Modified respirator to shield myself and others from COVID

Summary: I have tried many types of masks and respirators during the 2020 pandemic. My recommendation is to use ‘elastomeric’ respirators common in industry, and to either filter or completely block off their exhalation valve. The result is a comfortable respirator that I believe offers a high level of protection against airborne diseases to myself and others. I am not an infectious disease expert.

Contents

  1. Elastomeric respirators
  2. CDC recommendation for exhalation valves
  3. Recommendation A: 3M 6500QL Series with KN95 and surgical mask
    1. Choice of respirator
    2. Choice of filters for a 3M respirator
  4. Recommmendation B: Miller LPR-100 with tape and surgical mask
    1. How we need to modify the respirator
    2. Exhalation valve
    3. Inhalation valves
    4. Also use a surgical mask
    5. Choice of respirator
  5. Potential concerns
    1. Repeated use
    2. CDC guidelines
    3. Concerns specific to tape method
      1. C02 rebreathing
      2. Exhaling through filter
      3. Discussion of tape technique by others
  6. Overall recommendation

Elastomeric respirators

The effectiveness of a mask can be broken down into two parts: how well the mask fits on your face, and the filtration efficiency of the mask. A further important consideration is comfort.

Surgical and cloth masks are comfortable but have poor fit and filtration efficiency. I believe it’s possible to do much better.

Respirators that meet the NIOSH N95 or N99 standards for filtration efficiency, such as the N95 respirators pictured below, are popular in healthcare settings.


3M N95 respirator (left), N95 respirator in a medical setting (right)

In my experience, these have two main downsides:

  • They may be scarce during pandemics. You should probably leave limited supplies for healthcare workers.
  • The tight elastic band that ensures a good fit also makes the respirators very uncomfortable for extended use.

KN95 and KF95 are respectively the Chinese and Korean-manufactured masks that claim to have the same efficacy as N95s. They come with ear loops rather than behind-the-head elastic bands, so they have a far looser seal than N95s. I suppose you could add your own elastic bands to them to improve the seal, but then they would be just as uncomfortable as N95s. Therefore, they are not a competitive option.


A KN95 mask

There also exist N95+ masks designed for industrial tasks that produce harmful airborne particles (such as welding or paint spraying). They are called elastomeric respirators, or sometimes industrial respirators.

High filtration efficiency elastomeric respirators for industrial use. Miller LPR-100 (left), 3M 6200 (right).

Compared to healthcare N95s, these respirators:

  • are more widely available
  • achieve superior fit by using elastomers shaped like a human face (there is no need to bend a metal nose bridge)
  • are much more comfortable, mainly because they:
    • spread the pressure onto a wider area of skin
    • come in multiple sizes
    • have adjustable straps
  • won’t fog up your glasses

A downside is that it’s more difficult to be audible through an elastomeric respirator than through an N95 or surgical mask. I am able to be understood by raising my voice, but smooth social interactions are not guaranteed. It’s probably not a great setup for spending time with your friends; you can use a KN95 for that.

The fatal flaw1 of these elastomerics when it comes to disease control is that they have an exhalation valve that allows unfiltered air to exit the mask. In PPE jargon, they do not provide source control. (This may be about to change in 2021, see this footnote2. I will try to keep this post updated.)

The exhalation valve opening on the Miller LPR-100

We can modify these respirators to filter their exhalation valve (recommendaton A), or completely close it off (recommendation B).

(If infection through the mucosal lining of the eyes is an important concern to you, and you don’t wear glasses, you should also wear safety googgles.)

CDC recommendation for exhalation valves

During the 2020 pandemic, the US CDC issued the following recommendation, in a blog post from August 8 20203:

If only a respirator with an exhalation valve is available and source control is needed, cover the exhalation valve with a surgical mask, procedure mask, or a cloth face covering that does not interfere with the respirator fit.

Recommendation A: 3M 6500QL Series with KN95 and surgical mask

If you want to follow something similar to CDC guidance, I recommend:

  • A 3M 6500QL series respirator
  • A part of a KN95/KF95 mask tightly covering the exhalation valve


3M 6502QL. Unmodified (left), KN95 material covering valve (middle and right)

You’ll likely want to add a surgical mask on top of that:

  • as a backup
  • for the very small amount of additional filtration it provides
  • to avoid misunderstandings with strangers


3M 6502QL with KN95 material covering valve and a surgical mask on top

Surgical masks are not primarily designed to filter aerosols4. It seems clear to me that KN95s and KF95s are superior to a surgical mask for covering an exhalation valve (let a alone a cloth mask). (There is a list of such respirators that have received an emergency use authorization from the FDA. There are probably many low-quality masks fraudulently marketed as KN95 and KF95 at the moment, so make sure you buy from an approved manufacturer.)

In the models I have seen, the material in KN95s is far more flexible than in N95s, making allowing you to shape it so that it tightly covers an exhalation valve. It’s slightly fiddly but definitely possible with a bit of dexterity and perseverance. Using a thinner surgical mask would be easier; but the KN95’s extra protection for third parties is well worth it.

Here are the steps you should follow (see video):

  • cut a KN95 in half along the fold
  • cut one half to size further
  • use a rubber band and tape to attach the material over the respirator valve. This is better explained with a video than in words. The main thing to know is that you should use the two small ridges in the plastic below the valve to secure the rubber band.
  • add tape on the upper end of the KN95 material

Instructions

Unfortunately, for any valve covering approach, there is a trade-off between a fit and the surface area usable for exhale filtration. I have not been able to achieve a good fit when placing a KN95 or surgical mask more loosely over the valve, which would give more surface area. In my setup a small rectangle of KN95 has to do all the filtration, which likely lowers the efficiency. However, the N95 specification is for a flow rate of 85 liters per minute, which is many times the 6 liters per minute breathed by an individual at rest5, so I am not very concerned.


Ridges on 6502QL. Note that in the real setup the KN95 will go below the elastic band.


Location of the valve underneath the KN95 material. View form below the respirator.

Choice of respirator

I have tried two industrial respirator models, the 3M 6502QL and the Miller LPR-100. I prefer the build quality and aesthetics of the Miller (see below), but its shape makes it almost impossible to get a good seal if you attempt to cover the valve with a surgical mask or KN95. So for this technique, I recommend the 3M 6500QL series.

I am aware of three 3M half-facepiece reusable respirator groups, the 6000 series, the 6500 series, and the 7500 series.

3M half-facepiece reusable respirators (3M.com)

Since I have only tried a respirator of the 6500 series, I do not have a strong view on which is preferable. I would recommend the 6500, mostly because I have already demonstrated that it’s possible to cover the valve. The 6000 series does not have a downward-facing exhalation valve and may be harder to work with. I’m agnostic about the relative merits of the 7500.

The 6500 series has a quick latch version (difference explained here), which is the one I used. I’d recommend the quick latch 6500QL series, because it seems that the latch makes the fit of the KN95 material to the respirator more secure (see video). By the way, attaching a mask on top of the valve makes the quick-latch mechanism much less effective; I never use it.

Each series comes in three sizes, large, medium and small. I am a male with a medium-to-large head, and I use a medium (the 6502QL).

Regarding whether airlines will accept this setup, I have heard both some positive anecdotes and one negative anecdote.

Choice of filters for a 3M respirator

I use the 3M 2097 P100 filters.

You should use lightweight filters that are rated N100, R100 or P100. The “100” means that 99.97% of particles smaller than 0.3 micrometres are filtered out. The letters N, R and P refer to whether the filter is still effective when exposed to oil-based aerosols, this should be irrelevant for our purposes.

The weight of the filters is a crucial determinant of comfort. I originally used the 3M respirator with the 3M 60926 cartridges, which filter gases and vapors as well as particles. This was a mistake, as filtering gases and vapors is irrelevant from the point of view of infectious disease, and these cartridges are much heavier than the 3M 2097 P100 filters. Switching to the lighter filters made a world of difference; now wearing the 3M doesn’t bother me at all.


The 3M 6502QL respirator weighs 395 g. with 3M 60926 cartridges, but only 128 g. with 3M 2097 filters, a 68% reduction.

Recommmendation B: Miller LPR-100 with tape and surgical mask

I believe that, in expectation, the previous method offers slightly worse protection to third parties than a well-fit valveless medical N95, because our makeshift exhalation valve filter may not be entirely effective.

This section details another technique which may be able to achieve the best of both worlds: the comfort and availability of industrial masks, and the third-party protection offered by valveless masks.

How we need to modify the respirator

Let’s look at how the valves in industrial masks work. I’ll be using the Miller LPR-100, but the 3M is built similarly.

The exhalation valve is at the front. There are also two inhalation valves, one on each side between the mouth and the filter. These only allow air to come into the mask from outside, forcing all the exhaled air to go through the exhalation valve (instead of some of it going back through the filter).

Miller LPR-100 valves

We need to disable both the inhalation and exhalation valves:

  • The unfiltered exhalation valve should be completely sealed off.
  • In order to allow the user to exhale, the inhalation valves need to be turned into simple holes that allow two-way air circulation.

This will mean that both inhaled and exhaled air will go through the P100 filters.

Exhalation valve

We can seal off the exhalation valve from the outside with tape6. On the Miller respirator, there is a little plastic cage covering the valve, and this cage can be taped over. Note that tape sticks very poorly to the elastomer (the dark blue material on the Miller). This is why I only place tape on the plastic; this seems to be sufficient.

Tape on exhalation cage

I am using painter’s tape because it’s supposed to pull off without leaving a residue of glue. It’s possible that it would be better to use tape with a stronger adhesive. (A friend of mine commented: “Some ideas for sealing off the exhalation valve: (1) Butyl tape/self-vulcanizing tape. Not so much a sticky tape as a ribbon of moldable putty, so no adhesive residue. This stuff is pretty much unparalleled if you need to make a fully gas- and watertight seal around an irregularly shaped opening in a pinch without making a mess. The fact that it has no adhesive does put some constraints on the geometry of the part you’re sealing off, but I think it would work (better than painter’s tape, at least) on the Miller. (2) Vinyl tape/electrical tape. It’s relatively water-resistant and can be stretched to some extent. The adhesive also sticks to polymers pretty well (although it does leave a lot of residue after some time, but you can clean that off with a bit of IPA).”)

You can check the seal of your tape by pressing the mask onto your face and attempting to exhale (with the inhalation valves intact). Air should only be able to escape through the sides of the mask.

Inhalation valves

The inhalation valves are removable and can be pulled out. They are very thin and feel like they might be about to break when you pull them out, but I have been able to pull four of them out without a problem.

Touching a valve (left), a valve after it has been pulled out (right)

The two inhalation valves (left), the filter now visible through the holes (right)

Pushing the valves back in is easy.

The tape can be removed and the valves re-inserted, making my modification fully reversible.

Also use a surgical mask

Even if you’re using the tape technique, I recommend also covering the respirator with a surgical mask, since this has no downsides and might have some benefit. The seal on the exhalation valve might not be perfect and may get worse over time, so an extra layer of filtration, however imperfect, is a good backup.

It’s also beneficial because it makes what you’re doing legible to others. You don’t want to explain this weird tape business to strangers, even if it’s for their protection.

Miller LPR-100. Unmodified (left), with tape (middle), with tape and surgical mask (right)

Choice of respirator

For this technique, I recommend the Miller LPR-1007.

I recommend the Miller over the 3M because:

  • its build quality feels superior to me
  • it looks better
  • it blocks less of your field of view

Since the Miller is better than the 3M, and 3M is such a huge player in this market, I think there’s a decent chance that the Miller is in fact one of the very best options that exists.

The Miller weighs 139 g., which is a negligible difference to the 3M’s 128 g

I also like the fact that you can buy a neat rigid case to hold the Miller respirator. The case is called the 283374.


Miller case, 283374

The Miller model comes with replaceable P100-rated filters, while the 3M can be used with many types of filters and cartridges.

If you want to implement this technique on the 3M, it should be possible; all steps will be similar.

Potential concerns

The 3M+KN95 method we discussed earlier can be seen as a simple adaptation of CDC guidelines, so I have fewer concerns about it.

However, the tape technique involves a more fundamental alteration. This might seem unwise. How do I know I haven’t messed up something crucial, endangering myself and others?

Before discussing the specific concerns, it’s useful to consider: what are the relevant alternatives to my recommendation?

My best guess is that constantly wearing a correctly fitted medical N95, with the really tight elastic bands, is very slightly safer for others in expectation than the tape method (due to risks of things going wrong, like the tape getting unstuck). However, it is not a likely alternative for everyday use. First, in my experience, N95s are more difficult to fit correctly than industrial masks. Second, for me, these respirators are prohibitively uncomfortable. I have seen few people use them. I think the realistic alternatives for most people are cloth and surgical masks. I am relatively confident that both of my techniques are an improvement on that, for both the user and third parties.

By the way, I am not an expert in disease control. I studied economics and philosophy and then worked as a researcher.

Repeated use

Healthcare N95s are supposed to be used only once before being decontaminated. However, I plan to use the same filters many times. Is this a problem?

Why are N95s supposed to be used once? According to this CDC guidance,

the most significant risk [of extended use and reuse] is of contact transmission from touching the surface of the contaminated respirator. … Respiratory pathogens on the respirator surface can potentially be transferred by touch to the wearer’s hands and thus risk causing infection through subsequent touching of the mucous membranes of the face. …

While studies have shown that some respiratory pathogens remain infectious on respirator surfaces for extended periods of time, in microbial transfer [touching the respirator] and reaerosolization [coughing or sneezing through the respirator] studies more than ~99.8% have remained trapped on the respirator after handling or following simulated cough or sneeze.

Since I plan to leave the respirator unused for hours or days between each use, and any viral dose on the exterior of the filters is likely to be very small, I don’t think this is a huge concern overall. I am very open to contrary evidence.

By the way, based on this guidance, it seems to me we should also worry less about reusing respirators and masks in general, even without decontamination. (Decontamination makes a lot more sense for health care workers who are exposed to COVID patients).

It’s good to remember to avoid touching the filters.

CDC guidelines

As explained above, the CDC recommends a surgical or cloth mask to cover the valve. There is no evidence that they considered either of the techniques I described above when issuing their blog post.

The tape method is a greater deviation from the CDC guidelines than the KN95-covering method, so if you care about following official guidance you could use the latter.

Concerns specific to tape method

I assign a relatively low chance that the tape method is worse than the CDC recommendation of covering the valve with a surgical mask (my views depend considerably on the tightness of the surgical mask seal). and a very low chance that it’s worse than a surgical mask alone. The probability mass I assign to harm is a combination of concerns about exhaling through the filter reducing its efficacy, and unknown unknowns.

C02 rebreathing

Without the valves, part of the air you inhale will be air that you just exhaled, which contains more C02. I have not personally noticed any effects from this.

Exhaling through filter

Could exhaling through the filter be a bad thing somehow? I wasn’t able to find any source making an explicit statement on this, but I think it’s unlikely to be a problem.

One reason to worry is that the founder of Narwall Mask has told me that, according to one filtration expert he spoke to, one-way airflow greatly prolongs the life of the filters compared to two-way airflow. However, based on my small amount of research, I don’t think the life of the filters would be affected to a degree that is practically important.

The MSA valveless elastomeric respirator that I mentioned in this footnote2 appears to have filters that can be used for more than 1 month of daily use during the workday; and moreover, we can see in the respirator’s brochure that these filters, with model number 815369, are the same as those that are used in MSA’s line of regular, valved elastomeric respirators (see here). From this I conclude that: two-way airflow through regular P100 filters was considered an acceptable design choice by MSA; and these filters can be used two-way for at least a month of hospital use.

In addition, healthcare N95s (without valves) are designed to be exhaled through. They are only rated for a day of use, but I believe this is not because the filter loses efficacy (see section on repeated use).

Exhaled air has a relative humidity close to 100%. Could exposure to humid air reduce the efficacy of the filters? In this study of N95 filters, the difference penetration rose from around 2% to around 4% when relative humidity went from 10% to 80%, and this effect increased with duration of continuous use. The flow rate was 85 L/min.


Combination of figures 3 and 5, Mahdavi et al.

Note that this study, which simulates inhalation of humid air, does not address (except very indirectly) the question of how the exhalation humidity affects the inhalation filtration.

Discussion of tape technique by others

  • This NIOSH study tested three modifications of valved respirators: covering the valve on the interior with surgical tape, covering the valve on the interior with an electrocardiogram (ECG) pad, and stretching a surgical mask over the exterior of the respirator.
    • They found that “penetration was 23% for the masked-over mitigation; penetration was 5% for the taped mitigation; penetration was 2% for the [ECG pad] mitigation”. I would be very interested in more discussion of why the ECG pad did so much better than the surgical tape, the authors don’t say much. One guess could be that the ECG pad has a more powerful adhesive, which would suggest that it’s important to choose a strongly adhesive tape if implementing my technique.
    • When discussing the choice of modification strategies, the authors wrote that “two concerns are that the adhesive could pull away from the surface, thereby not blocking airflow to the same degree over time, and that these adhesives could contain chemicals that have toxicological effects.” study
  • In an FAQ released by 3M, in response to the question of whether one should tape over the exhalation valve, they wrote “3M does not recommend that tape be placed over the exhalation valve”, but do not give any reasons for this beyond the fact that it may become “more difficult to breathe through … if the exhalation valve is taped shut”.
  • The state of Maine’s Department of Public Safety recommends against tape-covering, but merely because “this would be considered altering the device and violates the manufacturer’s recommendation”.

Overall recommendation

I think it’s about 50/50 which of my two methods is better all things considered. They’re close enough that I think the correct decision depends on how much you care about protecting yourself vs source control. If source control is a minor consideration to you, I’d go with the KN95 valve coverage method, otherwise the tape method.

(As I said in a previous footnote2, if a valveless elastomeric mask is widely available by the time you read this, that is absolutely a superior option to the hacks I have developed.)

(The Narwall Mask is a commercial solution based on a snorkel mask that may be appealing if you don’t mind (i) the lack of NIOSH-approval and (ii) buying from a random startup, and (iii) you don’t mind or even prefer the full-facepiece design.)

  1. Or is it fatal? I had always assumed it was a fatal flaw, until I found some experts arguing otherwise. In this commentary, the authors say: “Data characterizing particle release through exhalation valves are presently lacking; it is our opinion that such release will be limited by the complex path particles must navigate through a valve. We expect that fewer respiratory aerosols escape through the exhalation valve than through and around surgical masks, unrated masks, or cloth face coverings, all of which have much less efficient filters and do not fit closely to the face”.

    I have been able to find some data; this recent NIOSH study finds that valved N95s have 1-40% penetration. “some models … had less than 20% penetration even without any mitigation. Other models … had much greater penetration with a median penetration above 40%.” Note that for these tests, the flow rates of 25-85 L/min are higher than the 6 L/min of a person at rest, and that lower flow rates had lower penetration.

    Penetration rates of tens of percent are not very good, and not acceptable for my standards, but it’s less bad than I expected, perhaps competitive with surgical masks, and better than cloth masks!

    Niosh 

  2. In fact, as of November 25 2020, the company MSA Safety announced in a press release that the first elastomeric respirator without an exhalation valve has been approved by NIOSH. It’s called the Advantage 290 Respirator. The product page has some good documentation.

    This journal article from September 2020, although it does not mention MSA, appears to be about the Advantage 290. (This is based on the picture in Fig 1. resembling the picture in the press release, and the fact that the hospitals in the paper are in Pennsylvania and New York states, while MSA is headquartered in Pennsylvania). The article explains how it was rolled out to thousands of healthcare workers (a first wave had 1,840 users). They claim that the cost was “approximately $20 for an elastomeric mask and $10 per cartridge”, which is amazingly low.

    They write: “After more than 1 month of usage, we have found that filters have not needed to be changed more frequently than once a month”.

    Unfortunately, it seems to be difficult to get one’s hands on one of these right now. The website invites you to contact sales, and the lowest option for “your budget” is “less than $9,999”.

    Moreover, even if you were able to get the Advantage 290, it might be too selfish to do so, since this respirator is likely to otherwise be used by healthcare workers. On the other hand, the price signal you create would in expectation lead to greater quantities being produced, partially offsetting the effect. If you are able to get one by paying a large premium over the hospital price, this may even be net positive for others.

    If this respirator became available in large quantities, everything I say here would be obsolete.

    By the way, I am astonished that it took until this November 2020 for a PPE company to create a valveless elastomeric respirator, this seems to be a very useful product for any infectious disease situation.  2 3

  3. It’s unclear to me how much one should downweight this recommendation due to appearing on a CDC blog rather than as more formal CDC guidance. In the post, the recommendations are called “tips”. 

  4. The FDA says: “While a surgical mask may be effective in blocking splashes and large-particle droplets, a face mask, by design, does not filter or block very small particles in the air that may be transmitted by coughs, sneezes, or certain medical procedures.” 

  5. 3M claims that “85 liters per minute (lpm) represents a very high work rate, equivalent to the breathing rate of an individual running at 10 miles an hour”. These lecture notes say that a person has a pulmonary ventilation of 6 L/min at rest, 75 L/min during moderate exercise, and 150L/min during vigorous exercise. 

  6. I tried two other methods before I settled on using tape: gluing a thin silicon wafer over the valve on the inside of the mask, and applying glue to the valve directly. Both these methods are entirely inferior and should not be used. 

  7. The model number is ML00895 for the M/L size, and ML00894 for the S/M size. 

January 2, 2021

Efficient validity checking in monadic predicate logic

Monadic predicate logic (with identity) is decidable. (See Boolos, Burgess, and Jeffrey 2007, Ch. 21. The result goes back to Löwenheim-Skolem 1915).

How can we write a program to check whether a formula is logically valid (and hence also a theorem)?

First, we have to parse the formula, meaning to convert it form a string into a format that represents its syntax in a machine-readable way. That format is an abstract syntax tree like this:

Formula:
∀x(Ax→(Ax∧Bx))

Abstract syntax tree:
∀
├── x
└── →
    ├── A
    │   └── x
    └── ∧
        ├── A
        │   └── x
        └── B
            └── x

Writing the parser was a fun lesson in a fundamental aspect of computer science. But there was nothing novel about this exercise, and not much interesting to say about it.

The focus of this post, instead, is the part of the program that actually checks whether this syntax tree represents a logically valid formula.

To start with, we might try to evaluate the formula under every possible model of a given size. How big does the model need to be?

We can make use of the Löwenheim-Skolem theorem (looking first at the case without identity):

If a sentence of monadic predicate logic (without identity) is satisfiable, then it has a model of size no greater than \(2^k\), where \(k\) is the number of predicates in the sentence. (Lemma 21.8 BBJ).

A sentence’s negation is satisfiable if and only if the sentence is not valid, so the theorem equivalently states: a sentence is valid iff it is true under every model of size no greater than \(2^k\).

For a sentence with \(k\) predicates, every constant \(c\) in the model is assigned a list of \(k\) truth-values, representing for each predicate \(P\) whether \(P(c)\). We can use itertools to find every possible such list, i.e. every possible assignment to a constant.

>>> k = 2
>>> possible_predicate_combinations = [i for i in itertools.product([True,False],repeat=k)]
[(True, True), (True, False), (False, True), (False, False)]

The list of every possible assignment to a constant has a length of \(2^k\).

We can then ask itertools to give us, for a model of size \(m\), every possible combination of \(m\) such lists of possible constant-assignments. We let \(m\) be at most \(2^k\), because of the theorem.

>>> for m in range(1,2**k+1):
>>>     possible_models = [i for i in itertools.product(possible_predicate_combinations,repeat=m)]
>>>     print(len(possible_models),"possible models of size",m)
>>>     for model in possible_models:
>>>         print(list(model))

4 possible models of size 1
[(True, True)]
[(True, False)]
[(False, True)]
[(False, False)]

16 possible models of size 2
[(True, True), (True, True)]
[(True, True), (True, False)]
[(True, True), (False, True)]
[(True, True), (False, False)]
[(True, False), (True, True)]
[(True, False), (True, False)]
[(True, False), (False, True)]
[(True, False), (False, False)]
[(False, True), (True, True)]
[(False, True), (True, False)]
[(False, True), (False, True)]
[(False, True), (False, False)]
[(False, False), (True, True)]
[(False, False), (True, False)]
[(False, False), (False, True)]
[(False, False), (False, False)]

64 possible models of size 3
[(True, True), (True, True), (True, True)]
[(True, True), (True, True), (True, False)]
[(True, True), (True, True), (False, True)]
[(True, True), (True, True), (False, False)]
[(True, True), (True, False), (True, True)]
[(True, True), (True, False), (True, False)]
[(True, True), (True, False), (False, True)]
[(True, True), (True, False), (False, False)]
[(True, True), (False, True), (True, True)]
[(True, True), (False, True), (True, False)]
...

256 possible models of size 4
[(True, True), (True, True), (True, True), (True, True)]
[(True, True), (True, True), (True, True), (True, False)]
[(True, True), (True, True), (True, True), (False, True)]
[(True, True), (True, True), (True, True), (False, False)]
[(True, True), (True, True), (True, False), (True, True)]
[(True, True), (True, True), (True, False), (True, False)]
[(True, True), (True, True), (True, False), (False, True)]
[(True, True), (True, True), (True, False), (False, False)]
[(True, True), (True, True), (False, True), (True, True)]
[(True, True), (True, True), (False, True), (True, False)]
...

What’s unfortunate here is that for our \(k\)-predicate sentence, we will need to check \(\sum_{m=1}^{2^k} (2^k)^m =\frac{2^k ((2^k)^{2^k} - 1)}{2^k - 1}\) models. The sum is very roughly equal to its last term, \((2^k)^{2^k} = 2^{k2^k}\). For \(k=3\), this is a number in the billions, for \(k=4\), it’s a number with 19 zeroes.

So checking every model is computationally impossible in practice. Fortunately, we can do better.

Let’s look back at the Löwenheim-Skolem theorem and try to understand why \(2^k\) appears in it:

If a sentence of monadic predicate logic (without identity) is satisfiable, then it has a model of size no greater than \(2^k\) , where \(k\) is the number of predicates in the sentence. (Lemma 21.8 BBJ).

As we’ve seen, \(2^k\) is the number of possible combinations of predicates that can be true of a constant in the domain. Visually, this is the number of subsets in a partition of the possibility space:

If a model had a size of, say, \(2^k + 1\), one of the subsets in the partition would need to contain more than one element. But this additional element would be superfluous insofar as the truth-value of the sentence is concerned. The partition subset corresponds to a predicate-combination that would already be true with just one element in the subset, and will continue to be true if more elements are added. Take, for example, the subset labeled ‘8’ in the drawing, which corresponds to \(R \land \neg Q \land \neg P\). The sentence \(\exists x R(x) \land \neg Q(x) \land \neg P(x)\) is true whether there are one, two, or a million elements in subset 8. Similarly, \(\forall x R(x) \land \neg Q(x) \land \neg P(x)\) does not depend on the number of elements in subset 8.

Seeing this not only illuminates the theorem, but also let us see that the vast majority of the multitudinous \(\sum_{m=1}^{2^k} (2^k)^m\) models we considered earlier are equivalent. All that matters for our sentence’s truth-value is whether each of the subsets is empty or non-empty. This means there are in fact only \(2^{(2^k)}-1\) model equivalence classes to consider. We need to subtract one because the subsets cannot all be empty, since the domain needs to be non-empty.

>>> k = 2
>>> eq_classes = [i for i in itertools.product(['Empty','Non-empty'],repeat=2**k)]
>>> eq_classes.remove(('Empty',)*k**2)
>>> eq_classes
[('Empty', 'Empty', 'Empty', 'Non-empty'),
 ('Empty', 'Empty', 'Non-empty', 'Empty'),
 ('Empty', 'Empty', 'Non-empty', 'Non-empty'),
 ('Empty', 'Non-empty', 'Empty', 'Empty'),
 ('Empty', 'Non-empty', 'Empty', 'Non-empty'),
 ('Empty', 'Non-empty', 'Non-empty', 'Empty'),
 ('Empty', 'Non-empty', 'Non-empty', 'Non-empty'),
 ('Non-empty', 'Empty', 'Empty', 'Empty'),
 ('Non-empty', 'Empty', 'Empty', 'Non-empty'),
 ('Non-empty', 'Empty', 'Non-empty', 'Empty'),
 ('Non-empty', 'Empty', 'Non-empty', 'Non-empty'),
 ('Non-empty', 'Non-empty', 'Empty', 'Empty'),
 ('Non-empty', 'Non-empty', 'Empty', 'Non-empty'),
 ('Non-empty', 'Non-empty', 'Non-empty', 'Empty'),
 ('Non-empty', 'Non-empty', 'Non-empty', 'Non-empty')]

We are now ready to consider the extension to monadic predicate logic with identity. With identity, it’s possible to check whether any two members of a model are distinct or identical. This means we can distinguish the case where a partition subset contains one element from the case where it contains several. But we can still only distinguish up to a certain number of elements in a subset. That number is bounded above by the number of variables in the sentence1 (e.g. if you only have two variables \(x\) and \(y\), it’s not possible to construct a sentence that asserts there are three different things in some subset). Indeed we have:

If a sentence of monadic predicate logic with identity is satisfiable, then it has a model of size no greater than \(2^k \times r\), where \(k\) is the number of monadic predicates and \(r\) the number of variables in the sentence. (Lemma 21.9 BBJ)

By analogous reasoning to the case without identity, we need only consider \((r+1)^{(2^k)}-1\) model equivalence classes. All that matters for our sentence’s truth-value is whether each of the subsets has \(0, 1, 2 ...\) or \(r\) elements in it.

>>> k = 2
>>> r = 2
>>> eq_classes = [i for i in itertools.product(range(r+1),repeat=2**k)]
>>> eq_classes.remove((0,)*k**2)
>>> eq_classes
[(0, 0, 0, 1),
 (0, 0, 0, 2),
 (0, 0, 1, 0),
 (0, 0, 1, 1),
 (0, 0, 1, 2),
 (0, 0, 2, 0),
 (0, 0, 2, 1),
 (0, 0, 2, 2),
 (0, 1, 0, 0),
 (0, 1, 0, 1),
 (0, 1, 0, 2),
 (0, 1, 1, 0),
 (0, 1, 1, 1),
...
  1. I believe it should be possible to find a tighter bound based on the number of times the equals sign actually appears in the sentence. For example, if equality is only used once, e.g. in \(\exists x \exists y \neg(x =y) \land \phi\) where \(\phi\) does not contain equality, it seems clear that the number of variables in \(\phi\) should have no bearing on the model size that is needed. My hunch is that more generally you need \(n*(n-1)/2\) uses of ‘\(=\)’ to assert that \(n\) objects are distinct, so, for example if ‘\(=\)’ appears 5 times you can distinguish 3 objects in a subset, or with 12 ‘\(=\)’s you can distinguish 5 objects. It’s only an intuition and I haven’t checked it carefully. 

November 27, 2020

Protecting yourself from Vanguard's poor security practices

The index fund company Vanguard supports two-factor authentication (2fa) with SMS. SMS is known to be the worst form of 2fa, because it is vulnerable to so-called SIM-swapping attacks. In this type of attack, the malicious party impersonates you and tells your telephone company you’ve lost your SIM card. They request that your number be moved to a new SIM card that they possess. The attacker could call up the company and ask them to activate a spare SIM card that they’ve acquired earlier, or they could visit a store and ask to be given a new SIM card for your number. Then they can receive your security codes.

The security of SMS-based 2fa is only as good as your phone operator’s protections against SIM-swapping, meaning probably not very good. The attacker only needs to convince one mall telco shop employee that they’re you, and they can likely try as many times as they want.

Vanguard claims to also support hardware security keys as a second factor. These are widely regarded as the gold standard for 2fa. Not only are they a true piece of hardware that can’t be SIM-swapped, they also ensure you’re protected even if you get fooled by a phishing attempt (by sending a code that is a function of the URL you are on).

So good news, right? No, because Vanguard made the inexplicable decision to force everyone who uses a security key to also keep SMS 2fa enabled as a fallback option. This utterly defeats the point. The attacker can just click ‘lost security key’ and get an SMS code instead. Users who enable the security key feature actually make their account less secure, because it now has two possible attack surfaces instead of one.

People have been complaining about this for years, ever since Vanguard first introduced security keys. On the Bogleheads forum (where intense Vanguard fanatics congregate), this issue was recognized in this thread from 2016, this one from 2017, this one from 2018, and several others. There are plenty of complaints on reddit too. It’s fair to assume some of these people will have contacted Vanguard directly too.

It’s disappointing that a company with over 6 trillion dollars of assets under management offers its clients a security “feature” that makes their accounts less secure.

The workaround I’ve found is to use a Google Voice number to receive SMS 2fa codes (don’t bother with the useless security key). Of course, you must set the Google Voice number not to forward SMS messages to your main phone number, which would defeat the purpose. Then, the messages can only be read by being logged in to the Google account. A Google account can be made into an extremely hardened target. The advanced protection program is available for the sufficiently paranoid.

If you don’t receive the SMS for some reason, you can also receive the authentication code with an automated call to the same number.

You need to have an existing US phone number to create a Google Voice account.

By the way, using Google Voice may not work for all companies that force you to use SMS 2fa. I have verified that it works for Vanguard. This poster claims that “many financial institutions will now only send their 2FA codes to true mobile phone numbers. Google Voice numbers are land lines, with the text messaging function spliced on via a third-party messaging gateway”.

November 25, 2020

Eliciting probability distributions from quantiles

We often have intuitions about the probability distribution of a variable that we would like to translate into a formal specification of a distribution. Transforming our beliefs into a fully specified probability distribution allows us to further manipulate the distribution in useful ways.

For example, you believe that the cost of a medication is a positive number that’s about 10, but with a long right tail: say, a 10% probability of being more than 100. To use this cost estimate in a Monte Carlo simulation, you need to know exactly what distribution to plug in. Or perhaps you have a prior about the effect of creatine on cognitive performance, and you want to formally update that prior using Bayes’ rule when a new study comes out. Or you want to make a forecast about a candidate’s share of the vote and evaluate the accuracy of your forecast using a scoring rule.

In most software, you have to specify a distribution by its parameters, but these parameters are rarely intuitive. The normal distribution’s mean and standard deviation are somewhat intuitive, but this is the exception rather than the rule. The lognormal’s mu and sigma correspond to the mean and standard deviation of the variable’s logarithm, something I personally have no intuitions about. And I don’t expect good results if you ask someone to supply a beta distribution’s alpha and beta shape parameters.

I have built a tool that creates a probability distribution (of a given family) from user-supplied quantiles, sometimes also called percentiles. Quantiles are points on the cumulative distribution function: \((p,x)\) pairs such that \(P(X<x)=p\). To illustrate what quantiles are, we can look at the example distribution below, which has a 50th percentile (or median) of -1 and a 90th percentile of 10.


A cumulative distribution function with a median of -1 and a 90th percentile of 10

The code is on GitHub, and the webapp is here.

Let’s run through some examples of how you can use this tool. At the end, I will discuss how it compares to other probability elicitation software, and why I think it’s a valuable addition.

Traditional distributions

The tool supports the normal and lognormal distributions, and more of the usual distribution families could easily be added. The user supplies the distribution family, along with an arbitrary number of quantiles. If more quantiles are provided than the distribution has parameters (more than two in this case), the system is over-determined. The tool then uses least squares to find the best fit.

This is some example input:

family = 'lognormal'
quantiles = [(0.1,50),(0.5,70),(0.75,100),(0.9,150)]

And the corresponding output:

More than two quantiles provided, using least squares fit

Lognormal distribution
mu 4.313122980928514
sigma 0.409687416531683

quantiles:
0.01 28.79055927521217
0.1 44.17183774344628
0.25 56.64439363937313
0.5 74.67332855521319
0.75 98.44056294458953
0.9 126.2366766332274
0.99 193.67827989071688

Metalog distribution

The feature I am most excited about, however, is the support for a new type of distribution developed specifically for the purposes of flexible elicitation from quantiles, called the meta-logistic distribution. It was first described in Keelin 2016, which puts it at the cutting edge compared to the venerable normal distribution invented by Gauss and Laplace around 1810. The meta-logistic, or metalog for short, does not use traditional parameters. Instead, it can take on as many terms as the user provides quantiles, and adopts whatever shape is needed to fit these quantiles very closely. Closed-form expressions exist for its quantile function (the inverse of the CDF) and for its PDF. This leads to attractive computational properties (see footnote)1.

Keelin explains that

[t]he metalog distributions provide a convenient way to translate CDF data into smooth, continuous, closed-from distribution functions that can be used for real-time feedback to experts about the implications of their probability assessments.

The metalog quantile function is derived by modifying the logistic quantile function,

\[\mu + s \ln{\frac{y}{1-y}} \quad\text{ for } 0 < y < 1\]

by letting \(\mu\) and \(s\) depend on \(y\) instead of being constant.

As Keelin writes, given a systematically increasing \(s\) as one moves from left to right, a right skewed distribution would result. And a systematically decreasing \(\mu\) as one moves from left to right would make the distribution spikier in the middle with correspondingly heavier tails.

By modifying \(s\) and \(\mu\) in ever more complex ways we can make the metalog take on almost any shape. In particular, in most cases the metalog CDF passes through all the provided quantiles exactly2. Moreover, we can specify the metalog to be unbounded, to have arbitrary bounds, or to be semi-bounded above or below.

Instead of thinking about which of several highly constraining distribution families to use, just choose the metalog and let your quantiles speak for themselves. As Keelin says:

one needs a distribution that has flexibility far beyond that of traditional distributions – one that enables “the data to speak for itself” in contrast to imposing unexamined and possibly inappropriate shape constraints on that data.

For example, we can fit an unbounded metalog to the same quantiles as above:

family = 'metalog'
quantiles = [(0.1,50),(0.5,70),(0.75,100),(0.9,150)]
metalog_leftbound = None
metalog_rightbound = None
Meta-logistic distribution

quantiles:
0.01 11.968367580205552
0.1 50.000000000008185
0.25 58.750000000005215
0.5 70.0
0.75 100.00000000000519
0.9 150.00000000002515
0.99 281.7443263650518

The metalog’s actual parameters (as opposed to the user-supplied quantiles) have no simple interpretation and are of no use unless the next piece of software you’re going to use knows what a metalog is. Therefore the program doesn’t return the parameters. Instead, if we want to manipulate this distribution, we can use the expressions of the PDF and CDF that the software provides, or alternatively export a large number of samples into another tool that accepts distributions described by a list of samples (such as the Monte Carlo simulation tool Guesstimate). By default, 5000 samples will be printed; you can copy and paste them.

Approaches to elicitation

How does this tool compare to other approaches for creating subjective belief distributions? Here are the strategies I’ve seen.

Belief intervals

The first approach is to provide a belief interval that is mapped to some fixed quantiles, e.g. a 90% belief interval (between the 0.05 and 0.95 quantile) like on Guesstimate. Metaculus provides a graphical way to input the same data, allowing the user to drag the quantiles across a line under a graph of the PDF. This is the simplest and most user-friendly approach. The tool I built incorporates the belief interval approach while going beyond it in two ways. First, you can provide completely arbitrary quantiles, instead of specifically the 0.05 and 0.95 – or some other belief interval symmetric around 0.5. Second, you can provide more than two quantiles, which allows the user to query intuitive information about more parts of the distribution.


Guesstimate


Metaculus

Drawing

Another option is to draw the PDF on a canvas, in free form, using your mouse. This is the very innovative approach of probability.dev.3


probability.dev

Ought’s elicit

Ought’s elicit lets you provide quantiles like my tool, or equivalently bins with some probability mass in each bin4. The resulting distribution is by default piecewise uniform (the cdf is piecewise linear), but it’s possible to apply smoothing. It has all the features I want, the drawback is that it only supports bounded distributions5.


Elicit

Mixtures

A meta-level approach that can be applied to any of the above is to allow the user to specify a mixture distribution, a weighted average of distributions. For example, 1/3 weight on a normal(5,5) and 2/3 weight on a lognormal(1,0.75). My opinion on mixtures is that they are good if the user is thinking about the event disjunctively; for example, she may be envisioning two possible scenarios, each of which she has a distribution in mind for. But on Metaculus and Foretold my impression is that mixtures are often used to indirectly achieve a single distribution whose rough shape the user had in mind originally.

The future

This is an exciting space with many recent developments. Guesstimate, Metaculus, Elicit and the metalog distribution have all been created in the last 5 years.

  1. For the quantile function expression, see Keelin 2016, definition 1. The fact that this is in closed form means, first, that sampling randomly from the distribution is computationally trivial. We can use the inverse transform method: we take random samples from a uniform distribution over \([0,1]\) and plug them into the quantile function. Second, plotting the CDF for a certain range of probabilities (e.g. from 1% to 99%) is also easy.

    The expression for the PDF is unusual in that it is a function of the cumulative probability \(p \in (0,1)\), instead of a function of values of the random variable. See Keelin 2016, definition 2. As Keelin explains (p. 254), to plot the PDF as is customary we can use the quantile function \(q(p)\) on the horizontal axis and the PDF expression \(f(p)\) on the vertical axis, and vary \(p\) in, for example, \([0.01,0.99]\) to produce the corresponding values on both axes.

    Hence, for (i) querying the quantile function of the fitted metalog, sampling, and plotting the CDF, and (ii) plotting the PDF, everything can be done in closed form.

    To query the CDF, however, numerical equation solving is applied. Since the quantile function is differentiable, Newton’s method can be applied and is fast. (Numerical equation solving is also used to query the PDF as a function of values of the random variable – but I don’t see why one would need densities except for plotting.) 

  2. In most cases, there exists a metalog whose CDF passes through all the provided quantiles exactly. In that case, there exists an expression of the metalog parameters that is in closed form as a function of the quantiles (“\(a = Y^{−1}x\)”, Keelin 2016, p. 253. Keelin denotes the metalog parameters \(a\), the matrix \(Y\) is a simple function of the quantiles’ y-coordinates, and the vector \(x\) contains the quantiles’ x-coordinates. The metalog parameters \(a\) are the numbers that are used to modify the logistic quantile function. This modification is done according to equation 6 on p. 254.)

    If there is no metalog that fits the quantiles exactly (i.e. the expression for \(a\) above does not imply a valid probability distribution), we have to use optimization to find the feasible metalog that fits the quantiles most closely. In this software implementation, “most closely” is defined as minimizing the absolute differences between the quantiles and the CDF (see here for more discussion).

    In my experience, if a small number of quantiles describing a PDF with sharp peaks are provided, the closest feasible metalog fit to the quantiles may not pass through all the quantiles exactly. 

  3. Drawing the PDF instead of the CDF makes it difficult to hit quantiles. But drawing the CDF would probably be less intuitive – I often have the rough shape of the PDF in mind, but I never have intuitions about the rough shape of the CDF. The canvas-based approach also runs into difficulty with the tail of unbounded distributions. Overall I think it’s very cool but I haven’t found it that practical. 

  4. To provide quantiles, simply leave the Min field empty – it defaults to the left bound of the distribution. 

  5. I suspect this is a fundamental problem of the approach of starting with piecewise uniforms and adding smoothing. You need the tails of the CDF to asymptote towards 0 and 1, but it’s hard to find a mathematical function that does this while also (i) having the right probability mass under the tail (ii) stitching onto the piecewise uniforms in a natural way. I’d love to be proven wrong, though; the user interface and user experience on Elicit are really nice. (I’m aware that Elicit allows for ‘open-ended’ distributions, where probability mass can be assigned to an out-of-bounds outcome, but one cannot specify how that mass is distributed inside the out-of-bounds interval(s). So there is no true support for unbounded distributions. The ‘out-of-bounds’ feature exists because Elicit seems to be mainly intended as an add-on to Metaculus, which supports such ‘open-ended’ distributions but no truly unbounded ones.) 

August 28, 2020

Debugging surprising behavior in SciPy numerical integration

I wrote a Python app to apply Bayes’ rule to continuous distributions. It looks like this:

Screenshot

I’m learning a lot about numerical analysis from this project. The basic idea is simple:

def unnormalized_posterior_pdf(x):
    return prior.pdf(x)*likelihood.pdf(x)

# integrate unnormalized_posterior_pdf over the reals
normalization_constant = integrate.quad(unnormalized_posterior_pdf,-np.inf,np.inf)[0]

def posterior_pdf(x):
    return unnormalized_posterior_pdf(x)/normalization_constant

However, when testing my code on complicated distributions, I ran into some interesting puzzles.

A first set of problems was caused by the SciPy numerical integration routines that my program relies on. They were sometimes returning incorrect results or RuntimeErorrs. These problems appeared when the integration routines had to deal with ‘extreme’ values: small normalization constants or large inputs into the cdf function. I eventually learned to hold the integration algorithm’s hand a little bit and show it where to go.

A second set of challenges had to do with how long my program took to run: sometimes 30 seconds to return the percentiles of the posterior distribution. While 30 seconds might be acceptable for someone who desperately needed that bayesian update, I didn’t want my tool to feel like a punch card mainframe. I eventually managed to make the program more than 10 times faster. The tricks I used all followed the same strategy. In order to make it less expensive to repeatedly evaluate the posterior’s cdf by numerical integration, I tried to find ways to make the interval to integrate narrower.

You can follow along with all the tests described in this post using this file, whereas the code doing the calculations for the webapp is here.

Small normalization constants

Alt text

When the prior and likelihood are far apart, the unnormalized posterior takes tiny values.

It turns out that SciPy’s integration routine, integrate.quad, (incidentally, written in actual Fortran!) has trouble integrating such a low-valued pdf.

prior = stats.lognorm(s=.5,scale=math.exp(.5)) # a lognormal(.5,.5) in SciPy notation
likelihood = stats.norm(20,1)

class Posterior_scipyrv(stats.rv_continuous):
    def __init__(self,d1,d2):
        super(Posterior_scipyrv, self).__init__()
        self.d1= d1
        self.d2= d2

        self.normalization_constant = integrate.quad(self.unnormalized_pdf,-np.inf,np.inf)[0]

    def unnormalized_pdf(self,x):
        return self.d1.pdf(x) * self.d2.pdf(x)

    def _pdf(self,x):
        return self.unnormalized_pdf(x)/self.normalization_constant

posterior = Posterior_scipyrv(prior,likelihood)

print('normalization constant:',posterior.normalization_constant)
print("CDF values:")
for i in range(30):
    print(i,posterior.cdf(i))

The cdf converges to… 52,477. This is not so good.

Because the cdf does converge, but to an incorrect value, we can conclude that the normalization constant is to blame. Because the cdf converges to a number greater than 1, posterior.normalization_constant, about 3e-12, is an underestimate of the true value.

If we shift the likelihood distribution just a little bit to the left, to likelihood = stats.norm(18,1), the cdf converges correctly, and we get a normalization constant of about 6e-07. Obviously, the normalization constant should not jump five orders of magnitude from 6e-07 to 3e-12 as a result of this small shift.

The program is not integrating the unnormalized pdf correctly.

Difficulties with integration usually have to do with the shape of the function. If your integrand zig-zags up and down a lot, the algorithm may miss some of the peaks. But here, the shape of the posterior is almost the same whether we use stats.norm(18,1) or stats.norm(20,1)1. So the problem really seems to occur once we are far enough in the tails of the prior that the unnormalized posterior pdf takes values below a certain absolute (rather than relative) threshold. I don’t yet understand why. Perhaps some of the values are becoming too small to be represented with standard floating point numbers.

This seems rather bizarre, but here’s a piece of evidence that really demonstrates that low absolute values are what’s tripping up the integration routine that calculates the normalization constant. We just multiply the unnormalized pdf by 10000 (which will cancel out once we normalize).

def unnormalized_pdf(self,x):
    return 10000*self.d1.pdf(x) * self.d2.pdf(x)

Now the cdf converges to 1 perfectly (??!).

Large inputs into cdf

We take a prior and likelihood that are unproblematically close together:

prior = stats.lognorm(s=.5,scale=math.exp(.5))# a lognormal(.5,.5) in SciPy notation
likelihood = stats.norm(5,1)
posterior = Posterior_scipyrv(prior,likelihood)

for i in range(100):
    print(i,posterior.cdf(i))

At first, the cdf goes to 1 as expected, but suddenly all hell breaks loose and the cdf decreases to some very tiny values:

22 1.0000000000031484
23 1.0000000000095246
24 1.0000000000031442
25 2.4520867144186445e-09
26 2.7186998869943613e-12
27 1.1495658559228458e-15

What’s going on? When asked to integrate the pdf from minus infinity up to some large value like 25, quad doesn’t know where to look for the probability mass. When the upper bound of the integral is in an area with still enough probability mass, like 23 or 24 in this example, quad finds its way to the mass. But if you ask it to find a peak very far away, it fails.

A piece of confirmatory evidence is that if we make the peak spikier and harder to find, by setting the likelihood’s standard deviation to 0.5 instead of 1, the cdf fails earlier:

22 1.000000000000232
23 2.9116983489798973e-12

We need to hold the integration algorithm’s hand and show it where on the real line the peak of the distribution is located. In SciPy’s quad, you can supply the points argument to point out places ‘where local difficulties of the integrand may occur’, but only when the integration interval is finite. The solution I came up with is to split the interval into two halves.

def split_integral(f,splitpoint,integrate_to):
    a,b = -np.inf,np.inf
    if integrate_to < splitpoint:
        # just return the integral normally
        return integrate.quad(f,a,integrate_to)[0]
    else:
        integral_left = integrate.quad(f, a, splitpoint)[0]
        integral_right = integrate.quad(f, splitpoint, integrate_to)[0]
        return integral_left + integral_right

This definitely won’t work for every difficult integral, but should help for many cases where most of the probability mass is not too far from the splitpoint.

For splitpoint, a simple choice is the average of the prior and likelihood’s expected values.

class Posterior_scipyrv(stats.rv_continuous):
    def __init__(self,d1,d2):
        self.splitpoint = (self.d1.expect()+self.d2.expect())/2

We can now override the built-in cdf method, and specify our own method that uses split_integral:

class Posterior_scipyrv(stats.rv_continuous):
    def _cdf(self,x):
        return split_integral(self.pdf,self.splitpoint,x)

Now things run correctly:

22 1.0000000000000198
23 1.0000000000000198
24 1.0000000000000198
25 1.00000000000002
26 1.0000000000000202
...
98 1.0000000000000198
99 1.0000000000000193

Defining support of posterior

So far I’ve only talked about problems that cause the program to return the wrong answer. This section is about a problem that only causes inefficiency, at least when it isn’t combined with other problems.

If you don’t specify the support of a continuous random variable in SciPy, it defaults to the entire real line. This leads to inefficiency when querying quantiles of the distribution. If I want to know the 50th percentile of my distribution, I call ppf(0.5). As I described previously, ppf works by numerically solving the equation \(cdf(x)=0.5\). The ppf method automatically passes the support of the distribution into the equation solver and tells it to only look for solutions inside the support. When a distribution’s support is a subset of the reals, searching over the entire reals is inefficient.

To remedy this, we can define the support of the posterior as the intersection of the prior and likelihood’s support. For this we need a small function that calculates the intersection of two intervals.

def intersect_intervals(two_tuples):
    d1 , d2 = two_tuples

    d1_left,d1_right = d1[0],d1[1]
    d2_left,d2_right = d2[0],d2[1]

    if d1_right < d2_left or d2_right < d2_left:
        raise ValueError("the distributions have no overlap")
    
    intersect_left,intersect_right = max(d1_left,d2_left),min(d1_right,d2_right)

    return intersect_left,intersect_right

We can then call this function:

class Posterior_scipyrv(stats.rv_continuous):
    def __init__(self,d1,d2):
        super(Posterior_scipyrv, self).__init__()
        a1, b1 = d1.support()
        a2, b2 = d2.support()

        # 'a' and 'b' are scipy's names for the bounds of the support
        self.a , self.b = intersect_intervals([(a1,b1),(a2,b2)])

To test this, let’s use a beta distribution, which is defined on \([0,1]\):

prior = stats.beta(1,1)
likelihood = stats.norm(1,3)

We know that the posterior will also be defined on \([0,1]\). By defining the support of the posterior inside the the __init__ method of Posterior_scipyrv, we give SciPy access to this information.

We can time the resulting speedup in calculating posterior.ppf(0.99):

print("support:",posterior.support())
s = time.time()
print("result:",posterior.ppf(0.99))
e = time.time()
print(e-s,'seconds to evalute ppf')
support: (-inf, inf)
result: 0.9901821216897447
3.8804399967193604 seconds to evalute ppf

support: (0.0, 1.0)
result: 0.9901821216904315
0.40013647079467773 seconds to evalute ppf

We’re able to achieve an almost 10x speedup, with very meaningful impact on user experience. For less extreme quantiles, like posterior.ppf(0.5), I still get a 2x speedup.

The lack of properly defined support causes only inefficiency if we continue to use split_integral to calculate the cdf. But if we leave the cdf problem unaddressed, it can combine with the too-wide support to produce outright errors.

For example, suppose we use a beta distribution again for the prior, but we don’t use the split integral for the cdf, and nor do we define the support of the posterior as \([0,1]\) instead of \({\rm I\!R}\).

prior = stats.beta(1,1)
likelihood = stats.norm(1,3)

class Posterior_scipyrv(stats.rv_continuous):
    def __init__(self,d1,d2):
        super(Posterior_scipyrv, self).__init__()
        self.d1= d1
        self.d2= d2

        self.normalization_constant = integrate.quad(self.unnormalized_pdf,-np.inf,np.inf)[0]
    
    def unnormalized_pdf(self,x):
        return self.d1.pdf(x) * self.d2.pdf(x)

    def _pdf(self,x):
        return self.unnormalized_pdf(x)/self.normalization_constant

posterior = Posterior_scipyrv(prior,likelihood)

print("cdf values:")
for i in range(20):
    print(i/5,posterior.cdf(i/5))

The cdf fails quickly now:

3.2 0.9999999999850296
3.4 0.0
3.6 0.0

When the integration algorithm is looking over all of \((-\infty,3.4]\), it has no way of knowing that all the probability mass is in \([0,1]\). The posterior distribution has only one big bump in the middle, so it’s not surprising that the algorithm misses it.

If we now ask the equation solver in ppf to find quantiles, without telling it that all the solutions are in \([0,1]\), it will try to evaluate points like cdf(4), which return 0 – but ppf is assuming that the cdf is increasing. This leads to catastrophe. Running posterior.ppf(0.5) gives a RuntimeError: Failed to converge after 100 iterations. At first I wondered why beta distributions would always give me RuntimeErrors…

Optimization: CDF memoization

When we call ppf, the equation solver calls cdf for the same distribution many times. This suggests we could optimize things further by storing known cdf values, and only doing the integration from the closest known value to the desired value. This will result in the same number of integration calls, but each will be over a smaller interval (except the first). This is a form of memoization.

We can also squeeze out some additional speedup by considering the cdf to be 1 forevermore once it reaches values close to 1.

class Posterior_scipyrv(stats.rv_continuous):
    def _cdf(self,x):
        # exploit considering the cdf to be 1
        # forevermore once it reaches values close to 1
        for x_lookup in self.cdf_lookup:
            if x_lookup < x and np.around(self.cdf_lookup[x_lookup],5)==1.0:
                return 1

        # check lookup table for largest integral already computed below x
        sortedkeys = sorted(self.cdf_lookup ,reverse=True)
        for key in sortedkeys:
            #find the greatest key less than x
            if key<x:
                ret = self.cdf_lookup[key]+integrate.quad(self.pdf,key,x)[0]
                self.cdf_lookup[float(x)] = ret
                return ret
        
        # Initial run
        ret = split_integral(self.pdf,self.splitpoint,x)
        self.cdf_lookup[float(x)] = ret
        return ret

If we return to our earlier prior and likelihood

prior = stats.lognorm(s=.5,scale=math.exp(.5)) # a lognormal(.5,.5) in SciPy notation
likelihood = stats.norm(5,1)

and make calls to ppf([0.1, 0.9, 0.25, 0.75, 0.5]), the memoization gives us about a 5x speedup:

memoization False
[2.63571613 5.18538207 3.21825988 4.56703016 3.88645864]
length of lookup table: 0
2.1609253883361816 seconds to evalute ppf

memoization True
[2.63571613 5.18538207 3.21825988 4.56703016 3.88645864]
length of lookup table: 50
0.4501194953918457 seconds to evalute ppf

These speed gains again occur over a range that makes quite a difference to user experience: going from multiple seconds to a fraction of a second.

Optimization: ppf with bounds

In my webapp, I give the user some standard percentiles: 0.1, 0.25, 0.5, 0.75, 0.9.

Given that ppf works by numerical equation solving on the cdf, if we give the solver a smaller domain in which to look for the solutions, it should find them more quickly. When we calculate multiple percentiles, each percentile we calculate helps us close in on the others. If the 0.1 percentile is 12, we have a lower bound of 12 for on any percentile \(p>0.1\). If we have already calculated a percentile on each side, we have both a lower and upper bound.

We can’t directly pass the bounds to ppf, so we have to wrap the method, which is found here in the source code. (To help us focus, I give a simplified presentation below that cuts out some code designed to deal with unbounded supports. The code below will not run correctly).

class Posterior_scipyrv(stats.rv_continuous):
    def ppf_with_bounds(self, q, leftbound, rightbound):
        left, right = self._get_support()

        # SciPy ppf code to deal with case where left or right are infinite.
        # Omitted for simplicity.

        if leftbound is not None:
          left = leftbound
        if rightbound is not None:
          right = rightbound


        # brentq is the equation solver (from Brent 1973)
        # _ppf_to_solve is simply cdf(x)-q, since brentq
        # finds points where a function equals 0
        return optimize.brentq(self._ppf_to_solve,left, right, args=q)

To get some bounds, we run the extreme percentiles first, narrowing in on the middle percentiles from both sides. For example in 0.1, 0.25, 0.5, 0.75, 0.9, we want to evaluate them in this order: 0.1, 0.9, 0.25, 0.75, 0.5. We store each of the answers in result.

class Posterior_scipyrv(stats.rv_continuous):
    def compute_percentiles(self, percentiles_list):
        result = {}
        percentiles_list.sort()

        # put percentiles in the order they should be computed
        percentiles_reordered = sum(zip(percentiles_list,reversed(percentiles_list)), ())[:len(percentiles_list)] # see https://stackoverflow.com/a/17436999/8010877

        def get_bounds(dict, p):
            # get bounds (if any) from already computed `result`s
            keys = list(dict.keys())
            keys.append(p)
            keys.sort()
            i = keys.index(p)
            if i != 0:
                leftbound = dict[keys[i - 1]]
            else:
                leftbound = None
            if i != len(keys) - 1:
                rightbound = dict[keys[i + 1]]
            else:
                rightbound = None
            return leftbound, rightbound

        for p in percentiles_reordered:
            leftbound , rightbound = get_bounds(result,p)
            res = self.ppf_with_bounds(p,leftbound,rightbound)
            result[p] = np.around(res,2)

        sorted_result = {key:value for key,value in sorted(result.items())}
        return sorted_result

The speedup is relatively minor when calculating just 5 percentiles.

Using ppf bounds? True
total time to compute percentiles: 3.1997928619384766 seconds

Using ppf bounds? False
total time to compute percentiles: 3.306936264038086 seconds

It grows a little bit with the number of percentiles, but calculating a large number of percentiles would just lead to information overload for the user.

This was surprising to me. Using the bounds dramatically cuts the width of the interval for equation solving, but leads to only a minor speedup. Using fulloutput=True in optimize.brentq, we can see the number of function evaluations that brentq uses. This lets us see that the number of evaluations needed by brentq is highly non-linear in the width of the interval. The solver gets quite close to the solution very quickly, so giving it a narrow interval hardly helps.

Using ppf bounds? True
brentq looked between 0.0 10.0 and took 11 iterations
brentq looked between 0.52 10.0 and took 13 iterations
brentq looked between 0.52 2.24 and took 8 iterations
brentq looked between 0.81 2.24 and took 9 iterations
brentq looked between 0.81 1.73 and took 7 iterations
total time to compute percentiles: 3.1997928619384766 seconds

Using ppf bounds? False
brentq looked between 0.0 10.0 and took 11 iterations
brentq looked between 0.0 10.0 and took 10 iterations
brentq looked between 0.0 10.0 and took 10 iterations
brentq looked between 0.0 10.0 and took 10 iterations
brentq looked between 0.0 10.0 and took 9 iterations
total time to compute percentiles: 3.306936264038086 seconds

Brent’s method is a very efficient equation solver.

  1. It has a very similar shape to the likelihood (because the likelihood has much lower variance than the prior). 

July 1, 2020