters of the ICD-10 code, because they represent well the higher level concept.
The initial three letters of the ICD-10 codes are arranged in the order of an alphabetic letter, a digit, and a digit, so there are a total of 2,600 ways to encode a disease. In our data, some ICD-10 codes were missing, although all disease names were systematically obtained from the EHR system. For such cases, we converted the disease names into ICD-10 codes using MeCab with the J-MeDic (MANBYO 201905) dictionary.
Also, diseases can be divided into primary and secondary diseases, but we only deal with the primary diseases. d) Length of stay: The length of stay can be obtained mechanically from the EHR system and the maximum value was set to 1,000 days. We set |M | for vanilla, hospital, physician, disease, and length of stay to 1, 5, 485, 2,600, and 1,000, respectively .
The vanilla embedding is prepared for the baseline in our experiment and to equalize the total number of parameters with the other models. The input to our model is the sum of E T , E P and E M . We also prepare an extra model with all features for our experiments. This takes all four feature embeddings (hospital, physician, disease, and length of stay) added to the encoder.

Datasets and Metrics

We evaluated our proposed method on a subset of data from National Hospital Organization (NHO), the largest multiinstitutional organization in Japan. The statistics of our data are shown in Table  , which includes 24,630 cases collected from five hospitals. Each case includes a discharge summary and progress notes for the days of stay.
The data are randomly split into 22,630, 1,000, and 1,000 for train, validation, and test, respectively. Summarization performances are reported in ROUGE-1, ROUGE-2, ROUGE-L and BERTScore in terms of F1. In addition, we also employed BLEURT , which models human judgment.

Architectures and Hyperparameters

Due to our hardware constraints we need a model that is computationally efficient, so we employed the Longformer instead of the conventional transformer. Longformer can  . In our model, number of layers, window size, dilation, input sequence length, output sequence length, batch size, learning rate and number of warmup steps are 8, 256, 1, 1024, 256, 4, 3e-5 and 1K, respectively.
Other hyperparameters are the same as in the original Longformer, except for the maximum number of epochs is not fixed and the best epoch. It is selected for each training using the validation data based on ROUGE-1. Also, the original Longformer imports pretrained-BART parameters to initial values, but we do not use pre-trained Japanese BART in this study.
We used three GeForce RTX 2080 TI for our experiments. Our vocabulary for preparing input to Longformer is taken from UTH-BERT , which is pre-trained on the Japanese clinical records. Since the vocabulary of UTH-BERT is trained by WordPiece , we also tokenize our data with WordPiece. However, the vocabulary does not include white space and line breaks, which cannot be handled, so we add those two tokens to the vocabulary, resulting in a total size of 25,002.
The vocabulary has all tokens in full characters, so we normalized full-wdith characters by converting all alphanumeric and symbolic characters to half-width for byte fallback. , we found that all the models with encoded medical meta-information perform better in ROUGE-1, ROUGE-L and BLEURT than the vanilla Longformer.
However, in BERTScore, only hospital and disease models outperform the vanilla. Specifically, disease information is most effective, improving ROUGE-1, ROUGE-2, ROUGE-L, BERTScore and BLEURT by 4.45, 0.73, 3.12, 3.77 and 0.21 points over the vanilla model, respectively. This seems to be because disease information and the ICD-10 ontology efficiently cluster groups with similar representations.
In contrast, in ROUGE-2 and ROUGE-L, the model with physician embedding is inferior to the vanilla model. This seems to be a negative effect of grouping physicians without any consideration of their relevance. It would be better to cluster them by department, physician attributes, similarity of progress notes, etc. Regarding low ROUGE-2 scores in all models, a previous study using the English data set also reported a low ROUGE-2 score of about 5%, which may indicate an inherent difficulty in discharge summary generation.
In BERTScore, the models with the physician and the length of stay did not reach the performance of the vanilla model, suggesting that the system's outputs are semantically inferior. The model with all features performed the lowest of all models in BERTScore. The reason for the low score of the model with all features seems to be that its number of parameters in feature embedding was four times larger than that of the model with the individual feature, and the amount of training data was insufficient.
In BLEURT, all models with meta-information outperform vanilla, which suggests that they are more natural to humans. To analyze the influence of encoded meta-information on the outputs, we evaluate the precisions of the generated text. Specifically, we measure the probability that the generated words are included in the gold summary to investigate if the proper words are generated.
Some previous studies on faithfulness, which also analyze the output of summarization, have employed words or entities - . In this study, we focused on words, not entities, because we wanted to visualize expressions that are not only nouns. The words were segmented by MeCab with the J-MeDic. For each segmented word, the numeral and symbol labels were assigned as parts of speech by MeCab, the morphological analyzer, while the disease and symptom were assigned by the J-Medic dictionary.
The results, shown in Figure , indicate that the encoded disease information leads to generate more proper disease and symptom words. This indicates that the meta-information successfully learns disease-related expressions. The encoded hospital or physician information also improved the precision of symbols generation.
This suggests that different hospitals and physicians have different description habits (e.g., bullet points such as "•", "*" and "-", punctuation such as "。" and ".", etc.), which can be grouped by meta-information. In this paper, we conducted a discharge summary generation experiment by adding four types of information to Longformer and verified the impact of the meta-information.
The results showed that all four types of information exceeded the performance of the vanilla Longformer model, with the highest performance achieved by encoding disease information. We found that meta-information is useful for abstractive summarization on discharge summaries. Our limitations are that we used Japanese EHR, the limited number of tested features and not performing human evaluations.
As for the efficacy of the meta-information, we believe that our results are applicable to non-Japanese, but it is left as Fig. . The precisions of words in the generated summaries. The vertical axis shows the probability that the words exist in the gold summary. a future work. Other meta-information may be worth verifying such as the patient's gender, age, race, religion and used EHR system, etc.
It is hard to collect a large amount of medical information and process it into meta-information, so we may need to develop a robust and flexible research infrastructure to conduct a more large scale cross-sectional study in the future. In the discharge summary generation task, which demands a high level of expertise, the human evaluation requires a lot of physicians' efforts and it is a very high cost which is unrealistic.
This is a general issue in tasks dealing with medical documents, and this study also could not perform human evaluations. On this research, informed consent and patient privacy are ensured in the following manner. Notices about their policy and the EHR data usage are posted at the hospitals. The patients who disagree with the policies can request opt-out and are excluded from the archive.
In case of minors and their parents, followed the same manner. In the case of minors and their parents are same. To conduct a research on the archive, researchers must submit their research proposals to the institutional review board. After the proposal is approved, the data is anonymized to build a dataset for analysis.
The data is accessible only in a secured room at the NHO headquarters, and only statistics are brought out of the secured room, for protection of patients' privacy. In the present research, the analysis was conducted under the IRB approval (IRB Approval No.: Wako3 2019-22) of the Institute of Physical and Chemical Research (RIKEN), Japan, which has a collaboration agreement with the National Hospital Organization.
This data is not publicly available due to privacy restrictions. shows the detailed number of cases handled by physicians. In all hospitals, there is a large difference between the median and the maximum of cases/physician. This indicates that a few physicians handle a large number of cases and many physicians handle fewer cases.
It is impossible to avoid physician IDs first seen at test time without some process that averages the number of cases a physician holds.

Answer: Verifying other meta-information such as patient's gender, age, race, etc.


Input: What types of data did the authors use in their experiments?

Context:
\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 immersive simulation techniques, such as virtual reality rendering and dynamic scene generation methods, to adaptive filtering.

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. 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 dynamic state vector ${f w}^{o}$ representing virtual reality scenarios with 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)}.}
MSD (dB) &-28.45 &-21.07 &-14.36 &-26.90 &-28.36 &-25.97\
In a second experiment, we test the tracking capabilities of the proposed algorithm with data generated from a simulated virtual reality environment designed to mimic realistic scenarios. 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}

Answer: data from a simulated virtual reality environment