The Naive Bayes classifier is one of the simplest algorithms in machine learning, yet it’s surprisingly powerful.
It answers the question:
“Given some evidence, what is the most likely class?”
It’s naive because it assumes that features are conditionally independent given the class. That assumption rarely
holds in the real world — but the algorithm still works remarkably well for many tasks such as spam filtering, document
classification, and sentiment analysis.
At its core, Naive Bayes is just counting, multiplying probabilities, and picking the largest one.
Bayes’ Rule Refresher
First, let’s start with a quick definition of terms.
Class is the label that we’re trying to predict. In our example below, the class will be either “spam” or “ham”
(not spam).
The features are the observed pieces of evidence. For text, features are usually the words in a message.
P is shorthand for “probability”.
P(Class) = the prior probability: how likely a class is before seeing any features.
P(Features | Class) = the likelihood: how likely it is to see those words if the class is true.
P(Features) = the evidence: how likely the features are overall, across the classes. This acts as a normalising constant so probabilities sum to 1.
So both classes land on the same score — a perfect tie, in this example.
Python Demo (from scratch)
Here’s a tiny implementation that mirrors the example above:
fromcollectionsimportCounter,defaultdict# Training data
docs=[("spam","buy cheap"),("spam","cheap pills"),("ham","meeting schedule"),("ham","project meeting"),]class_counts=Counter()word_counts=defaultdict(Counter)# Build counts
forlabel,textindocs:class_counts[label]+=1forwordintext.split():word_counts[label][word]+=1defclassify(text,alpha=1.0):words=text.split()scores={}total_docs=sum(class_counts.values())vocab={wforcountsinword_counts.values()forwincounts}V=len(vocab)forlabelinclass_counts:# Prior
score=class_counts[label]/total_docstotal_words=sum(word_counts[label].values())forwordinwords:count=word_counts[label][word]# Laplace smoothing
score*=(count+alpha)/(total_words+alpha*V)scores[label]=score# Pick the class with the highest score
returnmax(scores,key=scores.get),scoresprint(classify("cheap project"))print(classify("project schedule"))print(classify("cheap schedule"))
As we predicted earlier, "cheap project" is a tie, while "project schedule" is more likely ham. Finally, "cheap schedule"
is noted as spam because it uses stronger spam trigger words.
Real-World Notes
Naive Bayes is fast, memory-efficient, and easy to implement.
Works well for text classification, document tagging, and spam filtering.
The independence assumption is rarely true, but it doesn’t matter — it often performs surprisingly well.
In production, you’d tokenize better, remove stop words, and work with thousands of documents.
Conclusion
Building a Naive Bayes classifier from first principles is a great exercise because it shows how machine learning can be
just careful counting and probability. With priors, likelihoods, and a dash of smoothing, you get a surprisingly useful
classifier — all without heavy math or libraries.
A Bloom filter is a tiny, probabilistic memory that answers “Have I seen this before?” in constant time. It never lies
with a false negative—if it says “no”, the item was definitely never added. But to save huge amounts of space versus
storing all items, it allows false positives—sometimes it will say “probably yes” when collisions happen.
The trick is simple: keep a row of bits and, for each item, flip a small handful of positions chosen by hash functions.
Later, check those same positions. Any zero means “definitely not”; all ones means “probably yes.” With just a few
bytes, Bloom filters help databases skip disk lookups, caches dodge misses, and systems answer membership queries
blazingly fast.
Pen & Paper Example
Let’s build the smallest possible Bloom filter: 10 bits, indexed 0–9.
We’ll use two playful “hash” functions:
h1(word) = (sum of letter positions) % 10 (a=1, b=2, …, z=26)
h2(word) = (len(word) * 3) % 10
To insert a word, flip bits h1(word) and h2(word) to 1.
To query, compute the same bits:
If any bit is 0 → definitely not present
If both are 1 → probably present
With these rules in place, we can start to insert some words.
Our toy version used silly hash rules. Real implementations use cryptographic hashes and multiple derived functions.
Here’s a slightly more realistic snippet using double hashing from SHA-256 and MD5:
This implementation will allow you to filter much more complex (and longer) content. The wider your bit field is, and the
more complex your hashing algorithms are, the better bit distribution you will get. This gives you a lower
chance of false positives, improving the overall performance of the data structure.
Conclusion
Bloom filters are elegant because of their simplicity: flip a few bits when adding, check those bits when querying.
They trade absolute certainty for massive savings in memory and time. They’re everywhere—from browsers to databases to
networking—and now, thanks to a handful of cat, dog, and cow, you know how they work.
If you’ve spent any time in Haskell or FP circles, you’ll have run into the terms Functor, Applicative, and
Monad. They can sound mysterious, but at their core they’re just design patterns for sequencing computations.
Python isn’t a purely functional language, but we can still capture these ideas in code. In this post, we’ll build a
full Maybe type in Python: a safe container that represents either a value (Some) or no value (Nothing). We’ll
compare it with the Haskell version along the way.
A full runnable demo of the code presented here is available as a gist up on GitHub.
Maybe
We start of with our box or context. In our case today, we might have a value in the box (Some) or the box
maybe empty (Nothing). Because both of these are derivatives of the same thing, we create a base class of Maybe.
Our Maybe class here defines all of the operations we want to be able to perform on this datatype, but does not
implement any of them; leaving the implementation to be filled in my the derivatives. You can expect the implementations
between these derived classes to be quite different to each other.
We should end up with something like this:
classDiagram
class Maybe {
+map(f)
+ap(mb)
+bind(f)
}
class Some {
+value: T
}
class Nothing
Maybe <|-- Some
Maybe <|-- Nothing
Functor: Mapping over values
A Functor is anything you can map a function over. In Haskell, the generic Functor version of this is called
fmap:
fmap(+1)(Just10)-- Just 11fmap(+1)Nothing-- Nothing
The flow of values through map (or fmap) looks like this:
A Monad takes things further: it lets us chain together computations that themselves return a Maybe.
In Haskell, this is the >>= operator (bind):
halfIfEven::Int->MaybeInthalfIfEvenx=ifevenxthenJust(x`div`2)elseNothingJust10>>=halfIfEven-- Just 5Just3>>=halfIfEven-- Nothing
Here we’re chaining a computation that itself returns a Maybe. If the starting point is Nothing, or if the
function returns Nothing, the whole chain collapses.
flowchart LR
S[Some x] --bind f--> FOUT[Some y]
S --bind g--> GOUT[Nothing]
N[Nothing] --bind f--> NRES[Nothing]
Notice how the “empty box” propagates: if at any point we hit Nothing, the rest of the chain is skipped.
You’ll also see a common pattern emerging with all of the implementations for Nothing. There’s no computation. It’s
simply just returning itself. As soon as you hit Nothing, you’re short-circuited to nothing.
Do Notation (Syntactic Sugar)
Haskell makes monadic code look imperative with do notation:
doa<-Just4b<-halfIfEvenareturn(a+b)
In Python, we can approximate this style using a generator-based decorator. Each yield unwraps a Maybe, and the
whole computation short-circuits if we ever see Nothing.
This isn’t strictly necessary, but it makes larger chains of monadic code read like straight-line Python.
Wrapping Up
By porting Maybe into Python and implementing map, ap, and bind, we’ve seen how Functors, Applicatives, and
Monads aren’t magic at all — just structured patterns for working with values in context.
Functor: apply a function inside the box.
Applicative: apply a function that’s also in a box.
Monad: chain computations that each return a box.
Haskell bakes these ideas into the language; in Python, we can experiment with them explicitly. The result is safer,
more composable code — and maybe even a little functional fun.
Kerberos is one of those protocols that sounds mysterious until you see it in action. The moment you type kinit, run
klist, and watch a ticket pop up, it clicks: this is Single Sign-On in its rawest form. In this post we’ll set up a
tiny realm on a Debian test box (koffing.local), get a ticket-granting ticket (TGT), and then use it for SSH without
typing a password.
What is Kerberos?
Born at MIT’s Project Athena in the 1980s, Kerberos solved campus-wide single sign-on over untrusted networks. It
matured through v4 to Kerberos 5 (the standard you use today). It underpins enterprise SSO in Windows domains
(Active Directory) and many UNIX shops.
Kerberos authenticates clients to services without sending reusable secrets. You authenticate once to the KDC, get
a TGT (Ticket Granting Ticket), then use it to obtain per-service tickets from the TGS
(Ticket Granting Service).
Services trust the KDC, not your password.
Core terms
Realm: Admin boundary (e.g., LOCAL).
Principal: Identity in the realm, like michael@LOCAL (user) or host/koffing.local@LOCAL (service).
KDC: The authentication authority. Runs on koffing.local as krb5kdc and kadmind.
TGT: Your “hall pass.” Lets you ask the KDC for service tickets.
Service ticket: What you present to a service (e.g., SSHD on koffing.local) to prove identity.
Keytab: File holding long-term service keys (like for sshd). Lets the service authenticate without storing a password.
Here’s a visual representation of how the Kerberos flow operates:
sequenceDiagram
participant U as User
participant AS as KDC/AS
participant TGS as KDC/TGS
participant S as Service (e.g., SSHD)
U->>AS: AS-REQ (I am michael)
AS-->>U: AS-REP (TGT + session key)
U->>TGS: TGS-REQ (I want ticket for host/koffing.local)
TGS-->>U: TGS-REP (service ticket)
U->>S: AP-REQ (here's my service ticket)
S-->>U: AP-REP (optional) + access granted
Ok, with all of that out of the way we can get to setting up.
Setup
There’s a few packages to install and a little bit of configuration. All of these instructions are written for a
Debian/Ubuntu flavour of Linux. I’m sure that the instructions aren’t too far off for other distributions.
Install the packages
We install the Key Distribution Servicekrb5-kdc, Administration Serverkrb5-admin-server, and some Client
Utilitieskrb5-user.
The fully qualified name of my virtual machine that I’m testing all of this out on is called koffing.local. These
values would change to suit your environment.
Edit /etc/krb5.conf and make sure it looks like this:
[libdefaults]
default_realm = LOCAL
rdns = false
dns_lookup_kdc = false
forwardable = true
[realms]
LOCAL = {
kdc = koffing.local
admin_server = koffing.local
}
[domain_realm]
.local = LOCAL
koffing.local = LOCAL
Make sure your host resolves correctly:
hostname-f# should print: koffing.local (for me)
getent hosts koffing.local
# If needed, add to /etc/hosts:# 127.0.1.1 koffing.local koffing
Create the KDC database
Now we initialize the database that will hold all of your principals, policies, realms, etc.
sudo mkdir -p /var/lib/krb5kdc
sudo kdb5_util create -s -r LOCAL
# set the KDC master password when prompted
This will show you which hostnames it resolves, which tickets it requests, and where it fails.
List all principals in the KDC database:
sudo kadmin.local -q"listprincs"
Clear your credential cache if tickets get stale:
kdestroy
The two most common pitfalls are:
Hostname mismatch
Realm mismatch (default realm not set in /etc/krb5.conf).
SSO
So, we’ve got the proof of concept going, but it would be good to see this in action. What we’ll cover in this next
section is getting the sshd service to trust our Kerberos tickets. This will allow for passwordless SSH for the
user.
Add the host service principal and keytab
In order to get KDC to vouch for services, those services need principal definitions. A principal is any Kerberos
identity. Users get user principals (as we saw above), services also need principals.
A keytab is a file that stores one or more Kerberos keys (like passwords, but in cryptographic form). Unlike users
(who can type passwords into kinit), services can’t type passwords interactively. So the KDC generates a random key
for host/koffing.local@LOCAL (-randkey) and you export it into /etc/krb5.keytab with ktadd.
Now sshd can silently use that keytab to decrypt tickets clients send it.
Enable GSSAPI in sshd
The global /etc/ssh/sshd_config needs a couple of flags flicked. The SSH daemon doesn’t implement Kerberos directly,
so it uses the GSSAPI library functions provided by MIT Kerberos (or Heimdal) to handle ticket validation. GSSAPI
isn’t a protocol itself; it’s an API or abstraction layer.
Once we’ve flipped these switches we are telling sshd“Accept authentication from any GSSAPI mechanism. In practice, this means Kerberos tickets.”.
This setup is obviously done on any server that you want to do this SSO style login with. It’s a bit confusing in my
example here, because everything is on the one machine.
Configure your SSH client
Conversely, we have configuration to do on the client side. For clients that want to connect with this type of
authentication, the following settings are required in their ~/.ssh/config:
If everything lines up, ssh should not prompt for a password. Your Kerberos TGT has been used to authenticate silently.
Where Kerberos Fits
Kerberos is ideal for LAN-based authentication: it provides fast, passwordless single sign-on for services like SSH,
Postgres, and intranet HTTP apps. But it isn’t designed for cross-organization web or mobile use.
Modern protocols like OIDC (OpenID Connect) build on OAuth 2.0 to provide authentication and federation across the
public internet. They use signed tokens, redirect flows, and JSON-based metadata — making them better suited for SaaS,
cloud apps, and mobile clients.
In short: Kerberos is the right tool inside the castle walls; OIDC is the right tool when your users are everywhere.
Wrap-up
We’ve stood up a Kerberos realm (LOCAL), issued a TGT for a user (michael), and used it for passwordless SSH into
the same box. That’s enough to demystify Kerberos: no secrets flying across the network, just short-lived tickets
granted by a trusted KDC.
There’s plenty more that we can accomplish here as we could create service principals for HTTP, Postgres, or
cross-realm trust.
FreeBSD Jails are one of the earliest implementations of operating
system-level virtualization—dating back to the early 2000s, long before Docker popularized the idea of lightweight
containers. Despite their age, jails remain a powerful, flexible, and minimal way to isolate services and processes on
FreeBSD systems.
This post walks through a minimal “Hello World” setup using Jails, with just enough commentary to orient new users and
show where jails shine in the modern world of virtualization.
Why Jails?
A FreeBSD jail is a chroot-like environment with its own file system, users,
network interfaces, and process table. But unlike chroot, jails extend control to include process isolation, network
access, and fine-grained permission control. They’re more secure, more flexible, and more deeply integrated into the
FreeBSD base system.
Here’s how jails compare with some familiar alternatives:
Versus VMs: Jails don’t emulate hardware or run separate kernels. They’re faster to start, lighter on resources, and simpler to manage. But they’re limited to the same FreeBSD kernel as the host.
Versus Docker: Docker containers typically run on a Linux host and rely on a container runtime, layered filesystems, and extensive tooling. Jails are simpler, arguably more robust, and don’t require external daemons. However, they lack some of the ecosystem and portability benefits that Docker brings.
If you’re already running FreeBSD and want to isolate services or test systems with minimal overhead, jails are a
perfect fit.
Setup
Let’s build a bare-bones jail. The goal here is simplicity: get a jail running with minimal commands. This is the BSD
jail equivalent of “Hello, World.”
# Make a directory to hold the jailmkdir hw
# Install a minimal FreeBSD userland into that directorysudo bsdinstall jail /home/michael/src/jails/hw
# Start the jail with a name, IP address, and a shellsudo jail -cname=hw host.hostname=hw.example.org \
ip4.addr=192.168.1.190 \path=/home/michael/src/jails/hw \command=/bin/sh
You now have a running jail named hw, with a hostname and IP, running a shell isolated from the host system.
192.168.1.190 is just a static address picked arbitrarily by me. For you, you’ll want to pick an address that is
reachable on your local network.
Poking Around
With your jail up and running, that means you can start working with it. To enter the jail, you can use the following:
sudo jexec hw /bin/sh
jexec allows you to send any command that you need to into the jail to execute.
sudo jexec hw ls /
Querying
You can list running jails with:
jls
You should see something like this:
JID IP Address Hostname Path
2 192.168.1.190 hw.example.org /home/michael/src/jails/hw
You can also look at what’s currently running in the jail:
ps -J hw
You should see the /bin/sh process:
PID TT STAT TIME COMMAND
2390 5 I+J 0:00.01 /bin/sh
Finishing up
To terminate the jail:
sudo jail -r hw
This is a minimal setup with no automated networking, no jail management frameworks, and no persistent configuration.
And that’s exactly the point: you can get a working jail in three commands and tear it down just as easily.
When to Use Jails
Jails make sense when:
You want process and network isolation on FreeBSD without the overhead of full VMs.
You want to run multiple versions of a service (e.g., Postgres 13 and 15) on the same host.
You want stronger guarantees than chroot provides for service containment.
You’re building or testing FreeBSD-based systems and want a reproducible sandbox.
For more complex jail setups, FreeBSD offers tools like ezjail, iocage, and bastille that add automation and
persistence. But it’s worth knowing how the pieces fit together at the core.
Conclusion
FreeBSD jails offer a uniquely minimal, powerful, and mature alternative to both VMs and containers. With just a few
commands, you can create a secure, isolated environment for experimentation, testing, or even production workloads.
This post only scratched the surface, but hopefully it’s enough to get you curious. If you’re already on FreeBSD, jails
are just sitting there, waiting to be used—no extra software required.