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  "body": "\n## Definition\n\nLet $S$ and $O$ be finite sets, the *hidden states* and the *observations*.\nLet $\\lambda$ be a probability distribution on $S$, and let $A = (a_{ij})_{i, j \\in S}$ be a [[transition matrix|0F462493EA]].\nLet $E = (e_i(o))_{i \\in S, o \\in O}$ be a matrix with $e_i(o) \\ge 0$ and $\\sum_{o \\in O} e_i(o) = 1$ for each $i$.\nA *hidden Markov model* with these parameters is a pair of processes $(X_n, Y_n)_{n \\ge 0}$ with values in $S \\times O$ such that:\n\n1. $(X_n)$ is a [[Markov chain|033TJBZDG0]] with initial law $\\lambda$ and transition matrix $A$;\n2. given $(X_n)_{n \\ge 0}$, the variables $Y_0, Y_1, \\ldots$ are [[conditionally independent|0THJ7416DX]];\n3. $P(Y_n = o \\mid (X_k)_{k \\ge 0}) = e_{X_n}(o)$ for all $n \\ge 0$ and $o \\in O$.\n\nOnly $(Y_n)$ is observed.\n\n## Remarks\n\nThe joint law is $P(X_0 = i_0, Y_0 = o_0, \\ldots, X_n = i_n, Y_n = o_n) = \\lambda_{i_0} e_{i_0}(o_0) \\prod_{k=1}^{n} a_{i_{k-1} i_k} e_{i_k}(o_k)$.\nExample: let $S = \\set{F, L}$ for a fair and a loaded die, with $a_{FL} = a_{LF} = 0.05$, and let $O = \\set{1, \\ldots, 6}$.\nTake $e_F(o) = 1/6$ for all $o$, $e_L(6) = 1/2$ and $e_L(o) = 1/10$ for $o \\le 5$; each row sums to $1$, so clauses 1 to 3 define a model.\nNon-example: if $Y_n$ is allowed to depend on $Y_{n-1}$ as well as on $X_n$, clause 3 fails.\nThe observed process $(Y_n)$ alone is in general not a [[Markov chain|0G48PYM96Q]].\nThe forward algorithm computes the likelihood $P(Y_0 = o_0, \\ldots, Y_n = o_n)$ in time of order $n \\abs{S}^2$, and the Viterbi algorithm finds a most likely hidden path.\n",
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