2019-12-31

A few days ago I found out that my previous blog post about how I fixed my laggy laptop was all wrong. Well, maybe those things did help a little bit. However, the thing that finally solved the problem once and for all (yes, that's actually true this time) was switching to a real-time kernel. My laptop is amazing again now. That shouldn't have been necessary, but I'm just glad it's fixed!

2019-12-05

My laptop is pretty decent given that it only cost $1000 NZD, but I've had an annoying mouse lag issue lately. I suspected just about every program I run of causing this (for example, Brave, Discord, and LBRY often showed high up on my top output, but I couldn't really single one of these out in a satisfying way. So I did what I probably should have done at the beginning, and searched for my issue.

I searched for lagging mouse and found a post about a wireless mouse, which matched my situation. The poster had fixed this issue by plugging the little dongle receiver in via a hub or USB extension cord instead of straight into the computer. I tried this today, and it worked!

It's a pretty weird fix, but I'll take it. Now the mouse only lags if the hard disk is getting thrashed, which is what I'd expect to happen anyway.

Update: After a few days, I think this helped but didn't totally solve the problem. I'm going to try other things as well. Turning off bluetooth and muting my mic input device seemed to make a bit of a difference (googling the issue finds some people for whom the problem was audio software).

Update 2: The answer by UberSteve here also helped a lot!

2019-11-10

In this post I will describe three two ways of looking at any given Bayesian inference problem. The three ways are all mutually compatible, but bring out different aspects of what is going on. This is a very condensed version of what will ultimately be several book chapters.

I call the first view the parameter space view. This is the one most statisticians are used to, and focus on the set of possible values of the parameter \(\theta\), i.e., the parameter space. In this view, a Bayesian update proceeds when you take the prior \(\pi(\theta)\) and multiply it by a likelihood function \(L(\theta)\), then re-normalise to obtain the posterior distribution which is proportional to \(\pi(\theta)L(\theta)\). This is the best viewpoint for getting the job done, but not the best for understanding the fundamental reasons why updating should be done this way.

The second view is the product space view, but that is perhaps not the best term for it, since you don't always need a product space. That's just the way to get from the parameter space view (if you're used to it) to here.

Before knowing the data, not only do you not know the value of \(\theta\), you also don't know the value of the data \(D\) (thanks to Ariel Caticha for pointing this out to me). Probabilities for statements about \(D\) are well-defined via \(p(D | \theta)\). The probability distribution that actually describes the prior uncertainty is the "joint prior": $$p(\theta, D) = p(\theta)p(D|\theta)$$ which defines prior probabilities for statements like "\(\theta\) is between 3 and 4 and \(D\) is either positive or equal to \(-1\)". In a standard Bayesian setup, the space over which probability distributions are assigned is usually this product space — the set of possible values of the pair \((\theta, D)\).

What does the update look like from this view? Trivial! If the value of the data is found to be \(D_0\), all other values of \(D\) are ruled out, and the posterior is proportional to $$p(\theta, D | D=D_0) \propto \left\{\begin{array}{lr}p(\theta, D), & D=D_0 \\ 0, & \textrm{otherwise.}\end{array}\right.$$ From this point of view, the Bayesian update is trivial elimination. When you learn some statements are false, set their probabilities to zero and renormalise. If you then found the marginal posterior for \(\theta\), you'd end up with the same result from the first view, except now it's clear why the likelihood is the correct modulating function (it's because the product rule is used to set up the joint prior).

This update also sets the probability of statements known to be true to 1, in alignment with common sense. This view is good for seeing (i) how trivial Bayesian updating actually is, (ii) how reliant on prior information it is (you better have prior dependence between \(\theta\) and \(D\) or the latter won't tell you anything about the former), (iii) how Approximate Bayesian Computation (which could have been called 'product space sampling'), works; and (iv) understanding how Bayesian inference and Maximum Entropy are totally compatible.

This second view is also operative when there is no product space present. For example, suppose \(\theta\) is a real number and your prior is \(\pi(\theta)\). You then learn that \(\theta\) is between two and five, so update to $$p(\theta | \theta \in [2, 5]) \propto \left\{\begin{array}{lr}\frac{\pi(\theta)}{P_\pi(\theta \in [2, 5])}, & \theta \in [2, 5] \\ 0, & \textrm{otherwise.} \end{array}\right.$$ If you want to think of this as multiplication by a likelihood function whose values are only ever zero or one, you can — and every other Bayesian inference can be thought of like this as well.

The third view I was going to tell you about is the powerset view, which is the set of all statements. This forms a structure called a Boolean lattice, and it is good for understanding why the rules of probability theory apply to both counting sets and to degrees of implication. But I'll save that for another time.

2019-10-24

I've started a newsletter and will be sending out the first issue soon. The intended frequency is once per fortnight but I might change that in the future depending on how things go. The newsletter will cover any topics of interest to me (and hopefully you). Please sign up here if you're interested. When you do this, you'll be sent a confirmation email to click on. Please check your spam folder just in case!

I am considering archiving the newsletters somewhere as well but I'm not 100% sure yet. Edit: I will publish them on my LBRY channel. The first issue, which has already been sent out, is there now.

2019-09-26

This afternoon I had my weekly meeting with PhD student Ollie Stevenson. It was a routine meeting where he showed me what he'd been working on and we discussed a few issues around it. Then he mentioned that our most recently submitted paper had received some feedback. It was pretty good feedback! Check this out:

Thanks, Nick! This was super pleasing to read. I just hope the journal referees feel the same way, or at least half as positively. But until we hear back from them you can get the preprint from the arxiv or (why not) from LBRY.

2019-08-20

Do people learn about lookup tables anymore? I don't know where I picked them up from, but I have the impression that they are an old-fashioned programming technique from the 1980s and 1990s. However, I find myself using them all the time even though it's 2019. I have no idea whether they're a good idea in languages like Python or R, but I use them a lot in C++ and I should try them in Haskell some time.

For example, in some work I'm doing at the moment, I am fitting profiles to spectra of singers singing a single note. The spectra contain many peaks, and some of the peaks look approximately Gaussian whereas others look more like a Cauchy distribution. To capture these different shapes, I'm using model functions that are proportional to the density function of a Student t-distribution. For a peak of amplitude 1 centered at position 0 with unit width and shape parameter \(\nu\), the functional form is

$$f(x) = \left(1 + \frac{x^2}{\nu}\right)^{-\frac{1}{2}(\nu + 1)}.$$

The profile can be moved around using a location, scale, and amplitude parameter. Since I have to evaluate fifty of these (there are lots of peaks) per MCMC iteration in my fit, it dominates the computational cost. Things are much faster if I make a grid of \(x\)-values, populated with evaluations of \(f(x)\) at the grid points. I also have several arrays to cover a range of values of \(\nu\). This happens once at the beginning of my run, and if I want to evaluate \(f(x)\) for the billionth time, I just figure out which element of my pre-computed arrays is closest to the input \(x\)-value and get the corresponding \(f(x)\) value out (actually I linearly interpolate from the two grid points that my \(x\) falls between).

Here's the result of one of my fits. You can really see the need for the \(t\)-like shape in the second harmonic. The red curve comes from a continuum component I used which had a very flexible prior, and the blue is the total model curve after adding all the \(t\)-shaped harmonic peaks. Click the image to enlarge.

2019-08-12

This morning I bought a muffin from the convenience store on campus. In front of the muffins there is a sign instructing us to use the tongs to pick up a muffin, instead of our possibly grotty hands. I didn't, though — I just grabbed a muffin with my hands, disobeying the sign. Why would the tongs be clean to touch? They're telling everyone to touch them!

Presumably someone could touch a muffin and then change their mind and put it back, contaminated. The tongs would prevent this, but then the same person would have touched the tongs. I could be way off here, but is the whole tong ritual just "hygiene theatre", similar to how many regard airport security as "security theatre"? I don't know for sure, but I'm going to continue not using the tongs.

2019-08-08

On my old blog I occasionally posted songs that I'm into, for no particular reason other than wanting to share them. I don't know if anyone listened or liked those, but I thought I'd just continue to do it anyway. So here are two of my recent favourites. First, Universal Sound, by country singer-songwriter Tyler Childers. I adore this song both technically and emotionally.

Second, Australian virtuoso singer Vanessa Amorosi (of Sydney Olympics/noughties pop fame) covering Aretha Franklin and doing a wonderful job of it. The guitarist with the funny hat is the guy from Eurythmics. Also, in case you're curious about Complete Vocal Technique lingo, most of what Amorosi is famous for is using the edge mode higher in the range than most singers do. One of the commenters called it an "upper chest belt" which I suppose is the normal terminology for what she does when everyone gets really impressed (or calls it screeching, as my Nan would :)). Much of the earlier part of the song is in neutral.

2019-07-22

Attentive readers and/or stalkers may have noticed a recent addition on the research page of this site: a paper accepted for publication in Nature. This was an interesting collaboration with my PhD supervisor Geraint Lewis, whom I visited last year. We intended to work on two other projects but those didn't work out. Instead, I ended up doing the Bayesian model comparison for the hypothesis that there are two rotational components in M31's globular cluster population (distinguished by whether or not the globalar cluster is associated with substructure or not) vs. the hypothesis of a single rotational component. It was a reasonably straightforward inference problem from my point of view, but with cool astronomical implications.

From a statistical point of view, it was a nice opportunity to try applying John Skilling's advice to future-proof results by presenting the marginal likelihood, so that future analyses can be trivially compared to ours without having to re-do ours. In addition to calculating the marginal likelihoods of six different models, I also calculated the marginal likelihood of the "top statement" \(\top\) of the whole paper, i.e., the logical disjunction of the six models, or the proposition that one of these six models is correct. I hope to make this a habit in future publications.

Getting a Nature paper is also a pretty nice thing from a career point of view. Journals are a bit of an anachronism and there are lots of well-worn complaints about them, but they still function as a way of measuring the status of a paper or a researcher with minimal time and other transaction costs. With Nature, you can also assess the results quickly using the joke that anything published there is eventually found to be incorrect. I'm looking forward to someone figuring out a way to replace journals (surely it is possible with the internet and blockchains and such), but until then, this is satisfying to have. Once the preprint is up I'll link to it. If you want to check out my inference code for the project, it is available here.

2019-07-06

It's been pretty quiet here since I've been busy again. Here are the two main reasons:

That was about half of the total exam marking I recently did. Once that was done, it was time to prepare for our daughter's first birthday party :-)

2019-06-21

In the last couple of months I've come across some pretty cool new programs that I thought I'd tell you all about (I know I have at least one reader — I got a really nice email from a former student who reads the blog).

First, Zstandard. It's a new-ish compression algorithm that was developed by a guy at Facebook. It seems to have really good properties, compressing both quickly and effectively — way faster than bzip2 and with better compression than gzip. I think you can use it with tar but I haven't figured out how yet. You can always do it in two steps though:

tar cvf myfile.tar mydir
zstd myfile.tar -o myfile.tar.zst

The command to unzip is a bit awkward: unzstd. But the performance is so impressive I'm willing to overlook that!

The other thing I had to do recently was delete a Bitcoin wallet from an old computer before getting rid of it. To do so, I used srm (secure rm), which does a bunch of fancy wiping to make sure it can't be read even by someone hardcore. It's very easy to use — you can just srm a file like you would rm one, and it uses sane and intense defaults to make sure it's not recoverable.

2019-06-09

I sometimes tell my graduate students that the most important section of their dissertation is the acknowledgements. This is because it will be read by far more people than the rest, by a wide margin. After the acknowledgements, the next most important part is to put in some interesting or fun quotes at the beginning of the chapters.

I really enjoyed choosing the quotes for my PhD thesis, and I'm enjoying doing it again for the book I'm writing. I thought I'd share a couple of the quotes here. There will be more!

James Clerk Maxwell: The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind.

Kevin H. Knuth and John Skilling: The set-based view is ontological in character and associated with Kolmogorov, while the logic-based view is epistemological in character and associated with Cox.

A colleague who may or may not remain unnamed: Jaynes has too much MaxEnt shit in it [BJB: I disagree with the colleague's dismissal of the subject].

C. S. Lewis: I doubt whether we are sufficiently attentive to the importance of elementary text books.

John Skilling: I think that the fundamentals of Bayesian analysis should be very simple and accessible to school children.

Ariel Caticha: Before the data information is available, not only we do not know \(\theta\), we do not know \(D\) either.

Newman: But you remember this. When you control the mail, you control information.

I use that last one on my Dad a lot (he's a postman), but it also works for the information theory chapter. I hope you enjoyed these!

2019-06-01

A month or two ago, Lianne and I bought a Yamaha MT-03 to help with commuting. It's been really fun riding again, though of course I have plenty of reasons for caution on the roads. Today I experienced a hazard I've never encountered before, though — hail.

I had taken the bike to the North Shore for its first service. It was clear weather when I left to head home, though the CBD seemed to be covered in a dark cloud of doom as I approached the harbour bridge. I didn't even get to the bridge before I got hailed on, though. Luckily the hail was quite small and didn't do any damage to me or the bike.

Funnily enough I think this was only the second worst riding weather I've ever had. Back in Sydney around 2008 or so I once had to pull over under an underpass on the M4 because I was afraid of being struck by lightning. I stopped for about half an hour before the lightning cleared up.

By the time I arrived home today, in true Auckland form, it was sunny for about 10 minutes. Then it rained heavily. The one upside is that this city knows how to make good rainbows (click for high resolution):

2019-06-01

As my loyal reader knows, I've been into LBRY for a while now after hearing about it from economist Alex Tabarrok. It still has a ways to go to be fully featured and get more adoption, but it's been a fun thing to learn about and play with. Today I wanted to tell you about a cool feature that isn't supported in the GUI app yet but can be done from the command line.

One of the fun things you can do in the app is send a tip to someone after (or before :)) watching or downloading their video or other file. Rather than going straight to the publisher's wallet, the tip gets associated with the item, which has a few benefits. First, the item will rank higher in search results, and secondly, it means that the publisher's claim to the short URL of the item (such as lbry://hello, as opposed to a full permanent URL such as lbry://hello#b981513aeb8ff0394cc2d1d4aef6c1fa5177e24a) gets stronger, in the sense that someone would need to spend more in order to take over the short URL. If the publisher does want to spend the tip, it needs to be "unlocked" from the item and transferred into the publisher's wallet.

Tips work fine in the GUI app already, but there's something even more cool that you can do only from the command line at present. You can use some of your credits to support someone else's channel or item without sending them a tip, and in a way that can be easily reversed if you actually want to spend the LBC or use it elsewhere. A tip is a special case of something which is more generally called a "support" — specifically, a tip is a support where you lose the LBC you tip and the publisher gains it. The other kind of support lets you put some of your LBC towards someone else's publication, but you retain the right to withdraw it at any time, rather than giving that right to the publisher.

To make one of these, first you need the unique identifier of the claim (channel or published file), called the claim_id. If you know the full URL already, the claim_id is just the part after the #. If you only know the short URL, you can go to the command line and type something like this

lbrynet resolve lbry://hello

and then look for the claim_id in the output. If the item is inside a channel, you'll get the channel's claim_id in the output too, so make sure you can tell which is which.

Once you know the claim_id, it's pretty easy to make a non-tip support. Here's how I would do it for the hello example once I knew the claim_id:

lbrynet support create --claim_id=b981513aeb8ff0394cc2d1d4aef6c1fa5177e24a --amount=10.0

If you wanted to create the support as a tip instead (like you can do in the app), just pass an extra --tip option to the command.

lbrynet support create --tip --claim_id=b981513aeb8ff0394cc2d1d4aef6c1fa5177e24a --amount=10.0

Obviously the amount can be changed depending on how much you want to use. If you decide to abandon the support (i.e., withdraw the LBC back to yourself), you don't need to use the command line — the transaction appears in the app with a little trash icon next to it which you can use to abandon the support:

Now that I know about this, there's little reason not to deposit most of my credits against stuff I like! Better to have them sitting there doing something cool as opposed to just sitting there!

2019-05-29

The initial flurry of posts here may have implied that I would be keeping up that rate long term. I can't promise that, but I will try to post regularly still. Things have been pretty busy with teaching STATS 731 for the first time (I always find it hard to teach a course for the first time), and with Mackenzie picking up viruses from daycare on a regular basis.

One thing I've done to 731 is to put in Nested Sampling and more on model comparison. I've been emphasising it a lot. If you're interested in that, you might enjoy this tutorial problem on the Jeffreys-Lindley "paradox", which includes a diagram explaining it. The Nested Sampling code to use is here.

2019-05-16

When I lived in Australia, I never really paid much attention to the popular pop/rock singers that Australia had produced. I think I thought of them (Jimmy Barnes, John Farnham et al) as cheesy. But after living in the USA for three years I became really nostalgic for them, and this was boosted when I realised they were excellent singers. They also fit into the eras of music where most of my favourites are from — commonly known as "dad music". I don't know why, but I've always been a few decades behind, except for when I liked some 90s artists in the late 90s.

My latest "discovery" along the same lines, who I've known about my whole life, is Iva Davies and Icehouse. I've been digging the album Man of Colours. It sounds a lot like 80s-era David Bowie (think "Let's Dance"), and I think they actually toured together around that time:

Davies's vocals are often in the tenor range which I always find exciting and challenging, and he provides more evidence that mullets assist with this. Here's "Crazy", which shows off the mullet and the harbour bridge. Iconic!

2019-05-09

Answer: 36. That's 36 people who don't understand the LBRY naming/URL system, which works using a form of auction that provides no incentive to squatting. [edit: found the easier explanation]. That's also 36 people who are going to be disappointed if LBRY becomes popular and then find out their reward for early-adopting and calling their channel @PewDiePie is basically nothing.

For technical people, the command I ran to get that was:

lbrynet claim list @pewdiepie | grep claim_id | wc -l

There'll be an upgrade soon where the relevant command will change to:

lbrynet claim search @pewdiepie | grep claim_id | wc -l

2019-05-09

A quote from statistician John Tukey that is a cliché for a reason is the best thing about being a statistician is that you get to play in everyone's backyard. Sometimes, other people invite you to their backyards, like geophysicist Mike Rowe did. Other times, you just invite yourself over.

The Complete Vocal Institute was founded by Cathrine Sadolin, who struggled to learn to sing until she reinvented the field in a way that made sense for herself. Singing teaching has historically been a lot more of an art than a science, and her approach is a lot more methodical than most. There are things you could critique about it of course, but when I discovered it years ago it grabbed my attention because it was the first time I'd seen someone actually write down an explicit model for the things voices can do and how to do each of those things. While the new me™️ is not a rationalist, singing instruction needed more of that on the margin. Actually connecting the model to practice was hard for me and it took a long time, and the model has been revised over the years anyway as they have been researching things and modifying it accordingly.

Speaking of research, I just emailed them a few weeks ago to register myself as a fan of their work and a potential collaborator if they need statistics, which it turns out they do! So I'm now looking for a graduate student who's interested in vocals to work with me on that. They have lots of data and there are still forms to sign, but there should be lots of possibilities here! To be honest, there's some chance it won't work out, but I am excited so felt the need to write this and tell y'all about it.

2019-05-06

One of the best and worst things about being an academic is that sometimes you are assigned to teach a course about something you aren't very good at, or in some cases, have never learned at all. As long as you can keep ahead of the students, it usually works out fine. I've been lucky so far that this hasn't happened much, and when it has, I've been grateful to have learned the new material.

One of the courses I've been teaching regularly for a few years is STATS 220, an introductory computing course for statistics students, which teaches a bit of HTML, file formats, XML, SQL, and R. The course is the brainchild of Paul Murrell, but our previous head of department wanted someone else to teach it so that more than one person knows the course. I was happy to do it since I like computing, and saying yes meant I was less likely to ever have to teach frequentism. When I started I knew a bit of HTML and R, and knew about file formats, but had never touched XML or SQL. At first I learned enough in order to teach the course, but not much more.

However, recently I've had occasion to actually use more of this stuff in actual projects. For instance, this blog is written in plain HTML, and STATS 220's little HTML/CSS section has helped a fair bit so that I understand what I'm looking at when I use HTML. Today I set up an RSS feed (see that little button at the top of this page), which is secretly just an XML file, which I hand-wrote based on this guide from W3Schools.

I've also recently enjoyed using SQL to query the LBRY blockchain and plot things such as the amount various creators have received in tips and the total number of publications over time. Those were done using simple SELECT commands as taught in STATS 220 — the only slightly fancy stuff being (i) an INNER JOIN of a table with itself, and (ii) figuring out how to submit the darned query to their MySQL server and handle the results that come back.

See, students, some of this stuff is useful!

2019-05-05

I'm a bit unusual among Bayesians in that I still think the principle of maximum entropy is both true and (occasionally) useful, and that its critics have thrown out the baby with the bathwater. One of the chapters of my upcoming book, if I finish it, will be about this issue, and related ones such as where MaxEnt constraints come from in the first place.

Some of the bad criticisms of MaxEnt focus on the fact that you can easily get poor predictions from a MaxEnt distribution, in an ex-post sense (i.e., a scoring rule could give the MaxEnt distribution a worse score than some other distribution that people might intuitively suggest). I don't know why this would surprise anybody. Any broad prior, especially on a high dimensional space, is usually bad in this sense. MaxEnt does minimal updating of probability distributions, and if you minimally update something that doesn't perform that well according to an ex-post criterion, you're likely to get something else that doesn't perform that well with respect to the same criterion.

One interesting feature of MaxEnt distributions is that they often lead to macroscopically homogenous predictions. For example, let \(\{x_i\}_{i=1}^N\) be a batsman's next \(N\) scores in cricket, so the set of possibilities is \(S = \{0, 1, 2, 3, ...\}^N\). Starting with a flat prior over \(S\) and then doing MaxEnt with an expected value constraint like

$$\left<\frac{1}{N}\sum_{i=1}^N x_i\right> = \mu,$$

gives a posterior distribution for the \(\{x_i\}\) which is a product of iid geometrics with means \(\mu\). If \(N\) is large, this implies a high posterior probability for the sequence of scores looking macroscopically homogeneous, because the constraint never put in any time-dependent information, and because almost every element of \(S\) looks macroscopically homogenous.

Now, the actual sequence of scores \(\{x_i\}_{i=1}^N\) may have a trend or whatever. That doesn't mean MaxEnt is invalid! It still correctly updates the flat prior over \(S\) in a minimal fashion, which is what it claims to do. Most possible sequences (elements of \(S\)) look macroscopically flat, so MaxEnt predicts that. Reality, being complicated, happened to produce a sequence that was atypical in that sense, but probably fairly typical in some other senses.

2019-05-01

This kind of thing makes my day:

Dear Prof. Brewer,

I work as a Data Science/ML professional in the US having done my grad study in Stats.
I came across your Bayesian Stats course you teach at University of Auckland. I must say I have never found such a cogent and coherent text on Bayesian Statistics ever.

Now I'm in a good mood. In related news, I've moved the notes here for online reading.

2019-04-30

I'm currently preparing to teach STATS 731, the graduate Bayesian course here. My first lecture was meant to be today but I had to cancel it due to hoarseness following a throat infection. Anyway, the existing course materials use a lot of the traditional (in statistics) wide priors, stuff like this for a location and scale parameter respectively:

beta ~ dnorm(0, 1.0E-4)

tau ~ dgamma(0.001, 0.001)

var <- 1/tau

sig <- sqrt(var)

In the 1990s, it was important to choose conjugate priors to speed up the MCMC sampling, and in many cases (e.g. regression) stuff like this was appropriate. It still works today, too. However, in my experience students find the choice of a prior a bit mysterious, and these are more obscure than they need to be. Who has an intuition for what that gamma prior looks like — even for tau, let alone sig (the coordinate that's usually more interpretable). Also, are we really so sure beta is between -200 and 200? Probably depends on the units of the problem.

In my lecture on Thursday I'm still going to use these priors but I'm also going to (a) plot them, and (b) compare them to uniform and log-uniform priors which they are fairly similar to over a decent range of their parameter spaces. Hopefully this will remove a bit of the shroud of mystery.

2019-04-30

I recently enjoyed this excellent talk by Dallas Willard, entitled The Nature and Necessity of Worldviews. If you're familiar with Jonathan Haidt's work, it strikes familiar notes, but Willard's personal angle as a frustrated Christian in the secular university comes through.

Regardless of what you think about issues of religion (I'm undecided myself), 90% of the talk can be filed under important things I didn't learn until my 30s, and can be applied to a great deal of other issues.

If you don't have time for the whole thing, you can also check out this brief clip.

2019-04-29

When I moved my blog over here, I was a bit worried that I wouldn't be able to type \(\LaTeX\) anymore. But that's not the case! Thanks to MathJax, I can now get equations in here very simply. Here are two of my favourites:

$$P(H \,|\, D, M) = \frac{P(H \,|\, M)P(D \,|\, H, M)} {P(D \,|\, M)}$$

$$H\left(\boldsymbol{p}; \boldsymbol{q}\right) = -\sum_i p_i \log (p_i/q_i)$$

While researching this, I also came across some PDF to HTML converters. That sounds like a horrible idea, but some worked a lot better than I thought. If you click on my CV at the top right you'll get what looks like a PDF document but is actually HTML! How cool.

2019-04-28

The other day I bought a domain name for my website, which was fun, and has motivated me to continue simplifying my internet life in a few ways. Over the last year or so I've successfully reduced my social media usage by about 80%, and am now using the internet like it's 1999 again (well, except for the crypto stuff). It's been a pleasant experience, despite sounding a bit hipster.

In addition, Wordpress has been annoying lately with too many promotional emails and generally being more complicated than I need. Therefore, I've decided that I'll no longer be posting at plausibilitytheory.wordpress.com. I'll leave it up so people can still read it if they want, but in general, any new posts will go straight to the blog page on brendonbrewer.com.