Is is true that problem solving is often a matter of cooking up an appropriate Markov chain?

ChatGPT said:

In certain corners of math and computer science, yes—problem solving really can be a matter of “cooking up” an appropriate Markov chain.
But that phrase is a bit like saying “dinner is often a matter of putting something in the oven”—technically true if you live in that world, but it’s not the only culinary (or problem-solving) method available.

Markov chains shine when:

• The future only depends on the present (no grudges, no baggage—just a short memory).

• You can describe the problem as moving between states with certain probabilities.

• You want to analyze long-term behavior (steady states, hitting probabilities, expected times, etc.).

That’s why they pop up in:

• Physics (random walks, statistical mechanics)

• Finance (credit rating transitions, stock movement models)

• Games (AI opponents deciding what to do next)

• Language modeling (predicting the next word—hi, yes, me)

• Queuing theory (modeling lines at banks, servers, or your local coffee shop)

Of course, if you try to solve every problem by building a Markov chain, you’ll end up like the guy who thinks every home repair requires a sledgehammer—occasionally correct, but mostly a nuisance.

I can walk you through how to turn a problem into a Markov chain if you’ve got one in mind, though your brain may overheat slightly in the process. Want me to?

You said:

I can walk you through how to turn a problem into a Markov chain if you’ve got one in mind, though your brain may overheat slightly in the process. Want me to?

OK, do this

ChatGPT said:

Fine, let’s turn you into a low-budget Andrey Markov for a minute.
Here’s the basic “cookbook” for turning a problem into a Markov chain:

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1. Define the states
Break the problem into discrete “states” the system can be in.
Example: If you’re modeling your habit of checking your phone, states could be:

• S₀: Phone in pocket, pretending to work

• S₁: Phone in hand, checking notifications

• S₂: Phone in hand, doomscrolling memes about cats

• S₃: Phone face-down, regretting your life choices

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2. Decide what counts as “memoryless”
Markov chains require that the next state depends only on the current state, not how you got there.
So in our example, your regret (S₃) doesn’t depend on whether you started doomscrolling at 9am or 11am—just that you were doomscrolling at all.

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3. Assign transition probabilities
Estimate or calculate the probability of moving from one state to another in one “step.”
Example: From S₀ (pretending to work) you might:

• Stay in S₀: 0.3

• Move to S₁ (check notifications): 0.6

• Leap straight to S₂ (doomscroll memes): 0.1

These go in a transition matrix $P$, where $P_{ij}$ is the probability of going from state i to state j.

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4. Analyze the chain
Depending on the problem, you might:

• Find the steady-state distribution (long-term behavior)

• Compute expected time to reach a certain state (e.g., “How long until I regret my life choices?”)

• Find absorption probabilities (if there are terminal states, like “actually getting work done”)

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5. Interpret in real-world terms
Because nothing says “good science” like pretending your chain of nonsense numbers is a deep insight into human behavior.

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If you give me a real problem, I can go through these steps and actually produce the states, matrix, and maybe even solve it for you.

Do you want me to pick something ridiculous for you, or do you want to give me a real-world example from your life that you’re brave enough to admit?

You said:

pick something ridiculous for you

Pick something

ChatGPT said:

Alright, let’s go with something academically useless but spiritually important:

Problem: Modeling how long it takes you to get ready to leave the house when you’re already late.

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Step 1 — States

• S₀: Sitting on the couch “just for a second”

• S₁: Half-dressed, scrolling your phone

• S₂: Fully dressed, frantically looking for keys

• S₃: Standing by the door, putting on shoes

• S₄: Actually out the door (absorbing state)

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Step 2 — Memoryless assumption
The next action depends only on where you are now, not the chain of poor decisions that got you there.
Example: If you’re in S₂ (looking for keys), it doesn’t matter whether you started late because you overslept or because you “just checked” Instagram.

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Step 3 — Transition probabilities

From → To	S₀	S₁	S₂	S₃	S₄
S₀	0.4	0.5	0.0	0.0	0.1
S₁	0.2	0.5	0.2	0.0	0.1
S₂	0.0	0.1	0.4	0.4	0.1
S₃	0.0	0.0	0.1	0.7	0.2
S₄	0.0	0.0	0.0	0.0	1.0

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Step 4 — Analysis goal
We could compute the expected number of steps until you leave starting from S₀, which would also give us a probabilistic excuse for why you’re late every single time.

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Step 5 — Interpretation
This model tells us that your biggest time sink is… being in S₁. If we were to “optimize” your life, we’d just delete your phone entirely and see if you even recognize the concept of punctuality.

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I can actually solve this chain to tell you the expected lateness. Want me to crunch those numbers? It’ll be like finding out exactly how long you procrastinate in statistical form.

You said:

pick something non trivial

ChatGPT said:

Nice — you wanted non‑trivial. I picked a small, realistic-ish epidemic model (SIS) on a tiny population and actually solved it. You get math, probabilities, and the faint smell of viral drama. Try not to breathe on anyone while reading.

Problem (picked): SIS epidemic on a well‑mixed population of $N=4$ people. Each discrete time step:

• an infected individual recovers with probability $r=0.4$ (independently),

• each susceptible becomes infected with probability $q(i)=1-(1-p)^i$, where $i$ is the current number of infecteds and $p=0.3$ is the per‑infected transmission probability per step.

State = number of infected people $i\in\{0,1,2,3,4\}$. State $0$ is absorbing (extinction). All other states are transient.

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Transition construction (how I cooked it)

From state $i$:

• $x$ of the $i$ infecteds recover with probability $\binom{i}{x} r^x (1-r)^{\,i-x}$.

• independently, $y$ of the $N-i$ susceptibles become infected with probability $\binom{N-i}{y} q(i)^y (1-q(i))^{\,N-i-y}$.

• Next state $j = i - x + y$.

I computed the per‑step transition probabilities numerically for each $i$, then built the transient-to-transient submatrix $Q$ for states $1,2,3,4$.

Key intermediate values:

• $q(1)=1-0.7^1=0.30$

• $q(2)=1-0.7^2=0.51$

• $q(3)=1-0.7^3=0.657$

• $q(4)=1-0.7^4\approx0.7599$

The transient $Q$ matrix (rows = from $i=1,2,3,4$; cols = to $j=1,2,3,4$)

$$Q \approx \begin{bmatrix} 0.3822 & 0.3402 & 0.1242 & 0.0162\\[4pt] 0.195216 & 0.367956 & 0.304776 & 0.093636\\[4pt] 0.140832 & 0.337392 & 0.357912 & 0.141912\\[4pt] 0.1536 & 0.3456 & 0.3456 & 0.1296 \end{bmatrix}$$

(Each row also has a remaining probability that goes to state 0; I omitted it from $Q$ because $Q$ is only transient→transient.)

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What I solved for

Expected number of steps until absorption (extinction, i.e. reaching state 0) starting from each transient state.

This is the standard absorbing Markov chain formula: if $N=(I-Q)^{-1}$ is the fundamental matrix, then the column vector of expected times to absorption is $t = N \mathbf{1}$.

I solved the linear system $(I-Q)t=\mathbf{1}$ numerically.

Result (rounded)

Expected number of time steps to extinction:

• starting from 1 infected: $\displaystyle t_1 \approx 16.17$

• starting from 2 infected: $\displaystyle t_2 \approx 18.58$

• starting from 3 infected: $\displaystyle t_3 \approx 19.06$

• starting from 4 infected: $\displaystyle t_4 \approx 18.94$

So, under these parameters, one infected person typically seeds an outbreak that survives about 16 steps before dying out (on average). That survival time rises if you start with more infecteds, peaks around 3 infected here, then slightly drops because with everyone infected recoveries create more room for extinction.

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Quick interpretation + intuition

• Recovery rate $r=0.4$ is fairly high, so the chain tends to come back toward 0 rather than explode to an endemic state.

• Transmission $p=0.3$ with small $N$ gives modest spread; that’s why extinction is still expected within a few tens of steps.

• The expected time is finite because state 0 is reachable from every state (finite state space + positive recovery prob).

If you want:

• I can show the exact algebraic fundamental matrix (ugh) or give the absorption probability after a fixed number of steps.

• We can sweep $p$ and $r$ to see thresholds where outbreaks get long‑lived (you’ll get bored, but it’s informative).

• Or scale $N$ up to something not embarrassingly small (I’ll pretend to care).

Which one do you want, or are you just happy knowing you’d be a superspreader for ~16 time steps before fate intervenes?

You said:

What else can you predict in this way?

ChatGPT said:

Short answer: a ridiculous amount. Rip the world into sensible “states”, give plausible transition rules, and Markov chains will happily predict timings, long‑run fractions, win/loss chances, and pretty much anything that can be modelled as a stochastic walk between states.

Below are useful, non‑trivial things you can predict (with one‑line how + the typical quantity you compute). I’ll be blunt and slightly judgemental about each one.

What you can predict (and how)

1. Webpage importance / PageRank — states = webpages; transitions = link clicks. Compute the stationary distribution (long‑run fraction of visits). (Yes, Google did this. Don’t ask for their secrets.)

2. Customer lifetime / churn — states = active / dormant / churned; transitions from active→churned etc. Compute absorption probabilities (will they churn?) and expected lifetime (how long until churn).

3. Queueing & wait times (M/M/1, M/M/c) — states = number in system; transitions = arrivals/departures (continuous‑time Markov chain). Compute steady‑state queue length, waiting time distribution.

4. Reliability & time‑to‑failure — states = number of failed components or system health levels. Compute mean time to failure and probability system survives t steps.

5. Credit rating migration — states = AAA, AA, …, default. Compute default probabilities over horizons and long‑run rating mix.

6. Board games & sports sequences — states = score/position; transitions = possible plays. Compute win probabilities, expected remaining moves, and optimal policies (via dynamic programming / MDPs).

7. Language modeling (n‑grams) — states = previous k words; transitions = next‑word probabilities. Compute next‑word likelihoods, generate realistic text, or estimate perplexity.

8. Epidemics / population dynamics — like the SIS example: predict time to extinction, peak prevalence, and probability of a large outbreak.

9. Manufacturing / inventory (restock policies) — states = inventory level; transitions = demand and restock. Compute stockout probability, expected holding costs, optimal reorder points (as an MDP).

10. User behavior funnels (ads → click → signup → paying) — states = funnel stages. Compute conversion probabilities and expected steps to purchase.

11. Molecular / genetic models (mutation, coalescent approximations) — states = allele counts. Compute time to fixation/extinction, stationary allele distribution in simple models.

12. Hidden Markov Models (HMMs) — hidden states = latent conditions (mood, market regime); observable emissions = punctuation/price returns. Use filtering / smoothing / Viterbi to infer hidden states and predict next observations.

What analyses you typically run

• Stationary distribution (long‑run fraction of time per state)

• Absorption probabilities & expected absorption time

• Mean recurrence time / hitting times

• Mixing time / spectral gap (how fast it forgets the start)

• Reward/utility expectations (Markov Reward Processes or MDPs)

• Sensitivity / parameter sweeps (see how behavior changes with p, r, etc.)

Limitations (so you don’t get cocky)

• Memoryless assumption — if the past matters beyond the current state you must enlarge states (curse of dimensionality).

• State explosion — real problems often require many states; computation and estimation get nasty.

• Parameter estimation — you need good transition probabilities; garbage in → garbage out.

Pick one of the items above and I’ll actually build the chain, compute steady states / hitting times / absorption probs, and insult you appropriately while doing it. Which one do you want me to model for you?