Bayesian evaluation for the likelihood of Christ's resurrection (edit 129)

I'm continuing to work on editing Bayesian evaluation for the likelihood of Christ's resurrection. I've edited things down to near the end of chapter 4.

The plan here is to go once-over the whole thing, then add in one more major double check, probably about extreme value distributions. Then I can call it "done" for the time being, and move on to other projects.

You may next want to read:
The Gospel: the central message of Christianity
Interpreting the Genesis creation story
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (edit 128)

I'm continuing to work on editing Bayesian evaluation for the likelihood of Christ's resurrection - coming back to it after an extended period working on other things.

For this week, I've only made minor changes, after reading it with fresh eyes due to the long break. I think I should be able to wrap this up fairly soon.

You may next want to read:

More unoriginal thoughts about the election

My kingdom is not of this world. (John 18:36)

Our citizenship is in heaven. (Phil.3:20)

We must obey God rather than men. (Acts 5:29)

It is better to take refuge in the LORD than to trust in man. (Ps. 118:8)

Do not put your trust in princes; in human beings, who cannot save. (Ps. 146:3)

God turns the king's heart wherever he will. (Prov. 21:1)

He brings princes to naught, and reduces the rulers of this world to nothing. (Isa. 40:23)

The nations are like a drop in a bucket. They are regarded as dust on the scales. (Isa. 40:15)

But blessed are those whose help is the God of Jacob, whose hope is in the LORD. (Ps. 146:5)

To the King eternal, immortal, invisible, the only God, be honor and glory for ever and ever. (1Tim. 1:17)

May his kingdom come, and his will be done, on earth as it is in heaven. (Matt 6:10)

You may next want to read:

Fuller house: a short review

Watching "Fuller House" has been one of the stranger things that's happened to me recently. It's affected me enough for me to write this review.

I can't necessarily say that this this was a "good" show in the standard sense. I don't know that I would recommend it generally. There's plenty of other reviews of this show out there. Much of the negative things they say about this show is true. 

And yet, the show's positive qualities are rare gems in today's media landscape. For instance, how many family-centric shows are out there now? The 80's and 90's were full of them - from the original "Full House", to "Growing Pains", "The Fresh Prince of Bel-Air", and even "Married with Children". I'm sure such shows are still around, but I don't know any of them. None have achieved the success of shows like "Game of Thrones" or "Rick and Morty". Back then, as a recent immigrant, shows like "Full House" had an instrumental role in my integration into the United States.

And such personal notes, of how "Fuller House" relates back to my own life, has been where it's been most impactful. DJ and Kimmy are my age - I kind of keep track of things like that. And seeing how these characters and their actors have changed with time tells a story of its own. Twenty years of life has happened to them, with its tragedies and triumphs. These mirror my own in some ways, and I expect that to be true for any viewer like me.

Of course, the culture at large has also changed - and this, too, comes though the show. It's especially salient in comparison to the original. It was natural and inevitable that this show was going to be much more female and adult-centric, given who the main characters are. The portray of family and childrearing is quite different too. But the show's heart lies in its invariants: The show is still about these women - both the characters and their actresses - and how they go though entire decades of personal and societal change, from their childhood into their middle age. In the end, they somehow manage to find family, fulfillment, and love. It is perhaps an expression of what can best be described as Grace.

Ultimately, the aim of art is to reflect truth, and "Fuller House" does so in an utterly unique way. Its truth echos back to us through time and history. It's encoded into the changing cultures and issues discussed in and around the show. And it's etched into reality itself, through the lives of its incredibly human and relatable actors. 

But for all I might say, perhaps there's no better expression of this truth than Kimmy's actress Andrea Barber's own comment, on her "Fuller House Final Bow" video:

    "Thank you for growing up with us, for letting your kids grow up with us, and for being part of our            Fuller family. Forever grateful for your love and support."

You may next want to read:

Presenting my argument for the resurrection to the McLean Bible Church

Last week, I presented my argument for Jesus's resurrection, in an hour-and-a-half talk given to MCB Apologetics - a ministry of the McLean Bible Church. And it went really, really well!

The organizer (who's a high school friend of mine) told me that he thought the talk went incredibly well, that it was really fantastic. Many of the attendees too, in very high proportions, said that the talk was great, fabulous, really interesting, and enjoyable!

Thank you to my friend the organizer, and to the MCB Apologetics ministry, for this opportunity to present my work!

I think there'll be a video of it posted some time later. Meanwhile, here's the slide deck for the presentation, and the full, book-length blog post that it's summarizing.

You may next want to read:
The Gospel: the central message of Christianity
Another post, from the table of contents

The Gospel: the central message of Christianity

Image: By Danielclauzier, on Wikipedia commons
There is only one story in the universe.

Occasionally I'll find a badly told story, which still attempts to convey the message of Christianity. I've said of such works that "you can't blame it for not being the universe" - for that is what is required to tell the story well. God is the original storyteller, and he uses every part of the entire universe to tell us this one story, to send us his one message. All other stories are but retellings of small snippets of this very long and very great story.

We call this story the Gospel: the one story in the universe, and the central message of Christianity.

I will attempt to explain this Gospel in this post. I apologize beforehand for it not being the universe.

The Gospel is simple, as it must be if it's the one message that God has for us. I can summarize it in three words: Christ saves sinners. The Gospel is also complicated, as it must be if it's a story told with the entire universe. Fighting, torture, escapes, true love, and miracles all have a part in it. So let me begin by expanding that three word summary ("Christ saves sinners"), starting with the easiest word to understand: "sinners".

By "sinners", I mean people like me. Immortal souls bound in ephemeral constructs of flesh and blood, aware of their physicality yet knowing that they're not just a lump of matter. In our bodies we are bound to the circles of this world, yet we have an inkling of something beyond it, of something more. For we were gifted by the One Father of All - the source of all goodness - with his breath of life. We were made in his image as his children, intended to grow up to become like him. But we - that is to say, I - have abused this very gift, and have deliberately chosen to defy God and disobey his divine laws. That is sin. Its consequence is death.

Why is the consequence so serious? What harm could one small imperfection cause? Catastrophe. In an otherwise perfect system designed for infinite growth, every flaw is fatal, especially in a critical subsystem like morality. God, in his love for us, created us with unlimited potential - to become like God. His laws were decreed to help us develop into that destiny. Sin is the rejection of all that: it's violating his laws, discarding that divine destiny, and becoming alienated from God himself. And because of our unlimited potential, sin also grows without limit. For a rocket with infinite thrust, any error in its heading sends it infinitely far off course. In a boundless forest, a small fire causes an everlasting conflagration. In an organism that grows forever, cancer results in infinitely large tumors. In humans, sin causes endlessly more sin, which is endless separation from God.

We would not have this problem if we were just rocks, or cows, or even space-faring aliens who are merely fated to rule the stars then perish with the universe. But we are humans, created to be like God. We were the chosen ones. We were supposed to be the pinnacle of creation, not fallen sinners. But precisely because of that greatness, when we fall we fall deep into darkness, deeper into sin, leading to death. Like a debtor who borrows to pay the interest on his debt, like an alcoholic who drinks to deal with his alcoholism, we keep sinning, even in our attempt to do good. Sin begets sin.

Without intervention, we are thereby forever separated from God and therefore from all goodness. Sin infects us and turns us evil. It alienates us from God and makes us into his enemies, the objects of his rightful wrath. But in our folly we have attempted to convince ourselves that it's not that bad, that we're good enough, that we're not really sinners. But this attitude is actually the very symptom of our sinfulness. Sin makes us numb and blinds us to further sin. Consider the people you know. Judge them, if you think yourself qualified. Are not the worst kinds of people convinced that they're good? God is the standard of goodness, and one of the first consequences of falling away from him is that we can no longer rightly discern what is good. Oblivious to our own evil, we blindly stumble from sin into more sin.

We - that is, I - thus find myself in this pit of despair: estranged from God, and therefore also from all good things that might possibly enable me to return to him. Fixed to a doomed trajectory that I'm powerless to change. Infected in my moral core with sin's ever growing evil. Even my attempts to be good are tainted with sin, and my righteous deeds are themselves like filthy rags. Who shall save me from this wretched body of death?

We thus reach "save", the second of the three words that summarize the Gospel ("Christ saves sinners"). What can we do to be saved from our sins? Sin is alienation from God, so our sin distances us from goodness itself. Sin is wickedness, so our evil makes us objects of God's rightful wrath. How shall we save ourselves? What righteousness, what achievement, what works of good deeds can we offer God, to convince him to take us back? Or what discipline or self-improvement can we undergo to empower ourselves and work our way back to him? We have none of these things, because God is the source of all such goodness, and the very nature of our predicament is that we have strayed from him. There is nothing we can do. Any good we think we have is his to begin with. Even the first desire to repent would need to come from him, for we don't have it in ourselves.

So there is no hope in any kind of equivalent exchange. We have nothing good apart from God: nothing to give, nothing to exchange, nothing to bribe God with, and nothing to improve ourselves with. This actually points the way to the solution: we need a nonequivalent exchange - no; an unmerited, outright gift - in order to be saved. We need a transcendent class of help that does for us what we could never do for ourselves - like lead being transmuted into gold, a wooden puppet becoming a person, a fictional character becoming real, or the dead coming back to life.

If God provides such help, it would not be because we deserved it, but because he simply loves us. Not because we are good, but to enable us to do good. Not because we're worth it, but to make us worthy. We could do nothing to contribute to our salvation, but only accept it, trusting in the help God provides. Trust, or faith, would therefore be the mechanism of this salvation in us. Note that faith, of itself, will not save you: a strong faith in a faulty climbing rope will kill you, whereas a weak faith in a good rope will keep you safe. It's the object of faith that's all important: thus we must place ourselves in God's hands. So faith is merely that act of trusting God - an expression of our understanding that it is God alone who does all the saving, including giving us that very faith to accept his salvation.

Furthermore, if God would provide such help, it would not merely be a one-time course correction. Rather it will take us all the way from our creation to our destination. Otherwise we could simply sin again. Don't think that God first created us as "plan A", then saw us going wrong and rescued us as "plan B". God foresaw our fall before he created us, and his plan to make us his children has always been to save us from our sins, once and for all time. This plan includes everything: his breath of life that made us alive, his own image in which he made us, and his foreknowledge and predestination for us before all time. In addition, he rescues us from our sins, restores our broken relationship with him, sanctifies us to make us holy, gifts us his own righteousness, and keeps us from falling, until he finally fulfills in us our destiny - to become like him as his grown children. This plan is everything good that God has for us, which is everything good, period. This is not "plan B", it is the complete fulfillment of God's eternal plan. This is our salvation.

He who began this good work in us will surely bring about its completion. We therefore no longer speak of what God would do, but of what God has been doing since before the foundation of the world, and will continue to do until our faith is made perfect. But what gives us this confidence? And how will God accomplish all this? For that matter, what about other questions like when, where, why, and who? All these questions have their answer in a single sentence: for God so loved the world, that he gave his one and only Son (Jesus Christ), that whoever believes in him shall not perish but have eternal life.

That finally brings us to "Christ", the first and the most important word in my simple summary of the Gospel ("Christ saves sinners"). If I had to condense those three words down to one - just one word to describe the Gospel and therefore all of existence -  it would be "Christ". Jesus Christ is God himself incarnated as a man. In God's act of true love for us, Christ came - God came as a man - to fulfill the plan for our salvation. For what power does anyone else have to stop the course of sin? To save us? To reach us, he humbled himself down to our level, and took on the human form that he first granted us. Like us, he was conceived, born, and raised, and became a man familiar with our sorrow, who experienced our pain. Despite being fully human, he remained morally perfect, serving as our perfect example and enabling the next key part of the plan - his crucifixion and resurrection.

I do not understand Jesus' death on the cross. There are theories of how it worked, but I doubt we have anything close to the full picture. This is only expected: the cross is nothing less than the intersection of all of existence. Everything in heaven and earth, visible and invisible, life and death, good and evil, sin and righteousness, God and his creation, story and Author - they all collide here. I think that a complete understanding of Christ's death and resurrection would require nothing short of the entirety of the mind of God. My telling of the story is utterly insufficient for it - nevertheless I will proceed.

Through Christ's great love for us, he became one with us. As he became part of humanity, we became parts of Christ's body. We see this happening to a small degree between humans: people become one, to the degree that they love one another. Thus an individual's loss or gain, their grief or joy, their ignorance or knowledge, and even their guilt or righteousness, are all suffered or enjoyed by their family and friends, to the degree that they've become one in love. But Jesus is God, and God is love. In his perfect love for us he became perfectly one with us, as only God himself, as only love itself can.

In being united with us, he took upon himself all our sins and its consequences, and was crucified. Because of his perfect love for us, the transfer of our sins is also perfect: we truly bear them no more, as if we had never sinned. Conversely, Christ truly carried our sins, and became truly sinful for us. There on the cross, he suffered all of sin's consequences. He was pierced for our transgressions, and crushed for our iniquities. The infinite separation from God that was our due was laid upon him, and he cried out, "My God, my God, why have you forsaken me?" God himself had never before experienced this separation - this sundering of his own being - because the Trinity had existed without sin in perfect fellowship. No human has ever experienced it, because Jesus took it upon himself instead. The fullness of Christ's passion is therefore incomprehensible to us and incomparable to any of our experiences.

The cross also demonstrates God's love for us, giving proof of that love in sacrifice. The punishment that brought us peace was upon him, and by his wounds we are healed. That's how we know he loves us. And because he first loved us, we can love him back. That's what allows us to believe in him and participate in his plan of salvation. In loving him, we thereby become one with Christ just as he became one with us, thus participating in his suffering and death. Like the sin transfer, our perfect oneness makes this participation perfect: we are truly crucified with Christ and truly die with him. Some think that becoming a Christian is trivially easy, that one can simply plan to "repent" after a lifetime of sin then "get into heaven". In reality the closest analog to becoming a Christian is dying: you must take up the instrument of your execution and follow Jesus to his death.

Thus at Calvary I find myself identifying with nearly every party in the passion story. I am the mockers, deriding Jesus in my disbelief. I am the women, devastated that my teacher should suffer such agony. I am the centurion, saying "Truly this was the Son of God". I am Pilate, waffling between my need for justification and my cowardice. I am the disciples, overcome with fear and unable to understand. I am the thief on the cross, asking to be remembered. I am chanting "crucify! Crucify!", for this Jesus is a sinner, full of my sins, and he must be crucified. And I am with Jesus - crucified, dead, and buried, as punishment for my sins.

So we must be crucified and die. This is difficult - impossibly so. How could we do any of this? Wasn't the whole point of all this that it was supposed to be easy? How could we share in Christ's suffering and death, and not be defeated by it? Would we not be crushed by the penalty for our own sins, the very same penalty that we could never bear ourselves?

But what is impossible for man is possible with God. We are not on our own: we are united with Christ in love. He first loved us, he bears the burden for us, he enables us to love him, he enables us to have faith, he makes us one with him, he does everything. What can separate us from the love of Christ? We will not be crushed or defeated while he still stands. If God is for us, who can be against us? In Christ, we are crucified, dead, and buried for our sins - and yet we live.

The resurrection is the proof of Jesus's perfect victory over sin and death. If Christ was not raised, that would mean Jesus failed in bearing our sins, and they would have then crushed us in turn. God himself would have been defeated, our faith would have been falsely placed, and we would be pitiable above all humanity. But by his resurrection from the dead, Jesus demonstrates that death has no hold on him, that his power exceeds the power of sin, that both sin and death have been ultimately nullified.

The same power that raised Christ from the dead now works in us, for we have been made one with Christ by loving and believing in him. His victory is our victory. If we died with him, we will also live with him - yet not us, but Christ who lives in us. Therefore, we who are in Christ are a new creation - dead to our old selves and made new to be like Christ. Death and sin have no power over us, no more than they have any power over Jesus. The resurrection therefore demonstrates that this whole plan of salvation worked.

Behold furthermore what lavish love God has given to us in Christ: we are not only freed from sin, but made the children of God! We are given Jesus's righteousness, adopted by God, and made co-heirs to God's glory, all on account of being resurrected and made new in Christ. Jesus is the Son of God, so we also become the children of God. We are in Christ, and Christ is one with God. This, finally, is the fulfillment of our destiny, the plan for which God first created us.

So, this is my story: Christ saves sinners. It is a small telling of the one story, the only story in all of existence. Although we have a royal heritage, sin has long kept us from it, in darkness. But when Christ comes to us, we at last see the light - that by loving, believing, and thereby becoming one with him, we can fully become a part of God's family.

I hope that you, too, will tell this one story as your story. The promise of salvation is for absolutely anyone who believes in the Lord Jesus Christ. Anyone can thereby have life in his name.

You may next want to read:
How physics fits within Christianity (part 2)
Bayesian evaluation for the likelihood of Christ's resurrection
Another post, from the table of contents

A book review: The Generations of Heaven and Earth

This is a review for The Generations of Heaven and Earth, by Jon Garvey.

This was a much harder read than I anticipated - in a good way.

Like the title suggests, this book is about the Genesis creation account - specifically about Adam and Eve. New science has shown that a couple doesn't need to be the first humans ever to be the ancestors to all humans alive today. If they had lived just a few thousand years ago, they could easily be the ancestors of us all, simply by having their descendants intermarry with everyone else. The idea of the genealogical Adam and Eve (GAE) is that the biblical Adam and Eve were just such a couple.

This is a revolutionary new idea. It completely reshapes the discussion on interpreting the Genesis creation account. It compromises nothing on any front. It is, in my opinion, the most scientific AND the most biblical interpretation of Genesis. There is another book - "The Genealogical Adam and Eve" - which approaches this idea from the scientific side. This book approaches it from the theological side.

Now, I am in the rather unique position of having arrived at this idea independently. I've written on it extensively in another post. So when I started reading this book, I figured that I'd breeze through it, as I was already aware of the main idea and its theological implications.

I was partly right. Certainly, much of Jon Garvey's position as expounded in this book are very similar to mine. Some issues, like the question of Cain's wife, have obvious answers in the GAE model. Huge chunks of difficult passages fall into place, and just these low-hanging fruits can get us very far in better understanding Genesis and some of its key doctrines. It was a pleasure to read these sections where our thoughts lined up: it's a good sign when multiple people independently arrive at the same idea, and work out the same implications from it. Truly, GAE is an idea whose time has come.

But Jon Garvey goes far above and beyond that - and explores many topics related to GAE from many different angles. Some of these were very new to me: that's what made this a much harder read than I anticipated, in a good way. I learned a lot. In this book he covers topics as varied as the possible locations for Eden, comparative ANE mythology, primeval 'monotheism' all over the world, theories of salvation, the meta-narrative of the Bible, the all-important "image of God", and more. Some of these topics are rather speculative. Others are tangential. But they're all interesting, and the author carefully navigates through each of them, connecting them all back to the GAE model.

In so doing, this book shows that the GAE model is not just plausible or defensible from a theological standpoint, but actively helpful. Reading this book, and navigating through its many topics, has sparked many new insights for me - in theology, but also in its implications in the real world, in my personal life. These real-world associations for me include things as disparate as my career progression, interpretations of Frozen II, and Jordan Peterson's talking points. Truly, this book covers a wide range of topics and connects them all.

So, this book has further illuminated the ideas of GAE to me, even though I was already quite familiar with it. It did so in a way that actively enriched my theological understanding, which then carried over into topics in my personal life. In short, it has made my faith more whole - and there is no better class of recommendation I can give than to say that.

You may next want to read:
Interpreting the Genesis creation story
Another post, from the table of contents

Many places can reopen now, but "how" matters more than "when"

Different parts of the country are now starting to open back up after the coronavirus lockdowns - and this is drawing a lot of different reactions. Some are protesting for more reopening and greater freedoms, while others are warning that opening too soon will trigger an exponential flare-up and cause thousands of more deaths. Throw in the political polarization in our society, and it seems that people are being driven to a simplistic, binary yes/no position on reopening the country.

The reality is, of course, more complex and subtle. There are two major corrections we must make to a binary understanding. The first is that the United States is a large country, and local conditions vary greatly - and it's the local conditions that will dictate when a particular city or county can reopen. Many places are safe enough now, at least when we look at just the infection numbers. Other places still have a severe infection, and they need to get their numbers down lower. The decision of "when" will need to be made state by state, county by county, and city by city. There is no one-size-fits-all solution. This doesn't mean that I think every locality is doing the right thing: as far as I can tell from just the numbers, Georgia is being a little reckless in reopening too early, and the San Francisco Bay Area is being too cautious in extending its shelter-in-place order until the end of May.

But these concerns pale in comparison to the second, and by far the more important, correction to the reopening question: HOW we reopen matters far more than WHEN. Remember, we need to keep R0 under 1. We need enough social distancing, personal protection, disinfecting, and other such measures, so that each infected person causes less than one additional infection. That's the "how". The question of "when" almost doesn't matter in comparison. If we really get the "how" right, much of the country can open now, or very soon. If we get it really wrong, then it won't matter how long we wait - we'll never be able to safely lift the lockdowns. That's why we must reopen very carefully and deliberately. This is the trickiest part of the whole plan, the part where the "how" is most likely to go wrong.

But more on the "how" later - for now, let's focus back on "when", and look at the data from the different locations.

This is a graph of deaths and cases for my home state of California. You see that the numbers have been growing, although they may have plateaued starting around late April. All the graphs in this post use the latest data as of May 4th.

The numbers in the title are each trailing 7-day averages of their per day values. So there have been 1636 new cases and 72.1 deaths per day on average, over the last 7 days.

The most important of these numbers is the "deaths per 1M: 1.8". This says that there have been 1.8 deaths per day per million people in the population, when averaged over the last 7 days. This is the number that tells you the level of risk for the average person in the state of California.

One important point of reference for this number is 0.3 deaths per day per 1M - the risk of dying in a car accident. This is a level of risk that we're quite comfortable accepting. We won't shut down the economy to prevent people from dying in car accidents, so we should accept a similar level of risk from the coronavirus.

In fact, we can actually take on a bit more risk, because we know that the virus mostly kills the elderly, or those with pre-existing conditions. So if you're young and healthy, your risk of death drops by an order of magnitude or more - meaning that you would be able to tolerate numbers up to 3 deaths per day per 1M. With this in mind, I would say that any given locality should be safe to start reopening (slowly, carefully) if they're at or below 1 deaths per day per 1M. Up to 3 deaths per day per 1M may be acceptable, but places with values above that should seriously consider staying locked down until the numbers come down further.

In summary, if the deaths per day per 1M is:
Under 1: Safe to reopen
Between 1 and 3: Be cautious
Greater than 3: Should probably stay in lockdown

This is just a rule of thumb from some simple calculations: local leaders and experts will have more context and data, and you should defer to any such extra information they provide. Furthermore, this doesn't take into account how easy it would be to transmit the disease - even if you're young and healthy and the economy reopens, you should still try not to catch it, because you may pass it on to someone who's more at risk. But this has to do with the "how" of reopening - which, as I said, is far more important, and which I'll discuss later.

Looking at the graph for California, I'd be hesitant about opening up the whole state. The numbers are not yet on a clear downward trend, and the 1.8 deaths per day per 1M people is still a bit too high. But we can do better than looking at the whole state: when we drill down to the county level for the San Francisco Bay Area, we see this:

We see that many counties are below 1 death per day per 1M, with a few exceptions. The trends, too, all seem to be flat or decreasing. There are a few worrisome counties, but I would say that the Bay Area as a whole is a good candidate for reopening - although I wouldn't mind waiting another week or two.

What about some other counties in California? Let's look at the Sacramento and Los Angeles counties:

Sacramento is basically fine. With only 0.2 deaths per day per 1M, the coronavirus poses little danger in my state's capital. The trend, too, seems to be distinctly downwards. There was a protest there a few days ago calling for a reopening, and I'm inclined to be quite sympathetic towards them, especially if the reopening is just for the city of Sacramento and not for the whole state.

Los Angeles, on the other hand, should not reopen yet. 4.5 deaths per day per 1M is probably too high of a death rate, and the trend may still be increasing. In fact, we see that Los Angeles is basically responsible for much of the numbers for California as a whole.

We can run similar kinds of analysis for various states as well. Of course, New York and a bunch of other states in the northeast should not reopen yet:

Their deaths per day per 1M are still horrendously bad, and some of their trends are still upwards.

On the other hand, there are also states which are nearly unaffected by the virus:

These states are generally safe and they should be able to reopen without putting their people at too bad of a risk, as long as they get the "how" right.

Here are the graphs all 50 states, and their deaths per day per 1M value:

But like we saw with California, all states should really be looked at on a county-by-county, city-by-city level. Furthermore, while this is an important number, it still remains only one of the many factors which enter into when your state can reopen.

So that's the question of "when". Many states are quite safe and can reopen now, and many counties or cities in more iffy states are also safe. There are also areas that are still quite dangerous, which need to stay in lockdown. So the question of "when" needs to be approached with locally. There isn't a binary yes/no answer for the whole country.

But, remember, "how" matters far more than "when". Nothing I said about what areas are safe matter at all, if we get the "how" wrong. And the answer to "how" is simpler: we need to keep R0 under 1. This means doing all the things that you already know about: wash your hands. Don't touch your face. Keep 6 feet of distance from others. Stay home if you're sick. Wear a face mask.

Also consider what additional things you and your communities can do: face shields, in addition to face masks, probably help. You can work from home whenever possible, and you should avoid crowds. A temperature check can be required before entering a building or meeting someone. Hand sanitizers can be placed near commonly used surfaces, like door handles or elevator buttons.

As we progress further we will hear more about other measures that we can take: for instance, it seems that sunlight, heat, and humidity can all help us further fight the virus. If so, then these can all be incorporated into things you can do personally to fight the pandemic.

Some other "how" measures require more of a top-down approach from higher up, like testing (which we still need more of) and contact tracing. But much of the "how" question can be answered at an individual level, with personal responsibility. You are personally responsible for not catching the virus, and for not passing it on if you do.

And that's how we'll beat this thing. Especially as we reopen the economy, remember that the "how" matters more than "when".

You may next want to read:
Quick takes on the plan to re-open the country
Coronavirus endgame: how we get back to normal
Another post, from the table of contents

Re-analyzing the Stanford COVID-19 antibody study

Re-analyzing the Stanford covid antibody study
Introduction and results

This is a re-analysis of Stanford's antibody study in Santa Clara County, which reported a population prevalence of 2.5% to 4.2% for COVID-19 antibodies, and a corresponding infection fatality rate of 0.12% to 0.2%. This result, if true, would have huge implications, as the lower fatality rate would dramatically change the calculus on important policy decisions, like when and how we should reopen the economy. However, this study has also received numerous criticisms, most notably for the results being inconsistent with the false positive rate of the antibody test.

Here, I attempt to derive what the results ought to have been, under a better methodology. I will be using a Bayesian approach, employing beta distributions to model probabilities.

The results I get at the end are as follows:
Antibody prevalence in study participants: 
1.0% (95% CI: 0.16% to 1.7%)

Antibody prevalence in Santa Clara 
(speculative, due to missing data):
1.9% (95% CI: 0.3% to 3.2%)

Infection fatality rate implied from above:
0.27% (95% CI: 0.17% to 1.6%)
This fatality rate is quite uncertain on its own, but it is in broad agreement with other similar kinds of studies.
Methodology and code

Alright, let's begin. First, let's import some packages:
In [1]:
import numpy as np
import pandas as pd
from scipy.stats import beta
%matplotlib inline
Next, let's decide on a prior for the prevalence of antibodies in the study:
In [2]:
p_prior = beta(1.1, 60)
x = np.linspace(0, 0.2, 1000)
pd.Series(p_prior.pdf(x), index=x).plot()
<matplotlib.axes._subplots.AxesSubplot at 0x1f908dd0c88>
In [3]:
p_prior.mean(), p_prior.median(), p_prior.interval(0.95)
 (0.0006173369513848543, 0.06281995654253102))
Note that this prior is quite favorable to the results of the study. It piles on the prior probability right on top of where the study's results turned out to be (1.5%), with the mean and the median of the distribution falling right into the interval cited by the study (1.1-2%). So we are making an assumption a priori that the results of the study are correct. Of course, in a good Bayesian analysis this doesn't matter much in the end, as the prior should get overwhelmed by the evidence from the data.

Next, let's model the specificity and sensitivity of the antibody test they used. Pooling together the numbers they provided in the paper, we get:
In [4]:
sensitivity = beta(78 + 25 + 0.5, 7 + 12 + 0.5)
specificity = beta(30 + 369 + 0.5, 0 + 2 + 0.5)
We next run a simulation using random samples from these beta distributions, then only look at the results which actually return the empirical results of the study. This satisfies Bayes rule, which says that the probability values which best predict the outcome should be favored. It's consistent with the adage that “when you have eliminated the impossible, whatever remains, however improbable, must be the truth”. It undergirds Bayesian hierarchical modeling, and it's the same methodology I used in my argument for the resurrection of Jesus Christ.

The data from the study says that there were 50 positive cases out of 3330.
In [5]:
n_sim = 1000000
total_cases = 3330
positive_cases = 50

df = pd.DataFrame()
df["p_population"] = p_prior.rvs(n_sim)
df["sensitivity"] = sensitivity.rvs(n_sim)
df["specificity"] = specificity.rvs(n_sim)
df["detected_positives"] = (
    # true positives
    df["p_population"] * df["sensitivity"] 
    # plus false positives
    + (1 - df["p_population"]) * (1 - df["specificity"])
).apply(lambda x: round(x * total_cases))

#"eliminate the impossible":
data_df = df[df["detected_positives"] == positive_cases]

p_population sensitivity specificity detected_positives
81 0.005393 0.909420 0.989789 50
226 0.003434 0.781180 0.987629 50
455 0.011168 0.891503 0.994876 50
The distribution of true prevalence of the antibodies can then simply be read off from the "p_population" column:
In [6]:
<matplotlib.axes._subplots.AxesSubplot at 0x1f9090fce80>
The relatively smooth distribution shows that the number of simulations was sufficient. Note that the distribution runs right up to 0, as you'd expect with a specificity with a lower bound of about 98%. The actual prevalence of antibodies in the study can then be characterized as follows:
In [7]:
In [8]:
data_df["p_population"].quantile([0.025, 0.975])
0.025    0.001608
0.975    0.016619
Name: p_population, dtype: float64
So we have an antibody prevalence of 1.0% (95% CI: 0.16% to 1.7%) for the study participants.
Discussion on demographic reweighing

This now needs to be re-weighed to match the demographics of Santa Clara county. The paper mentions that it re-weighed the samples by zip code, race, and sex.

In order to do this calculation, we would need every zip-race-sex combination in the study, along with the detected number of positives and negatives in that combination. Unfortunately, the paper doesn't provide that data - presumably for privacy reasons. Fair enough.

However, I will note that this re-weighing is highly unlikely to reduce the uncertainty in our metric. That is to say, if we had a perfectly random sample, then our study would perfectly reflect the population - and even so our uncertainty already spans a whole order of magnitude (0.16% to 1.7%). Could deviating from this ideal scenario make our final answer MORE certain? Any deviation, and the required adjustment, is far more likely to add uncertainty rather than certainty.

If I may engage in a bit of speculation here, I suspect this is where the study went wrong. As far as I can tell without the missing data, they calculated a point estimate for the prevalence in the study participants, then performed the reweighing to get the population-adjusted point estimate. This step nearly doubled the prevalence, from 1.5% to 2.8%. Meaning that, at this point, they were effectively thinking of the positives in their samples not as 50/3330, but 94/3330 - nearly doubling the number of positive samples artificially. Only after this adjustment did they correct for sensitivity and specificity - with the result that the inflated positive numbers were now able to outpace the false positive rate.

Now, it's not impossible for the correct procedure to end up doing something similar. After all, the individual demographic information is additional data, so adding in that data could add more certainty, and that certainty could work to increase the prevalence. But this would require a rare, specific set of circumstances, and very strong assumptions. So I'm not saying that the original paper was necessarily wrong - but I should like a release of the data itself, or at least a discussion of it, before I believed the results. Such rarities in the data would, in all likelihood, point to a flaw in the sampling rather than additional certainty.

And indeed there ARE flaws in the sampling, even apart from this issue. Two points stand out: first, the participants for the study were recruited via Facebook - which would naturally select for those who had higher reason to believe that they had been infected at some point. Second, the demographic combination mentioned above explicitly doesn't correct for age, which means that compared to the country, the study systematically underrepresents the elderly (65 or older). Of course, this is the demographic which has the most to lose from an infection, and so would be most cautious in trying not to catch it. So underrepresenting this group would cause the reported prevalence rate to be inflated.

Both of these effects would artificially increase the calculated prevalence rate - which means that we should be VERY cautious about the demographic reweighing further increasing our results. In particular, it is very difficult to justify the lower bound increasing from 0.16% in the study participants, to 2.5% in the population - which is the value that's given in the paper. Nor is that 2.5% lower bound from a proper 95% interval: it is rather just the smallest value among three constructed scenarios - where two of the scenarios have their own lower bounds BELOW this 2.5% value.
Results, comparison to other studies, and conclusion

Given all this, and the fact that the required data are simply not provided, perhaps the best we can do for the demographic reweighing is to simply increase our unweighed results proportionately, which would keep the relative uncertainty the same. In the study itself, the demographic reweighing increased the prevalence from 1.5% to 2.8% - a factor of 1.87. Doing the same to our results cited above gives:

Antibody prevalence of 1.9% (95% CI: 0.3% to 3.2%) for the Santa Clara County.

Using the same number of deaths as in the study (100), this translates to:

infection fatality rate of 0.27% (95% CI: 0.17% to 1.6%).

This is a tentative result obtained by fudging around the missing data, and the uncertainty range is broad and not particularly helpful for policy setting - and yet it provides some measure of insight in conjunction with other similar studies.

For example, There's a German antibody study that reports an infection fatality rate of 0.37% (with a 14% prevalence, which makes it robust against false positives).

An antibody test in New York gives a rate of 0.5% (with a 10-20% prevalence, again making it robust against false positives).

Preliminary results from two other antibody tests have also been released: USC conducted a test in LA county, with an infection mortality rate of 0.13% to 0.27%, and a prevalence rate of 4.1%. University of Miami conducted a study of the Miami-Dade county, which gave a prevalence rate of 4.4% to 7.9%. With some 300 deaths in the county, that translates to an infection mortality rate of 0.14% to 0.24%. These seem to share much of the same characteristics as the Santa Clara study: blood tests reporting low prevalence. In addition, they have smaller sample sizes. It's not yet known whether they share the same flaws.

A study for the Diamond Princes cruise ship gives an adjusted infection fatality rate of 0.6%, with a 95% CI of 0.2% to 1.3%.

A group of pregnant women were also tested in New York City. About 14% of them tested positive for the virus. This is an atypical demographic group who were tested with a different method, so their results cannot be expanded to the whole population. But the results here are roughly consistent with the previously cited New York study, reinforcing the 5% number.

Lastly, Iceland performed a sampling from their whole population, testing for the active virus itself. They reported that about 0.7% of their population actively had the virus in the 20 days leading up to April 4th. Given that their total cases number about 3 times the average active number they had during those dates, this roughly translates to a 2.1% prevalence rate and a 0.13%. But there are huge uncertainties associated with this number - it's not known how many tests were performed at which points in the infection's progress, and Iceland is a small country - they only have 10 deaths and a total population of 360 thousand people.

So looking at these previous results, we can say that our Santa Clara study, when re-analyzed as above, is in line with the rest. Though there are still large uncertainties, these studies seem to be converging roughly in the ballpark of 0.2% to 0.6% for the infection fatality rate.

You may next want to read:
Quick takes on the plan to re-open the country
Keeping score: my coronavirus predictions
Another post, from the table of contents