\documentclass[11pt]{article}
\usepackage{mathnotes}
\discipline{UCLA COMPUTER SCIENCE FIELD NOTES}
\course{CS C121}{Probabilistic Models in Computational Genomics}{Probability, genomic data, sequence models, latent variables, inference, and responsible interpretation}

\begin{document}
\makemathcover
\coursefrontmatter{
\begin{curriculumtable}
\unitline{1}{Genomic data as a probabilistic object}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{2}{Conditional probability and Bayesian genomics}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{3}{Markov chains and sequence dependence}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{4}{Hidden Markov models and dynamic programming}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{5}{Mixtures, latent classes, and expectation maximization}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{6}{Probabilistic alignment and evolutionary models}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{7}{Inference, uncertainty, and numerical stability}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{8}{Model criticism, population structure, and reproducibility}{Explain the model, apply it to a concrete case, and defend the relevant correctness or design decision.}
\unitline{9}{Integration workshop}{Connect at least three chapters in one end-to-end problem and explain where their contracts meet.}
\unitline{10}{Synthesis and project review}{Audit assumptions and explain a complete solution from interface to failure handling.}
\end{curriculumtable}
}{\curriculumsource{https://catalog.registrar.ucla.edu/course/2025/COMSCIC121}}

\section{Genomic data as a probabilistic object}

\begin{idea}
Genomic analysis begins with biological questions, sequence and variant representations, noisy measurement, population sampling, and a model for how observed data were generated. A probability model separates latent biological state from sequencing and sampling error.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Genomic analysis begins with biological questions, sequence and variant representations, noisy measurement, population sampling, and a model for how observed data were generated. A probability model separates latent biological state from sequencing and sampling error. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of genomic data as a probabilistic object. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach genomic data as a probabilistic object on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from the course foundations to conditional probability and bayesian genomics. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Genomic analysis begins with biological questions, sequence and variant representations, noisy measurement, population sampling, and a model for how observed data were generated.
\item \textbf{Mechanism.} A probability model separates latent biological state from sequencing and sampling error.
\item \textbf{Boundary.} The useful test is whether I can apply genomic data as a probabilistic object to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when genomic data as a probabilistic object is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I write down what was observed, what is hidden, and which part of the story is noise before I touch an equation.
\end{studenttake}

\begin{plainbox}
Genomic analysis begins with biological questions, sequence and variant representations, noisy measurement, population sampling, and a model for how observed data were generated.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: Why is a reference genome not a probability model? The conclusion is A reference supplies coordinates and a comparison sequence, but it does not specify variation, measurement error, ancestry, or sampling. A probabilistic model must assign likelihoods to possible observations under explicit latent states.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} Why is a reference genome not a probability model?

\textbf{Solution.} A reference supplies coordinates and a comparison sequence, but it does not specify variation, measurement error, ancestry, or sampling. A probabilistic model must assign likelihoods to possible observations under explicit latent states.

\medskip
\textbf{Practice 2.} Give a short oral explanation of genomic data as a probabilistic object that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct genomic data as a probabilistic object from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a genomic data as a probabilistic object problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Conditional probability and Bayesian genomics}

\begin{idea}
Bayes' rule combines a prior over biological hypotheses with a likelihood for observed genomic data. Conditional independence reduces complicated joint distributions, while posterior odds make prior assumptions and evidence visible.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Bayes' rule combines a prior over biological hypotheses with a likelihood for observed genomic data. Conditional independence reduces complicated joint distributions, while posterior odds make prior assumptions and evidence visible. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of conditional probability and bayesian genomics. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach conditional probability and bayesian genomics on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from genomic data as a probabilistic object to markov chains and sequence dependence. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Bayes' rule combines a prior over biological hypotheses with a likelihood for observed genomic data.
\item \textbf{Mechanism.} Conditional independence reduces complicated joint distributions, while posterior odds make prior assumptions and evidence visible.
\item \textbf{Boundary.} The useful test is whether I can apply conditional probability and bayesian genomics to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when conditional probability and bayesian genomics is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I use Bayes as bookkeeping for assumptions and evidence, not as a formula that makes weak data certain.
\end{studenttake}

\begin{plainbox}
Bayes' rule combines a prior over biological hypotheses with a likelihood for observed genomic data.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: A variant prior is 0.01 and a positive read has sensitivity 0.99 and false-positive rate 0.05. Find the posterior after one positive read. The conclusion is The numerator is 0.99 times 0.01, or 0.0099. Total positive probability is 0.0099 plus 0.05 times 0.99, or 0.0594. The posterior is about 0.1667, showing why a rare prior still matters.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} A variant prior is 0.01 and a positive read has sensitivity 0.99 and false-positive rate 0.05. Find the posterior after one positive read.

\textbf{Solution.} The numerator is 0.99 times 0.01, or 0.0099. Total positive probability is 0.0099 plus 0.05 times 0.99, or 0.0594. The posterior is about 0.1667, showing why a rare prior still matters.

\medskip
\textbf{Practice 2.} Give a short oral explanation of conditional probability and bayesian genomics that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct conditional probability and bayesian genomics from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a conditional probability and bayesian genomics problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Markov chains and sequence dependence}

\begin{idea}
A Markov model factors a sequence through local transition probabilities. Initial distribution, transition matrix, stationarity, order, and state meaning determine which dependence the model can express and which long-range structure it misses.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: A Markov model factors a sequence through local transition probabilities. Initial distribution, transition matrix, stationarity, order, and state meaning determine which dependence the model can express and which long-range structure it misses. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of markov chains and sequence dependence. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach markov chains and sequence dependence on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from conditional probability and bayesian genomics to hidden markov models and dynamic programming. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} A Markov model factors a sequence through local transition probabilities.
\item \textbf{Mechanism.} Initial distribution, transition matrix, stationarity, order, and state meaning determine which dependence the model can express and which long-range structure it misses.
\item \textbf{Boundary.} The useful test is whether I can apply markov chains and sequence dependence to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when markov chains and sequence dependence is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I label each state as a biological claim, not just a matrix row; otherwise a clean computation can answer the wrong question.
\end{studenttake}

\begin{plainbox}
A Markov model factors a sequence through local transition probabilities.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: For a first-order DNA model, what conditional independence is assumed? The conclusion is Given the current base or state, the next base is independent of all earlier bases. The sequence probability factors into an initial probability and one transition probability per adjacent pair.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} For a first-order DNA model, what conditional independence is assumed?

\textbf{Solution.} Given the current base or state, the next base is independent of all earlier bases. The sequence probability factors into an initial probability and one transition probability per adjacent pair.

\medskip
\textbf{Practice 2.} Give a short oral explanation of markov chains and sequence dependence that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct markov chains and sequence dependence from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a markov chains and sequence dependence problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Hidden Markov models and dynamic programming}

\begin{idea}
A hidden Markov model adds latent states with transition probabilities and emissions for observations. Forward computes total likelihood, backward supplies suffix evidence, Viterbi finds the most probable path, and posterior decoding answers a different marginal question.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: A hidden Markov model adds latent states with transition probabilities and emissions for observations. Forward computes total likelihood, backward supplies suffix evidence, Viterbi finds the most probable path, and posterior decoding answers a different marginal question. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of hidden markov models and dynamic programming. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach hidden markov models and dynamic programming on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from markov chains and sequence dependence to mixtures, latent classes, and expectation maximization. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} A hidden Markov model adds latent states with transition probabilities and emissions for observations.
\item \textbf{Mechanism.} Forward computes total likelihood, backward supplies suffix evidence, Viterbi finds the most probable path, and posterior decoding answers a different marginal question.
\item \textbf{Boundary.} The useful test is whether I can apply hidden markov models and dynamic programming to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when hidden markov models and dynamic programming is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I keep 'best complete path' separate from 'best state at each position' because they are not interchangeable.
\end{studenttake}

\begin{plainbox}
A hidden Markov model adds latent states with transition probabilities and emissions for observations.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: Why can posterior decoding differ from the Viterbi path? The conclusion is Viterbi maximizes one joint state sequence. Posterior decoding maximizes each position's marginal state probability after summing over all paths, so independently best states need not form the globally best path.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} Why can posterior decoding differ from the Viterbi path?

\textbf{Solution.} Viterbi maximizes one joint state sequence. Posterior decoding maximizes each position's marginal state probability after summing over all paths, so independently best states need not form the globally best path.

\medskip
\textbf{Practice 2.} Give a short oral explanation of hidden markov models and dynamic programming that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct hidden markov models and dynamic programming from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a hidden markov models and dynamic programming problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Mixtures, latent classes, and expectation maximization}

\begin{idea}
Mixture models represent heterogeneous samples through latent component assignments. Expectation maximization alternates posterior responsibilities with parameter updates, increases likelihood locally, and requires initialization and identifiability checks.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Mixture models represent heterogeneous samples through latent component assignments. Expectation maximization alternates posterior responsibilities with parameter updates, increases likelihood locally, and requires initialization and identifiability checks. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of mixtures, latent classes, and expectation maximization. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach mixtures, latent classes, and expectation maximization on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from hidden markov models and dynamic programming to probabilistic alignment and evolutionary models. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Mixture models represent heterogeneous samples through latent component assignments.
\item \textbf{Mechanism.} Expectation maximization alternates posterior responsibilities with parameter updates, increases likelihood locally, and requires initialization and identifiability checks.
\item \textbf{Boundary.} The useful test is whether I can apply mixtures, latent classes, and expectation maximization to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when mixtures, latent classes, and expectation maximization is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I treat a discovered cluster as a model component until biology independently supports the label.
\end{studenttake}

\begin{plainbox}
Mixture models represent heterogeneous samples through latent component assignments.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: What does the E-step compute in a two-component mixture? The conclusion is For each observation it computes posterior membership probabilities using current mixing weights and component likelihoods. Those fractional responsibilities become weights in the M-step.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} What does the E-step compute in a two-component mixture?

\textbf{Solution.} For each observation it computes posterior membership probabilities using current mixing weights and component likelihoods. Those fractional responsibilities become weights in the M-step.

\medskip
\textbf{Practice 2.} Give a short oral explanation of mixtures, latent classes, and expectation maximization that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct mixtures, latent classes, and expectation maximization from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a mixtures, latent classes, and expectation maximization problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Probabilistic alignment and evolutionary models}

\begin{idea}
Sequence comparison needs a model for substitutions, insertions, deletions, background composition, and evolutionary distance. Log-odds scoring compares a related-sequence model with a null model, while dynamic programming organizes alignments.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Sequence comparison needs a model for substitutions, insertions, deletions, background composition, and evolutionary distance. Log-odds scoring compares a related-sequence model with a null model, while dynamic programming organizes alignments. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of probabilistic alignment and evolutionary models. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach probabilistic alignment and evolutionary models on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from mixtures, latent classes, and expectation maximization to inference, uncertainty, and numerical stability. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Sequence comparison needs a model for substitutions, insertions, deletions, background composition, and evolutionary distance.
\item \textbf{Mechanism.} Log-odds scoring compares a related-sequence model with a null model, while dynamic programming organizes alignments.
\item \textbf{Boundary.} The useful test is whether I can apply probabilistic alignment and evolutionary models to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when probabilistic alignment and evolutionary models is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
My shorthand for this chapter is: an alignment score only means something relative to the model that produced its match, mismatch, and gap terms.
\end{studenttake}

\begin{plainbox}
Sequence comparison needs a model for substitutions, insertions, deletions, background composition, and evolutionary distance.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: Why use log-odds substitution scores? The conclusion is A log likelihood ratio adds across positions and rewards pairs more likely under the related model than under background. Positive values support relatedness and negative values support the null.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} Why use log-odds substitution scores?

\textbf{Solution.} A log likelihood ratio adds across positions and rewards pairs more likely under the related model than under background. Positive values support relatedness and negative values support the null.

\medskip
\textbf{Practice 2.} Give a short oral explanation of probabilistic alignment and evolutionary models that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct probabilistic alignment and evolutionary models from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a probabilistic alignment and evolutionary models problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Inference, uncertainty, and numerical stability}

\begin{idea}
Maximum likelihood, posterior inference, intervals, likelihood ratios, permutation tests, and multiple-testing control answer different uncertainty questions. Log probabilities, log-sum-exp, and scaled recursions prevent underflow.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Maximum likelihood, posterior inference, intervals, likelihood ratios, permutation tests, and multiple-testing control answer different uncertainty questions. Log probabilities, log-sum-exp, and scaled recursions prevent underflow. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of inference, uncertainty, and numerical stability. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach inference, uncertainty, and numerical stability on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from probabilistic alignment and evolutionary models to model criticism, population structure, and reproducibility. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Maximum likelihood, posterior inference, intervals, likelihood ratios, permutation tests, and multiple-testing control answer different uncertainty questions.
\item \textbf{Mechanism.} Log probabilities, log-sum-exp, and scaled recursions prevent underflow.
\item \textbf{Boundary.} The useful test is whether I can apply inference, uncertainty, and numerical stability to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when inference, uncertainty, and numerical stability is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
I report uncertainty beside the biological conclusion because a single best call hides how fragile the evidence may be.
\end{studenttake}

\begin{plainbox}
Maximum likelihood, posterior inference, intervals, likelihood ratios, permutation tests, and multiple-testing control answer different uncertainty questions.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: Why do forward probabilities underflow on long sequences? The conclusion is They multiply many numbers below one and eventually round to zero. Work in log space with log-sum-exp or rescale each recursion step while tracking the scale factors.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} Why do forward probabilities underflow on long sequences?

\textbf{Solution.} They multiply many numbers below one and eventually round to zero. Work in log space with log-sum-exp or rescale each recursion step while tracking the scale factors.

\medskip
\textbf{Practice 2.} Give a short oral explanation of inference, uncertainty, and numerical stability that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct inference, uncertainty, and numerical stability from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a inference, uncertainty, and numerical stability problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Model criticism, population structure, and reproducibility}

\begin{idea}
Genomic models can fail through population structure, batch effects, relatedness, reference bias, leakage, selection bias, and incorrect independence assumptions. Reproducible analysis records provenance, software, parameters, uncertainty, and subgroup behavior.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Genomic models can fail through population structure, batch effects, relatedness, reference bias, leakage, selection bias, and incorrect independence assumptions. Reproducible analysis records provenance, software, parameters, uncertainty, and subgroup behavior. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of model criticism, population structure, and reproducibility. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach model criticism, population structure, and reproducibility on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from inference, uncertainty, and numerical stability to the cumulative course model. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{What I need to be able to do}
For this chapter, I should be able to define the objects and state involved, state the governing invariant or contract, trace a small example without hand-waving, and explain which assumption makes the method valid. I should also be able to compare this model with the nearest alternative and say what failure would make me switch approaches.

\subsection{Deep notes}
\begin{itemize}
\item \textbf{Model.} Genomic models can fail through population structure, batch effects, relatedness, reference bias, leakage, selection bias, and incorrect independence assumptions.
\item \textbf{Mechanism.} Reproducible analysis records provenance, software, parameters, uncertainty, and subgroup behavior.
\item \textbf{Boundary.} The useful test is whether I can apply model criticism, population structure, and reproducibility to a new case and explain the assumption that makes the answer valid.
\item \textbf{Decision test.} I should be able to say when model criticism, population structure, and reproducibility is the right model, when its assumptions fail, and what evidence would change my choice.
\end{itemize}

\begin{studenttake}
My shorthand for this chapter is: a tiny p-value does not rescue a biased sample or a model that confused ancestry with causality.
\end{studenttake}

\begin{plainbox}
Genomic models can fail through population structure, batch effects, relatedness, reference bias, leakage, selection bias, and incorrect independence assumptions.
\end{plainbox}

\begin{examplebox}
A useful way to work this chapter is to make the smallest concrete instance, write the state or contract explicitly, and then scale the reasoning only after that instance behaves correctly. The worked question below makes that reasoning concrete: A classifier predicts disease well only within one ancestry group. What should be checked? The conclusion is Audit sampling, prevalence, population structure, calibration and error by subgroup, batch and site effects, leakage, and transport to external cohorts. Restrict claims until independent data support broader use.
\end{examplebox}

\begin{warningbox}
The common mistake is to memorize the visible procedure while dropping its precondition. I write the assumption beside the method and test one boundary case before trusting the result.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice.} A classifier predicts disease well only within one ancestry group. What should be checked?

\textbf{Solution.} Audit sampling, prevalence, population structure, calibration and error by subgroup, batch and site effects, leakage, and transport to external cohorts. Restrict claims until independent data support broader use.

\medskip
\textbf{Practice 2.} Give a short oral explanation of model criticism, population structure, and reproducibility that includes its model, one assumption, one failure mode, and one comparison with a neighboring method.

\textbf{Solution.} A strong answer begins with the model in the blue box, names the assumption that licenses the method, uses the common miss above as the failure mode, and chooses the comparison according to the decision test in the deep notes. The goal is not one memorized sentence; it is a four-part explanation I can reproduce on a new example.

\medskip
\textbf{Mastery practice A.} Reconstruct model criticism, population structure, and reproducibility from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a model criticism, population structure, and reproducibility problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section*{Final study check}
I should be able to close this packet and reconstruct the core models, not merely recognize their names. My final check is to explain one complete problem aloud: what the inputs mean, which invariant or contract controls the work, why the method is legal, what it costs, how it can fail, and what evidence would convince another engineer.

\end{document}
